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Operator Algebras and Their ModulesAn operator space approach$

David P. Blecher and Christian Le Merdy

Print publication date: 2004

Print ISBN-13: 9780198526599

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198526599.001.0001

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C*-modules and operator spaces

C*-modules and operator spaces

Chapter:
(p.296) 8 C*-modules and operator spaces
Source:
Operator Algebras and Their Modules
Author(s):

David P. Blecher

Christian Le Merdy

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198526599.003.0008

Abstract and Keywords

This chapter has three main goals. First, to examine Hilbert C*-modules (and their W*-algebra variant, W *-modules) as operator modules. It aims to show that the theory of C *-modules fits comfortably into the operator module framework. Second, to consider space X. In particular, it will discuss the noncommutative Shilov boundary ☐(X) of X. TRO methods and this Shilov boundary provide important insights into the structure of X. Third, to illustrate how C*-module and TRO methods can lead to interesting results about operator spaces. Notes and historical remarks are presented at the end of the chapter.

Keywords:   Hilbert C*-modules, noncommutative Shilov boundary, W*-modules, C*-module maps, operator space multipliers, operator modules

This chapter only depends on (parts of) Chapters 1–4 of out text. It has several goals. The first is to study Hilbert C*-modules (and their W*-algebra variant, W*-modules) as operator modules. We aim to show that the theory of C *-modules fits comfortably into the operator module framework Indeed the operator space viewpoint will lead us in a streamlined way through several aspects of the theory of C*-modules. Although we will not say much about this here, our methods also permit the generalization of these modules to the nonselfadjoint operator algebra case (see the Notes section of references). In contrast, Banach module methods are not generally compatible with C*-module constructions, and indeed completely break down when one attempts the aforementioned nonselfadjoint generalization. The second goal of our chapter is to consider some important TROs or C *-modules which are associated to every operator space X. In particular, we will discuss further here the noncommutative Shilov boundary 𝒯(X) of X. TRO methods, and this Shilov boundary, provide important insights into the structure of X. Third, we will illustrate how C*-module and TRO methods can lead to interesting results about operator spaces.

Thus there is a profound two-way interaction between C*-modules and operator spaces, which has attracted much interest in recent years. Because of limitations of space, we cannot reproduce here many of the operator space applications of C*;-module theory which appear in the literature. Instead, our more modest goal, and this is the fourth purpose of our chapter, is to lay out, in a systematic way, most of the basic concepts, theory, and connections, which are needed for such applications.

There are many different ‘pictures’, or ways of at, looking at, C* -modules, as well as many routes through this theory. We will start from scratch, moving quickly, and later we will begin to add the operator space and TRO perspectives. Our presentation is essentially selfcontained. However, we will concentrate on material not in the standard sources on C*-modules (for exmple, see [35,173, 239, 296, 356, 421, 423]). We refer the reader to those texts for topics not covered here.

We will mostly state out results for right modules. However, as we shall see, there is a striking ‘left-right symmetry’ to the theory; in particular every right C*-module is also, canonically, a left C*-module over another C*-algebra. When we need to apply to a module which is being viewed as a left module, a result established earlier for right modules, we often refer to the ‘other-handed version’ (p.297) of the earlier result.

8.1 HILBERT C*-MODULES—THE BASIC THEORY

Throughout this section and the next, A and B are C*-algebras.

8.1.1 (The definition)

A (right) C*-module over A is a right A module Y, together with a map < | > : Y × Y A , which is linear in the second variable, and which also satisfies the following conditions;

  • (1)y|y〉 ≥ 0 for all yY,

  • (2)y|y〉 =0 if and only if y =0,

  • (3)y|za 〉 = 〈y|za for all y, zY, aA,

  • (4)y|z〉* = 〈z|y〉 for all y, zY,

  • (5) Y is complete in the norm y = y | y 1 2 .

We call | the A-valued inner product on Y. It follows from (3) and (4) that 〈ya|z〉 = a* 〈y|z〉 for all y, zY, aA.

In (5), the fact that ‖ · ‖ is a norm follows just as for Hilbert spaces from the following Cauchy–Schwarz inequality: ‖〈y|z〉‖≤ ‖y‖ ‖z‖ for y, zY. This follows from the relation

y | z z | y z 2 y | y ,
which in turn may be proved by using the fact that 0 ≤ 〈y + zb|y+ zb〉, and taking b = − 〈z|y〉/‖z2. The last calculation is an easy exercise using the fact that c* ac ≤ ‖ac*c for any cA, aA +.

It the linear span of the range of the inner product | is dense in A, then Y is called full.

Left C*-modules are defined analoguously. Here Y is a left module over a C*-algebra A, the A-valued inner porduct id linear in the first variable, and condition (3) in the above is replaced by 〈ay|z〉 = ay|z〉, for y, zY, aA. Note that if A=C, then these are exactly the Hilbert spaces. On the other hand, and right C*-module over C is also Hilbert space in the usual (mathematical) sense, with the ‘reversed inner product’ ζ × η ↦ 〈η|ζ〉.

If Y is right C* module over A, then there is a canonical left C*-module Y ¯ over A, which is simply the conjugate vector space of Y with left action a y ¯ = y a * ¯ , and inner product y ¯ | z ¯ = y | z . This is usually called the conjugate C*-module in the literature. We will favour the term adjoint module instead.

8.1.2 (Equivalence bimodules)

If Y is an A-B-bimodule, then we say that Y equivalence bimodule, if Y is full right C*-module over B, and a full left C*-module over A, and the two inner products are compatible in the sense that xy|z〉 = [x|Y]z, for al x, y, zY. Here we have written [ | ] for the A-valued product. Equivalence bimodules are also sometimes called imprimitivity bimodules, or strong Morita equivalence A-B-bimodules, or equivalence A-B-bimodules. If there exists such an equivalence bimodule, we say that A and B are strongly Morita equivalent.

(p.298) Good examples of equivalence bimodules are furnished by the ternary rings of operators, or TROs, encountered in 4.4.1, and in Example 3.1.2 (6). If Z is a TRO, then we recall that D = Z Z and C= ZZ are C*-algebras. Clearly Z is both a full right C*-module over D, and a full left C*-module over C. Indeed such Z is clearly and equivalence C-D-bimodule.

Thus we have one direcltion of the correspondence between TROs and C*-modules. We will return to the other direction later, in 8.1.19 and 8.2.8.

8.1.3 (C*-modules are Banach modules)

If Y is a right C*-module over A, then Y is a nondegenerate Banach A-module (see A.6.1). Indeed,

y a 2 = y a | y a = a * y | y a a 2 y 2 ,
for yY and aA. If (et)t is a positive cai for A, then
y - y e t | y - y e t = y | y e t y | y y | y e t + e t y | y e t 0
Thus ye ty for all yY. Hence Y is a nondegenerate Banach A-module.

8.1.4 (The ideal I)

Throughout this chapter we reserve the symbol I for the closure of the linear span fo the C*- algebra valued inner product on a right C*-module. If Y is a right C*-module over A then it is clear that I is a closed two-sided ideal in A, and that Y is a full right C*-module over I. We make some simple but important remarks:

(1) The canonical map IB(Y), taking aI to the operator y↦ya, is a linear isometry. Indeed it is clearly contractive. If Ya = 0 then we have that 〈Y|Ya〉 = 〈Y|Ya =0. This implies that I a = 0, so that a*a0. Hence a = 0. Thus the map is one-to-one. It follows from A.5.9 (or from the other-handed vesion of the later result 8.1.15, for example) that this map is isometric.

(2) Since Y is a C*-module, and hence a nondegenerate Banach module, over I, we have Y I = Y by Cohen's factorization theorem A.6.2. In particular, the linear span of terms of the form xy|〉, for x, y, zY, is dense in Y.

(3) If Zis a Banach A-module then B A (Y, Z) = B I (Y, Z). Indeed, suppose that uB I(Y, Z), xY and aA. By (2) and A.6.2, we may write x = ya′, with a′ ∈ I, yY. Then u(xa) = u(yaa) = u(y)aa = u(ya′)a = u(x)a.

(4) Conversely to the first paragraph of 8.1.4, if Y is a right C*-module over A, and if C is a C*-algebra containing A as an ideal, then Y is also a right C*-module over C. The module action here is the canonical one: (ya)c = y(ac), for yY, aA, and cC. The fact that this action is well declined, and that 8.1.1 (3) holds, follows from the relation 〈z|y(ac)〉 = 〈z|yac = 〈z|yac, for zY.

Lemma 8.1.5

Let u: YZ be a bounded A-module mop between right C*-modules over A. Then 〈u(y)|u(y)〉 ≤ ‖u‖2〈y|y〉, for all y ∈ Y.

Proof We may suppose that ‖u‖ ≤ 1, and (by 8.1.4(4)) that A is unital. Then the result follows by a trivial modification of the argument for (4.10). □

(p.299) Propsiotion 8.1.6

Suppose that A is a C*-subalgebra of B(H). The norm and inner product on a right C*-module Y over A are related by the formula:

y | y ζ , ζ = sup { f ( y ) ζ 2 : f Ball ( B A ( Y , A ) ) } , y Y , ζ H .
Proof For f ∈ Ball(BA (Y, A)), yY, and ζ ∈ H, we have
f ( y ) ζ 2 = f ( y ) * f ( y ) ζ , ζ y | y ζ , ζ ,
by 8.1.5. Next, set f = 〈z n|·〉, where z n = y ( y | y + 1 n ) 1 2 . Here n ∈ N. We have z n | z n = ( y | y + 1 n ) 1 2 y | y ( y | y + 1 n ) 1 2 1 . Hence f ∈ Ball(BA(Y, A)). Note that f ( y ) * f ( y ) = < y | y > ( < y | y > + 1 n ) - 1 < y | y > . By spectral theory,
lim n y | y ( y | y + 1 n ) 1 y | y = y | y .
Putting these facts together gives the desired equality. □

Proposition 8.1.6, together with the polarization identity (1.1), shows that the inner product on a C*-module Y is completely determined by, and may be recovered from, the Banach module structure of Y.

8.1.7 (Adjointable maps)

If Y, Z are right C*-modules over A, then we write B A(Y, Z) (or simply B(Y, Z)) for the set of adjointable maps from Y to Z, that is, the set of maps T: YZ such that there exists a map S: ZY with

T ( y ) | z = y | S ( z ) , y Y , z Z .
It is easy to see that such an S is unique; it is denoted by T*. It follows immediately from the centered equation that (i) T is right A-module map, (ii) T is bounded (use the closed graph theorem), (iii) T* is a bounded right A-module map, and (iv) T** = T. It is also easy to verify relations such as ( T 1 T 2 ) * = T 2 * T 1 * . Also, B A(Y, Z) is a norm closed subspace of B A(Y, Z). Indeed, if (Tn)n is Cauchy in B A(Y, Z), then ( T n * ) n is Cauchy too. The limits of these two sequences evidently are the adjoints of each other. Writing B A (Y)=B A(Y, Y), it is easy to verify that B A (Y) is a C*-algebra with respect to the usual norm of a bounded operator. For example, the more difficult direction of the C*-identity ‖T*T‖ = ‖T‖2 follows immediately from the fact that is y ∈ Ball(Y), then
T y 2 = T ( y ) | T ( y ) = T * T ( y ) | y T * T
We define K A(Y, Z) (or simply K(Y, Z) to be the closure, is B(Y, Z), of the linear span of the ‘rank-one’ operators |z〉 〈y|, for yY, zZ. We are using ‘braket’ notation here; thus |z〉 〈y| is the operator which takes an xY to zy|x〉. It is easy to verify familiar relations such as (|z〉〈y|)* = |y〉 〈z|, and that the product of ‘rank one’ operators is ‘rank one’: (|y〉 〈z|) (|y′〉 〈z′|) = |y(〈z|y′〉)〉〈z′|. Also |ya*〉 〈z| = |y〉 〈za| if aA. From these relations, it is easy to see that (p.300) K A(Y) = K A (Y, Y) is a C* -subalgebra of B A (Y). Indeed K A (Y) is a closed ideal of B A (Y), as may be seen from the simple relations T(|y〉 〈z|) = |T(y)〉 〈z| and (|y〉 〈z|) S = |y〉 〈S* z| for TB A(Y) and S ∈ B A (Y) Similarly,
(8.1)
K A ( Y , Z ) is canonically a B A ( Z ) B A ( Y ) bimodule .
We say that a map u: YZ between right C*-modules over A is unitary if u is a surjective A-module map such that 〈uy|uy〉 = 〈y|y〉, for all yY. By the polarization identity (1.1) this is equivalent to: 〈uy|uz〉 = 〉y|z〉 for all y, zY. it is clear that any unitary map is adjointable. If there exists such a unitary then we say YZ unitarily. If u is unitary then u* = u −1. If Y = Z then u is unitary if and only if it is a unitary in the C*-algebra B A(Y).

Corollary 8.1.8

An A-module map u: YZ between right C* -modules over A is unitary if and only if u is isometric and surjective.

Proof The one direction is obvious. The other direction follows from 8.1.5 applied to both u and u−1.

8.1.9 (The direct sum)

If {Yi: iI} is a collection of right C*-modules over A, then we define the direct sum C*-module by

i c Y i = { ( y i ) i I Y i : i y i y i converges in norm in A } .
With the canonical inner product 〈 (y i)|(zi)〉 = ∑iyi|zi〉 (which converges by the polarization identity (1.1)), and obvious A-module action, i c Y is a right C*-module over A. We omit the routine proof. In passing, we remark that there are alternative ways to define the direct sum. See, for example, 8.2.14.

For a cardinal I, C I(Y) denotes the C* -module direct sum of I copies of Y. We will see in 8.2.3 (4) that there is no conflict with earlier operator space notation: viewed as an operator space, C I(Y) means exactly what it meant before.

It is important, and easily seen, that the canonical inclusion and projection maps between i c Y i and its summands Y i are adjointable.

We say that a right C*-module Z is the internal orthogonal direct sum of closed submodules Y and W, if Z = Y + W and YW (i.e. 〈y w〉 =0 for all yY, wW). In this case, Z Y c W unitarily, quite clearly. We say that Y is orthogonally complemented in Z if there exists such a W. It is clear that Y is orthogonally complemented in Z if and only if Y is the range of a projection (i.e. a selfadjoint idempotent) P in the C* -algebra B A(Z).

8.1.10 (A characterization of module maps)

Result 8.1.5 may be improved. In fact, a linear map u: YZ between C* -modules over B is a contractive B-module map, if and only if 〈u(y) u(y)〉 ≤ 〈y|y〉, for all yY. This is also equivalent to ‖(u(y), z)‖≤ ‖(y, z)‖, for all yY and zZ, where the norms here are those of Zc Z and Yc Z respectively. We omit the proofs of these assertions since we shall not need them (see [302, 56]). We remark that the latter condition is the analogue for C*-modules of condition 4.5.2 (ii).

(p.301) Proposition 8.1.11

Suppose that Y is a right C*-model over A.Then:

  • (1) Y≅K A(A, Y) isometrically.

  • (2) Y̅ ≅ K A(Y, A) isometrically.

  • (3) K A(C n (Y))≅M n(K A(Y)) and B A(C n(Y))≅M n(B A(Y)) as C*-algebras.

Proof (1) That the canonical map L: YBA(A, Y) is isometric, with range contained inside K A(A, Y), is a simple modification of the proof of 3.5.4 (1).Since that range is a closed vector subspace which contains every ‘rank one’ operator it must equal K A(A, Y).

(2) Define Φ: Y̆ → B A (Y, A) by Φ(y̅)(z)=<y|z>, for y, zy. Clearly Φ is isometric and linear. We show that Φ maps onto K A (Y, A). Indeed, by A.6.2 We may write any yY as ya, for y′ ∈ Y, aA. Then Φ(ȳ) = a*〈y′|·〉, the generic ‘rank one’ operator in K A(Y, A). We conclude as in (1).

(3) There is a canonical homomorphism θ: M n(K A(Y))→ B A(C n(Y)). It is easy to check that θ(a*)= θ(a*), so that θ is a *-homomorphism into B A(cn(Y)). Clearly θ is one-to-one. We leave it as an exercise that Ran(θ)= K A(C n(Y))(using the (adjointable) canonical inclusion and projection maps between C n(Y) and its summands and (8.1)). The other assertion is similar. □

Lemma 8.1.12

Let Y be a right C*-module over A, and suppose that T is a linear map on Y. Then T ∈ B A(Y)sa (resp. T ∈ B A(Y)+) if and only ifTy|y〉 is selfadjoint (resp. 〈Ty|y〉≥ 0) for all yY.

Proof We sketch a proof of the difficult implication. If 〈Ty|y〉 is selfadjoint for all yy, then 〈Ty|y〉 = (〈TY|y〉)* = 〈y|Ty〉. By the polarization identify (1.1), T is adjointable on Y, with T* = T. If, further, 〈Ty|y〉≥0, for yY, then T is selfadjoint, by the above. To see that T ≥ 0, suppose that AB(H), and apply the following simple general fact about C*-algebras to the collection of positive functionals y | y ζ , ζ on B A ( Y ) , for  y Ball ( Y ) , ζ Ball ( H ) . Namely, let B be a C*-algebra, and let S be a set of positive contractive functionals on B with

b = sup { ϕ ( b ) : ϕ S } , for all b B + .
Then if bB sa and if φ(b)≥0 for all φ∈S, then bB +. To prove this general fact, suppose that the endpoints of the spectrum of b are α and β, with α ≤ β. The centered equation applied to β1 − b, gives
β α = β 1 b = sup { β ϕ ( 1 ) ϕ ( b ) : ϕ S } β .
From this it is clear that α ≥ 0. Hence b ≥ 0 □

Corollary 8.1.13

if a = [ a i ] M n ( A ) , t h e n a n equals

sup { a b C n ( A ) : b Ball ( C n ( A ) ) } = sup { i , j c j * a i j b j : b , c Ball ( c n ( A ) ) } .
Moreover, the following are equivalent:

(p.302)

  • (i) a is positive in Mn (A),

  • (ii) i , j b i * a i j b j 0 in A, for all b 1,…, bnA,

  • (iii) a is a finite sum of matrices of the form [ a i * a j ] , , with a 1, …, anA.

Proof Define θ: Mn(A) → BA(cn(A)) by the canonical left action of Mn(A) on Cn(A). Clearly θ is a one-to-one *-homomorphism into the C*-algebra B A(Cn(A)) of 8.1.7. Then the first statement is clear from A.5.8. The equivalence of (i) and (ii) follows from 8.1.12. The equivalence with (iii) is a simple exercise. □

8.1.14 (C*-modules are equivalence bimodues)

Any right C*-module Y over B is canonically also a full left C*-module over K B(Y), using |·〉〈·| for the inner product. When checking this, the only nontrivial point is that y y 0 , which follows quite easily from 8.1.12. It is now clear that if Y is a right C*-module over B, then Y is also an equivalence K B(Y)-I-bimodule. Here I is as in 8.1.4.

Note also that the norm on Y induced by this new inner product corresponding to this left C*-module action, is the same as the old norm. In fact it is evident that y y y 2 . The reverse inequality follows from the fact that

| y y 2 | y y | ( z ) 2 = z | y y | y y | z , z Ball ( Y ) .
Setting z = y /‖y‖ yields the desired inequality.

The proof of the following result shows, conversely, that given an equivalence A-B-bimodule Y, then A≅K B(Y) *-isomorphically, and via this isomorphism the action of A on Y corresponds exactly to the canonical K B(Y) action, and the A-valued inner product corresponds to the K B(Y)-valued inner product. Thus we see that full right C*-modules are essentially the same things as strong Morita equivalence bimodules. This gives another ‘picture’ of C*-modules, as the strong Morita equivalence bimodules.

For the next result, we write λ for the canoncial map from A into B(Y), for a left Banach A-module Y.

Lemma 8.1.15

If Y is an equivalence A-B-bimodule, then λ(A)= K B(Y). Indeed, A≅K B(Y)*-isomorphically, via the map λ above. Also, the two norms defined on Y via each of the two inner products coincide.

proof Writing [·|·] for the A-valued inner product, we have

x a y | z = [ x | a y ] z = [ x | y ] a * z = x y | a * z , x , y , z Y , a A .
Thus by 8.1.4(1), we have 〈ay|z〉= 〈y|a*z〉. Hence λ(A) ⊂ B B(Y), and λ is a *-homomorphism into B B(Y). Clearly λ([y|z])= |y 〉 〈 z|, so that it follows by continuity and density that the range of λ is K B(Y). By the left module version of 8.1.4(1), λ is one-to-one. Thus λ is an isometry, by A.5.8.

For the last assertion, since λ is isometric, and by the second paragraph of 8.1.14, we have [ y | y ] = y y = y | y .

(p.303) Proposition 8.1.16

  • (1) If Y is an equivalence A-B-bimodule then the canonical map from the multiplier algebra M (A) to B(Y) (see 3.1.11) is isometric, and indeed is a *-isomorphism onto B B(Y). This *-isomorphism extends the canonical isomorphism between A and K B(Y) (see 8.1.15).

  • (2) Further, the *-isomorphism in (1) extends to an isometric isomorphism between LM(A) and B B(Y).

  • (3) If Y is a right C*-module over B, then BB(YLM(K(Y)) isometrically (as Banach algebras), and B B(Y) ≅ M(K B(Y))*-isomorphically.

  • (4) B B(Y) is the span of the Hermitian elements (see A.4.2) of BB(Y) .

Proof Suppose that Y is an equivalence A-B-bimodule, and that (et)t is a positive cai for A. By 3.1.11, Y is a Banach LM(A)-module, and we have a corresponding contractive homomorphism θ:LM(A)→BB(Y)given by

(8.2)
θ ( η ) ( y ) = lim t η ( e t ) y = η ( a ) y , y , y Y , a A , such that y = a y .
It follows from this, and 8.1.4(1), that θ is a one-to-one homomorphism extending λ. Next fix η ∈ M(A). By (8.2) and 8.1.15,
θ ( η ) y | z = lim t η ( e t ) y | z = lim t y | η ( e t ) * z , y , z Y
writing z = a z for a A , z Y , , the latter quantity equals
lim t y | e t η * ( a ) z = y | η * ( a ) z = y | θ ( η * ) z
using (8.2) again. Thus θ restricts to a *-homomorphism from M(A) into B B(Y).

Suppose that Y is simply a right C*-module. We may assume, by 8.1.4 (3), that Y is full, and we apply the above with A = K B(Y). By (8.1), we may define a contractive homomorphism ρ: BB(Y),→LM(K B(Y))  by ρ(S)(T) = ST, for SBB(Y),T ∈ K B(Y). Here we are viewing LM(A) as the right A-module maps on A. It is easy to see that θ ∘ ρ is the identity map. Hence θ is surjective, and ρ is isometric. Also, by 2.6.8, it is easy to check that ρ takes B B(Y) into M(K B(Y)). Thus we have proved (3).

In the situation of (2), by (3) and 8.1.15 we have isometric isomorphisms

L M ( A ) L M ( K B ( Y ) ) B B ( Y ) .
It is easy to check that the composition of these maps is the map θ in (8.2). This proves (2), and (1)is similar. Assertion (4) follows from (3) and 2.6.9. □

8.1.17 (The linking C*-algebra)

Suppose that Y is a right C*-module over B, and write A = K B(Y). We define the linking C*-algebra ℒ(Y) to be the set of 2 × 2 matrices:

L ( Y ) = [ A Y Y ¯ B ] .
We turn this set into an algebra, using the usual product of 2 × 2 matrices, and using the inner products and module actions. for example, the product yz̄, of a (p.304) term yfrom the 1-2-conner, and a the z̄ from the 2-1-corner, is taken to mean |y〉〈zA = K B(Y). Or the product z̄a, for aA, is given by a * z ¯ . We define the involution of one of these 2 × 2 matrices in the obvious way.

Define a map π: ℒ(Y) → B(Yc B) by the obvious action (i.e. viewing an element of Yc B as a column with two entries, and formally multiplying a 2 × 2 matrix and such a column. It is easy to check that π(m)∈B B(Yc B) for each matrix m ∈ ℒ(Y), and moreover that π is a *-homomorphism into B B(Yc B). Also, one can quickly check that π is one-to-one, and that π is an isometry when restricted ro each of the four corners of ℒ(Y). Hence the range of π is closed. We give ℒ(Y) a norm by pulling back the norn form B B(Yc B) via π, thus ℒ(Y) is a C* -algebra (-isomorphic to the range of π. Indeed we may regard B B(Yc B) as a 2 × 2 matrix C*-algebra consisting of matrices t = [tij] whose four entries are adjointable maps. To see this, note that tij are defined in terms of t and the (adjointable) projection and inclusion maps between Yc B and its two summands. With this in mind, it is clear from 8.1.11 and (8.1), that K B(Yc B) is exactly the range of π. Hence the linking C *-algebra may simply be thought of as K B(Yc B).

By the last fact, and 8.1.16 (3), we see that the multiplier algebra M(ℒ(Y)) is B B(Yc B). We define the unitized linking C*-algebra1(Y) of Y to be the linear span within M(ℒ(Y)) of K B(Yc B) and the two diagonal idempotent matrices p = 1 ⊕0 and q = 0 ⊕ 1. The last two 1's may be viewed as the identities of the unitizations of A and B respectively (where we take the unitization of a unital algebra to be itself). Then ℒ1(Y) is a unital C*-algebra with identity 1= p - q. Clearly Y is the 1-2-corner of both ℒ1(Y and ℒ(Y). In particular,

(8.3)
Y p L ( Y ) ( 1 p )

If we take a general C*-algebra A, and if Y is an equivalence A-B-bimodule, then we may form ℒ(Y) as above, but using A instead of K B(Y). Of course by 8.1.15 this is essentially the same thing; that is, the resulting linking algebras will be *-isomorphic. In this case we say that ℒ(Y) is the Morita linking algebra of Y. We will also use this terminology even when A is not specified, taking A = K B(Y), however we insist that Y be full over B in this case.

Corollary 8.1.18

If Y is right C*-module over B, then L(Y) is strongly Morita equivalent to B (via the equivalence bimodule Y ⊕c B).

Proof Clearly Yc B is a full B-module. Since K B(Yc B)≅ℒ(Y), we have by 8.1.14 that ℒ(Y)is strongly Morita equivalent to B, via Yc B.

8.1.19 (C*-modules and corners)

One great advantage of the linking C*-algebra of a C*-module Y, is that the inner products and module actions have been replaced by concrete multiplication of elements in a C*-algebra. To see this, we employ the completely isometry in (8.3). This is simply the ‘corner map’ c, taking yy to the matrix in ℒ(Y) with y in the 1-2-corner and zeroes elsewhere. If we identify B with the 2-2-corner in a similar way, then 〈y|z〉 is simply the (p.305) product c(y)* c(z) in the C*-algebra ℒ(Y). Indeed it is convenient, and usually leads to no difficulties, to suppress the ‘c’ map and simply write y*z for the last expression above. Similarly, if we write xy*z the reader will have no difficulty in seeing that what is meant is the element xy|z〉, or equivalently |xy|(z). Similar concentions apply to longer such products.

Thus, by (8.3), any right C*-module Y is a corner in a C*-algebra, in the sense of 2.6.14. Conversely, any corner pAq in a C*-algebra A, is clearly a right C*-module over qAq. This gives another ‘picture’ of C*-modules, as the corners of C*-algebras, In the language of 4.4.1, any C*-module may be viewed as a TRO, namely, as a subtriple of ℒ(Y). This, togerther with the second paragraph of 8.1.2, gives abother ‘pictre’ of C*-modules, as the TROs. We will tighten up this observation further in 8.2.8.

If Y is an equivalence A-B-bimodule, then one can also view LM(A), and its action on Y, in terms of the linking algebra. Indeed, LM(A) ⊂ A** ⊂ ℒ(Y)**. The composition of these canonical inclusions is easily seen (since Y = AYB) to have range within LM(ℒ(Y)). That is, we may regard as subalgebras:

L M ( A ) L M ( L ( Y ) ) L ( Y ) * * .
Similar assertions hold for RM(B).

We turn to some more corollaries of 8.1.16:

Corollary 8.1.20

If A and B are strongly Morita equivalent, then the centers of their multlier algebras are *-isomorphic, via a *-isomorphism θ satisfying

θ ( η ) y = y η , f o r a l l y Y , η Z ( M ( B ) )
Here Y is the associated equivalence A-B-bimodule.

Proof By 3.1.11 Y is a right Z(M(B))-module, with action yη = limt yη(et), for yY,η∈Z(M(B)). Here (et)t is a cai for B. As in A.6.1, this defines a contractive unital homomorphism π: Z(M(B)) → B(Y). Clearly π maps into B B(Y). By 8.1.16 (4), together with A.4.2, π maps Z(M(B))sa, and hence also Z(M(B)), into B B(Y). Thus by 8.1.16 (1), there exists a unique ν∈M(A) such that νy = yη for all yY. Clearly this implies that νay = ayν = aνy, if aA. This, and 8.1.4(1), implies that ν∈Z(M(A)). Moreover, if we define (η)=ν then θ is a homomorphism from Z(M(B)) to Z(M(A)). Now it is easy to see, by symmetry, that θ must be an isomorphism. By the last part of A.5.4, for example, θ is a *-ismorphism.

Corollary 8.1.21

  • (1) Let P be a contractive idempotent A-module map on a right C*-module Y over A. Then P is adjointable. Indeed P is an orthogonal projection in the C*algebra B(Y), and the range of P is an orthogonally complemented submodule of Y.

  • (2) Suppose that Y and Z are right C*-modules over A, and that α: YZ and β: ZY are contractive module maps with βα: = IY. Then these maps (p.306) are adjointable, with β=α*. Moreover, α is a unitary module map onto an orthogonally complemented submodule of Z

Proof(1) By 8.1.16 (3), P corresponds to a contractive idempotent in the operator algebra LM(K A(Y)). By 2.1.3, the last idempotent is Hermitian. Thus P is Hermitian, and is adjoinable by 8.1.16 (4). The rest follows from 8.1.9.

  • (2) Note that α is an isometry onto the closed submodule W* Ran (αβ) of Z. Since αβ satisfies the conditions of (1), it is adjoinable. By 8.1.9 we see that W is orthogonally complemented. By 8.1.8 we have that α is unitary, and hence adjointable as a map into W. It is than easy to see that α is adjointable as a map into Z. Indeed,

β z | y = ( α β ) z | α y = z | ( α β ) α y = z | α y ,
for yY, zZ. □

From 8.1.21, one may deduce a universal property of the direct sum:

Proposition 8.1.22

Suppose that {Yi: i∈I} is a collection of right C*-modules over A, and that Y is a right C*-module over A, such that there exist contractive module maps ε i: YiY and Pi: YYi such that P i ε j = δ i I Y , , and such that ε i P i converges strongly on Y. Then there exists an orthogonally complemented submodule W of Y such that Y ( i c Y i ) c W unitarily. If ε i P i converges strongly to I y, then W =0.

Proof By 8.1.21, each εi and P i are adjointable, with P i = ε i * , and Q i = εi o Pi is an adjoinatable projection on Y. Moreover, the Q i are mutually orthogonal, and of course positive, elements of the C*-algebra B(Y). Set R(y) = ∑i Q i(y), for yY. Then R is a module map on Y, and

0 R ( y ) | y = lim J j J Q j y | y y | y , y Y ,
the limit over finite subsets j of I. Thus R is contractive and positive(by 8.1.12). Clearly RQi = Qi, which implies that R idempotent. Hence, by 8.2.21, R and I Y R are adjointable projections on Y. Clearly IR is orthogonal to each Q i. If W = R a n ( I R ) and W = R a n ( R ) , , then Y W c W unitarily. Define u : W i c Y i by u ( y ) = ( Q i y ) . It is easy to check that u y | u y = y | y , so that u is an isometry. Since Ran (u) is norm dense, u is a unitary.

8.1.23 (Finite rank approximation)

If Y is a right C *-module over B, then we claim that the C *-algebra K B(Y) has a cai (et)t of the form

(8.4)
e t = k = 1 n ( t ) | x k t > < x k t | ,
for elements x k t in Y. To obtain this, first pick a cai (f t)t from the dense ideal of ‘finite rank’ operators in K B(Y). Let e t = f t * f t is also a cai for K B(Y). (p.307) If f t = i = 1 n | y i z i | , then by relations in the second paragraph of 8.1.7 we have f t * f t = i , j = 1 n | z i a i j z j | , where [ a i j ] = [ < y i | y j > ] . The latter is a positive matrix P with entries in B (this may be seen by the criterion 8.1.13(ii), for example). Factoring P as the square of its square root, and regrouping, we obtain(8.4).

Since K B (Y) acts nondegenerately on Y (by 8.1.3 and 8.1.4), for any yY we have that k = 1 n ( t ) x k t x k t | y y . From this we deduce the following:

Corollary 8.1.24

Let Y be a right C*-module over B. Then there exists a net ( n ( t ) ) t of positive integers, and contractive B-module maps α t : Y C n ( t ) ( B ) and β t : C n ( t ) ( B ) Y , such that β t ( α t ( y ) ) y for every yY. Indeed this can be done with α t * = β t .

Proof We use the notation above. For yY, define αt(y) ∈ C n(t)(B) to have kth entry x k t | y . Also, define β t ( b ) = k = 1 n ( t ) x k t b k . Here b has kth entry b kB. We have that β t ( α t ( y ) ) = k = 1 n ( t ) x k t x k t | y = e t y y , as we saw immediately above the corolly. It is easily checked that αt, βt are adjointable, with α t * = β t , so that they have same norm. We have:

α t ( y ) 2 = k y | x k t x k t | y = y | k x k t x k t | y = y | e t y y 2
Thus α t is contractive, and hence so is β t. □

8.1.25 (Asympototic factorization)

Thus any right C*-module Y over B ‘factors asymptotically’ through spaces of columns over B. In passing, we remark that a simple modification of the last proof, using 8.1.14 and the left-handed version of 8.1.23, shows that if Y is a full C*-module over B, then B factors asymptotically (in the sense of the last result) through spaces of the form C n(t)(Y). Similarly, one can show that K B(Y) factors asymptotically (via completely contracttive linear maps) through spaces of the form M n(t)(B). Since we shall not use these, we omit the details (see [46, p. 391]).

The converse of 8.1.24 is also true: if Y is a right Banach B-module for which there exist nets of contractive B-module maps as in the lemma, then Y is a right C*-module over B. This gives a characterization of C*-modules among the Banach B-modules. Indeed we have:

Theorem 8.1.26

Suppose that B is a C*-algebra and that Y is a right Banach B-module. Suppose further that there is a net (Yt)t of right C*-modules over B, and contractive B-module maps αt: → Y tY, such that βtt(y))→ y for every yY is a right C*-module over B, and the norm on Y coincides with the norm induced by the inner product(see 8.1.1(5))).

The inner product on Y is given by the formula

y | z = lim t α t ( y ) | α t ( z ) , y , z Y
The limit here is is the norm topology of B

(p.308) A page long proof of this result, which uses only the triangle and Cauchy–Schwarz inequalities, and 8.1.5, may be found in [65, p.41].

For finitely generated or countably generated C*-modules one may improve on the last result considerably. By an algebraically finitely generated B-module we mean a module Y for which there exists y 1,…,y nY such that the map f: C n(B)→Y given by f ( ( a k ) ) = k y k a k , is surjective.

Theorem 8.1.27

  • (1) If Y is an algebraically finitely generated right C*-module over B, then Y is unitarily isomorphic to an orthogonally complemented submodule of Cn(B), for some n ∈N.

  • (2) A right C*-module Y over B is algebracally finitely generated if and only if K B (Y) is unital. In this case, K B(Y,Z) contains all B-module maps from Y to Z, for any C*-module Z over B.

Proof (1) Let f be the above 8.1.27, which is easily seen to be adjointable. By [423] 15.3.8 (or rather the obvious varient of that result to maps between two possibly diffrent C*-modules over B), there is a polar decomposition f = u|f|; W= Ran(|f|) is a closed orthogonally complemented submodule of C n(B); and Y is untitarily isomorphic to W via the partial ismoetry u.

(2) For the first part, it is easy to see by A.6.2 that Y is algebraically finitely generated over B if and only if it is also finitely generated over B 1. Thus we may assume that B is unital. If K B(Y) is unital, then since Y is a nondegenerate K B(Y)-module (by 8.1.3 and 8.1.14), this identity is IY. Given ε>0, we can find e t = k | y k y k | as in 8.1.23, with e t I Y < ε . Hence e t is invertible with inverse S say, so that I Y = k | S ( y k ) > < y k | , from which it is immediate that (S(y k))k generates Y.

Conversely, if Y is algebraically finitely generated, let u: C n(B) → Y be the surjective partial isometry in the proof of (1). We have uu* = I Y. Put y k = u(e k), where e k has 1 B in the kth entry, and is zero elsewhere. Then

k | y k y k | = u ( k | e k e k | ) u * = u u * = I Y .
Hence I Y ∈ K B(Y).

Finally, any B-module map υ: YZ into a Banch B-module Z satisfies

υ ( y ) = υ ( k = 1 n y k y k | y ) = k = 1 n υ ( y k ) y k | y , y Y .
Thus υ is bounded, and the stated assertion is clear. □

8.2 C*-MODULES AS OPERATOR SPACES.

8.2.1 (C*-modules are operator spaces)

If Y is a right C*-module over B, and if η ∈ N, then M n is a right C*-module over M n(B), with inner product (p.309)

(8.5)
[ y i j ] | [ z i j ] = [ k = 1 n y k i | z k j ] , [ y i j ] , [ z i j ] M n ( Y ) .
One way to see this is to identify Y with the 1-2-corner of the linking C*-algebra ℒ(Y), and B with the 2-2-corner, as in 8.1.19, so that y | z = y * z for y, ZY. If we do this then, first, M n(Y) may be identified with a corner of the C*-algebra M n(ℒ(Y), and the canonical inner product inherited from the latter C*-algebra is y * z = [ k = 1 n Y k i * z k j ] , for y = [y ij], z = [z ij] ∈ Mn(Y). This gives (8.5). second, Y inherits a canonical operator space from the C*-algebra ℒ(Y). We call this the canonical operator space structure on Y. It is given explicitly by
(8.6)
[ y i j ] n = [ k = 1 n y k i | y k j ] 1 2 , [ y i j ] M n ( Y ) ,
as may be seen by using the C*-identity in M n(ℒ(Y)). When we consider a C*-module as an operator space, it will always be with respect to this structure.

The formula (8.6) is also valid for nonsquare matrices. For instance, for a column y = [y 1y n]tC n(Y) = M n,1(Y), we have y = k = 1 n y k | y k 1 2 . Viewing Y as a subspace of the linking C*-algebra, we also have

y = y y * 1 2 = [ y i y j * ] 1 2 = [ | y i y j | ] 1 2 .
Note that this shows that C n(Y) is isometric to the C*-module direct sum of n copies of Y. Similarly, if [x 1x n] ∈ R n(Y), then
(8.7)
[ x 1 x n ] R n ( Y ) = [ x i | x j ] 1 2 = k = 1 n | x k x k | 1 2 .

Proposition 8.2.2

for C*-modules Y and Z over B, every bounded B-module map u: YZ is completely bounded, withu‖ = ‖ ucb. If u is unitary then it is a complete isometry.

Proof This may be seen in many ways. For example, assume that ‖u‖ ≤ 1, and that x i,…,x nY,b 1,…,b nB. Set z = i x i b i . Then

i , j b i * ( x i | x j u x i | u x j ) b j = z | z u z | u z 0 ,
using 8.1.5.By 8.1.13(ii) it follows that [〈x i | x j 〉]− [〈ux i| ux j〉] ≥ 0. Then the first result follows from an easier varient of the implication ‘(v) implies (ii)’ of Theorem 4.5.2. we leave the second as an exercise. □

Henceforth, we give B B(Y,Z) = CB B(Y,Z) the operater space structure from 3.5.1. We assign to B B(Y,Z) and K B the operator space structures which they inherit as subspaces of CB B(Y,Z). We shall see in 8.2.3 (7) below that if Y = Z, the latter operator space structures coincide with their canonical C*-algebra operator space sstructure.

(p.310) 8.2.3 (Operator space variants of C*-module facts)

Most of the results in section 8.1 have operator space variants. We list the key points below; some of these will be used often in the rest of the chapter.

(1) Any right C*-module Y over B is a right operator B-module. Indeed this follows immediately from 3.1.2 (5) and (8.3). By 3.1.11, Y is also a right opreator module over M(B), or over RM(B). By symmetry, Y is also a left operator module over K B(Y), and (using also 8.1.16(3)) over B B(Y). Thus Yis an operator B B(y)-B-bimodule. Similarly, if Y is an equivalence A-B-bimodule, then Y is an operator A-B-bimodule, and an operator M(A)-M(B)-bimodule.

(2) Let Y be an equivalence A-B-bimodule. Viewing Y as the 1-2-corner of the lkinking C*-algebra, and the ‘adjoint module’ Ȳ as the 2-1-corner, one sees the canonical operator space structure on Ȳ, as exactly the adjoint operator space structure Y* from 1.2.25. Note that Ȳ is an operator B-A-bimodule (see 3.1.16).

(3) If i c Y i is a direct sum of right C*-modules over B, equipped with as canoncial operator space structure, then M n ( i c Y i ) i c M n ( Y i ) unitarily as M n(B)-modules. We leave this to the reador.

(4) For any cardinal I, the right C*-module direct sum of I copies of Y, is completely isometrically isomorphic to the opreator space C I(Y) defined in 1.2.26. To see this, recall the canonical complete isometry c:YL(Y) from 8.1.19. The amplification CI,1 of c is a completely isometric embedding from C I(Y) into C I(ℒ(Y))(see 1.2.26).It is clear that

c I , 1 ( ( y i ) ) 2 = i c ( y i ) * c ( y i ) = i y i | y i , ( y i ) C I ( Y ) .
This proves the isometric case of our result. The complete isometry may be deduced from the isometric case together with (3) above.

(5) There is an ‘operator space version’ of 8.1.26, with we may state as follows. Let Y be an operator space and a right B-module, and suppose that there exist maps αt and βt satisfying all the conditions in 8.1.26. If αt and βt are completely contractive, then in addition to the conclusions of 8.1.26, the given matrix norms on M n(Y) coincide with the norm (8.6) induced by the inner product. To prove this, notice that the amplifications (αt)n and (βt)n are yields the desired assertion.

(6) We consider the oppearator space version of 8.1.16. By (1)above, any right C*-module Y over B, is a left operator A-module, where A = K B(Y). By 3.1.11 it is also a left operator LM(A)-module. Thus by 3.1.5(1), the isomorphism θ: LM(A)→ B B(Y) given by (8.2), is a completely contractive homomorphism into CB B(Y). Also, the map ρ: CB B(y) → LM(A) in the proof of 8.1.16, is clearly completely contractive (for example, because CB(Y) is a matrix normed algebra, as we observed in 2.3.9). Thus LM(A) ≅ CB B(Y) completely isometrically isomorphically. Variants of several of the facts in 2.6.6 hold for C*-modules. For example, there is a canonical isomorphism from M n(CB B(Y)) onto B B(C n(Y)). Indeed M n(CB B(Y)) ≅ M n(LM(K B(Y))) by the above, and (p.311)

M n ( L M ( K B ( Y ) ) ) L M ( M n ( K B ( Y ) ) ) L M ( K B ( C n ( Y ) ) ) B B ( C n ( Y ) ) ,
using 2.6.6 (3), 8.1.11 (3), and 8.1.16 (3).

(7) The canonical operator space structure on the C*-algebra B B(Y) coincides with the inherited operator space structure from CBB(Y). Indeed, by (6), we have Mn(CBB(Y)) ≅ BB(Cn(Y)). On the other hand, by 8.1.11 (3), we have Mn(B B(Y)) ≅ B B(Cn(Y)) ⊂ BB(C n(Y)).

By the discussion in (6) and (7) above, together with 8.1.16 (4), we have:

Corollary 8.2.4

If Y is a right C*-module over B, then CB B(Y) is a unital operator algebra completely isometrically isomorphic to LM(K B(Y)). Moreover, Δ(CBB(Y)) = B B(Y) (see 2.1.2 for this notation).

8.2.5 (Countably generated modules)

We will use operator space column and row notation (see 1.2.26) to lead us through the important ‘stabilization theorems’. We say that a Banach B-module X is countably generated if there is a sequence (xn) in X such that Span{bxn: bB, n ∈ N} is dense in X. By a (countable) right quasibasis of a right C*-module Y over B, we mean a row [yk] ∈ Rw(Y) (see 1.2.26 for this notation), such that

(8.8)
y = k = 1 y k y k | y , y Y ,
the sum converging in norm. Clearly if there exists a right quasibasis, then Y is countably generated over B. Indeed, clearly if Y has a right quasibasis, then K B(Y) has a countable approximate identity. By 8.1.15, this is equivalent to A having a countable approximate identity, if Y is an equivalence A-B-bimodule. It is also easy to see that if K B(Y) has a countable approximate identity, then Y is countably generated over B. We shall not use this fact, but conversely, if Y is a countably generated right C*-module, then K B(Y) has a ‘strictly positive element’ (see [65, Proof of 7.13]), and hence a countable approximate identity [320, Proposition 3.10.5]; following the proof of [75] Lemmas 2.1–2.3, one sees that Y has a right quasibasis (see e.g. [46, Theorem 8.2]).

We claim that if Y has a right quasibasis, then Y is unitarily isomorphic to an orthogonally complemented submodule of C(B). Here of course C(B) = CI(B) (see 8.2.3 (4)) when I = N. This is a variant of Theorem 8.1.27 (1). To prove this, note that by (8.8) we have y | y = k = 1 y | y k y k | y , for all yY. This permits us to define an isometric B-module map α: YC(B), by α(y) = (〈yk|y〉)k. By a simple calculation analoguous to that in 1.2.27, there is a well defined contractive B-module map β: C(B) → Y given by β((bk))= ∑k ykbk. Clearly β ∘ α = IY. Our claim then follows from 8.1.21 (2).

A left quasibasis for Y is a column (zk) ∈ Cw(Y) with y = k = 1 y z k | z k for all yY. If Y is full, then the latter condition is equivalent to the same condition, but for yB, since in that case B = Y * Y and Y = YB. Taking adjoints, we see the latter is also equivalent to (p.312)

(8.9)
k = 1 z k | z k b = b , b B .
By the other-handed version of an assertion made in the second last paragraph, (8.9) is also equivalent to B having a countable approximate identity. We shall avoid using this though, since we have not proved it.

In the following, to avoid a notational conflict, we write K (Y) for the space we wrote as K(Y) in 1.2.26. That is, K (Y) ≅ K ⊗min Y.

Corollary 8.2.6 (Brown-Kasparove stabilization)

Suppose that Y is a right C*-module over B. Then (using the notation above):

  • (1) C(B)⊕cYC(B) unitarily, if Y has a right quasibasis.

  • (2) C(B)⊕cC(Y) ≅C(B) unitarily, if Y has a right quasibasis.

  • (3) C(B)⊕cC(Y) ≅C(Y), if Y is full, and has a left quasibasis.

  • (4) C(B) ≅ C(Y). under the hypotheses of both (2) and (3).

  • (5) If Y is an equivalence A-B-bimodule satisfying the hypotheses of both (2) and (3), then K (B) ≅ K (Y) ≅K (A) linearly completely isometrically.

Proof For (1), by the ‘claim’ proved in 8.2.5, we may write C(B) ≅ Y (B)⊕cW for a submodule W of C(B). By the ‘associativity’ of the C*-module sum, we may employ the ‘Eilenberg Swindle’:

C ( B ) C ( B ) c C ( B ) c ( Y c W ) c ( Y c W ) c Y c ( W c Y ) c ( W c Y ) c Y c C ( B ) .
‘Associativity’ of the sum also gives (2). For example, we have using (1),
C ( B ) C ( C ( C ( B ) ) C ( C ( B ) ) c Y ) C ( C ( B ) ) c C ( Y ) C ( B ) c C ( Y ) .
For (3), suppose that (zk) is a left quasibasis. Define a map α: BC(Y) by the prescription α(b) = (zjb). That (zjb). That (zjb) ∈ C(Y) is easily seen from (8.9). Define β: C(Y) → B by β((yj)) = ∑jzj|yj〉. The latter sum converges by the argument in 1.2.27, and indeed this argument shows that ‖β‖ ≤ 1. Then α and β are contractive B-module maps which compose to the identity mapping on B, again by (8.9). Consequently, by 8.1.21, B is unitarily isomorphic to an orthogonally complemented B-submodule of C(Y). Then (3) follows by an argument similar to that of (1) and (2).

Item (4) is clear from (2) and (3). For (5) note that K (Y) ≅ R(C(Y)) (this is easily deduced from (1.37), for example). By (4) we deduce that

K ( Y ) R ( C ( Y ) ) R ( C ( B ) ) K ( B )
The assertion about A follows by symmetrical arguments (replacing C(B) above by R(A), and so on). □

(p.313) 8.2.7 (The Brown-Green-Rieffel stable isomorphism theorem)

Since this is in most of the cited C*-module texts, we will be quick here. Suppose that Y is an equivalence A-B-bimodule which has both a left and a right quasibasis (which occurs, as we mentioned in passing in 8.2.5, exactly when A and B both have countable approximate identities). We saw in 8.2.6 (5) that K (A) ≅ K (B) linearly completely isometrically. Thus by 4.5.13, K ≅ K (B) *-isomorphically. Conversely, if K (A) ≅ K (B) *-isomorphically, then it is easy to see that A and B are strongly Morita equivalent (see the hints in the Notes section).

8.2.8 (Representations of C*-modules)

Suppose that Y is an equivalence bi-module over A and B, and that we are given a nondegenerate *-representation π: ℒ(Y) → B(H) of the Morita linking algebra of Y (see 8.1.17). If p is the projection introduced above (8.3), then, using the notation and facts in 2.6.15, q = π ^ ( p ) is a projection in B(H), and we may decompose B(H) as a 2 × 2 matrix operator algebra. Indeed the i-j-corner of B(H) is simply B(Hj,Hi), where Ran(q) = H 1 and Ker(q) = H 2. By 2.6.15, π is corner-preserving, and we may decompose π as [πij]. The maps πij are complete contractions, which are complete isometries if π is faithful. Also π11 and π22 are *-representations of A and B on H 1 and H 2 respectively. In fact π11 and π22 are also nondegenerate. Indeed, if (b β) is a cai for ℒ(Y), then qπ(b β)(q ζ) → q ζ, for all ζ ∈ H. However qπ(bβ)q = π̂(pbβp), in the language of 2.6.12 and 2.6.15. Since pbβp is a cai for A, π11 is nondegenerate. A similar argument applies to π22. Note also that [π12(Y)H 2] = H 1, since we have H 1 = [π11(A)H 1] = [π2(Y21(Ȳ)H 1] ⊂[π12(Y)H 2]. A similar argument shows that [π12(Y)⋆H 1] = H 2.

If c is the corner map mentioned in 8.1.19, then we have

q π ( c ( y ) * ) q π ( c ( z ) ) q = q π ( c ( y ) * ) π ( c ( z ) ) q = q π ( c ( y ) * c ( z ) ) q .
From this, it follows immediately that
(8.10)
π 12 ( y ) * π 12 ( z ) = π 22 ( y | z ) , y , z Y .
Similarly,
π 12 ( x y * z ) = q π ( c ( x ) c ( y ) * c ( z ) ) q = π 12 ( x ) π 12 ( y ) * π 12 ( z ) , x , y , z Y .
Thus π12 is a triple morphism (see 4.4.1). Hence the range of π12 is a TRO inside B(H 2,H 1). if π is faithful, then we have represented Y completely isometrically as a TRO in B(H 2,H 1). In fact this must hold even if Y is only a C*-module, since we saw in 8.1.14 that every C*-module Y is an equivalence bimodule.

8.2.9 (Rigged C*-modules)

In C*-module theory, the most important tensor product is the so-called interior tensor product. We will discuss this tensor product momentarily; for now we will just say that it is formed from a right C*-module Y over A, and a so-called B-rigged A-module Z. By the latter term, we will mean a right C*-module Z over B together with a *-homomorphism θ: A → B B (Z), such that θ is nondegenerate in the sense that Z, considered as a (p.314) left Banach A-module in the canonical way (see A.6.1), is nondegenerate in the sense of A.6.1.

Lemma 8.2.10

Suppose that A and B are C* -algebras, and that Z is a right C*-module over B, which is also a left A-module. Then Z is a B-rigged A-module if and only if Z with its canonical operator space structure as a right C*-module (see (8.6)), is also a nondegenerate left operator A-module.

Proof If Z is a B-rigged A-module, then by the observations above, Z is certainly a nondegenerate left B-module. By 8.2.3 (1), Z is a left operator module over B B(Z). By 3.1.12, Z is an operator A-module.

If Z is a left operator A-module, then since Z is also a right operator module over B, we see by the last assertion in 4.6.7 that Z is an A-B-bimodule. Define θ: ACB B(Y) by θ(a)(Y) = ay. Then θ is a contractive homomorphism. Indeed θ is a *-homomorphism into Δ(CBB(Y)) = B B(Y), by the last assertion in 2.1.2 and 8.2.4. The rest is clear.

We emphasize that this lemma shows that the bimodules met with in the theory of C*-modules, are operator bimodules. By virtue of the lemma, it makes sense to define the module Haagerup tensor product (see Section 3.4) of a right C*-module over A and a B-rigged A-module.

Theorem 8.2.11

Suppose that Y is a right C*-module over A, and that Z is a B-rigged A-module. Then the module Haagerup tensor product YhA Z is a right C*-module over B, with B-valued inner product determined by the formula

(8.11)
y z | y z = z | y | y z , y , y Y , z , z Z .
Moreover, the usual operator space structure on YhA Z coincides with the canonical operator space structure induced by the inner product as in (8.6).

Proof By 8.1.24 and 8.2.2, there exist completely contractive A-module maps αt: YCnt (A) and βt: Cnt (A) → Y such that βt ∘ αtIY strongly on Y. By the functoriality of the module Haagerup tensor product (see 3.4.5), we obtain contractive B-module maps αtIZ: YhA ZCnt(A) ⊗hA Z and βtIZ: Cnt(A) ⊗hA ZYhA Z. By density of the elementary tensors, the net of maps (βtIZ) ∘ (αtIZ) converges strongly to the identity map on YhA Z. By 3.4.11, we have Cnt(A) ⊗hA ZCnt(Z), and the latter is a right C*-module over B. Via this isomorphism, it is easily checked that the induced inner product on Cn (A) ⊗ hA Z is given by the formula

a z | a z = z | θ ( a * a ) z , a , a C n ( A ) , z , z Z .
By 8.1.26, we conclude that YhA Z is a right C*-module over B. By 8.1.26 used twice more, the inner product on YhA Z is specified by its value on a pair of rank one tensors yz and y′ ⊗ z′, as follows:
lim t α t ( y ) z | α t ( y ) z = lim t z | θ ( α t ( y ) * α t ( y ) ) z = z | y , y z .
This, together with 8.2.3 (5), proves the result. □

(p.315) In the remainder of this section, we will simply write YA Z for YhA Z.

8.2.12 (Properties of the tensor product)

One may view the last result as the assertion that the well-known interior tensor product of Y and Z of C*-modules (see any of the texts cited at the start of this chapter), coincides with the module Haagerup tensor product. This is helpful in many ways, partly because the Haagerup tensor product has many useful properties. For example, one advantage of 8.2.11 is that it gives most of the important properties of the interior tensor product ‘for free’. For example:

(1) (Functoriality) If u: YY′ is a bounded right A-module map between right C*-modules over A, and if υ: ZZZ′ is a bounded A-B-bimodule map between B-rigged A-modules, then u⊗υ extends to a bounded right B-module map between the interior tensor products: YA ZYA Z′ (of norm ≤‖u‖‖υ‖). This follows immediately from the functoriality property of the Haagerup tensor product (see 3.4.5), and 8.2.2.

(2) (Associativity) We have (YZ) ⊗B W = YA (ZB W) unitarily, if Y is a right C*-module over A, if Z is a B-rigged A-module, and if W is a C-rigged B-module. This follows immediately from the associativity of the Haagerup tensor product (see 3.4.10).

(3) (Commutation with the direct sum) We have

(8.12)
i c ( Y i A Z ) ( i c Y i ) A Z unitarily ,
for right C*-modules Yi over A, and a B-rigged A-module Z. Also,
i c ( Y A Z i ) Y A ( i c Z i )
for a right C*-module Y over A, and B-rigged A-modules Zi. These relations both follow from the universal property of the direct sum in 8.1.22. To see (8.12), let εi and Pi be the canonical inclusion and projection maps between i c Y i and its summands. Define maps ε i = ε i I Z and P i = P i I Z between ( i c Y i ) A Z and Y i A Z . These are contractive by (1) above, and are easily checked to satisfy the hypotheses of 8.1.22.

The second centered relation above is almost identical, however one first should check that i c Z i is indeed a B-rigged A-module. Observe that the canonical left action of A on i c Z i is well defined since by 8.1.5,

j θ j ( a ) z j | θ j ( a ) z j a 2 j z j | z j , a A , z j Z j .
Here θj:A → B B(Zj) is the homomorphism associated with the left A-module actions on Zj. Furthermore, it is easy to check that this action of A on i c Z i is as ‘adjointable operators’, and that the action is nondegenerate. Hence i c Z i is a B-rigged A-module.

(4) (The adjoint module) By the definition of the module Haagerup tensor product in Section 3.4, it is easy to see using 8.2.3 (2) that the ' adjointC*-module' of  Y A Z is Z ¯ A Y ¯ , completely isometrically.

(p.316) Corollary 8.2.13

Suppose that Y is a right C*-module over B, and that θ is a nondegenerate *-representation of B on a Hilbert space H. Then:

  • (1) YB Hc is a Hc is a Hilbert space.

  • (2) If Y is a B-rigged A-module, then YB Hc is a Hilbert A-module. If θ and the canonical map from A into B(Y) are both one-to-one, then so is the canonical map from A into B(YB Hc).

Proof(1) This follows from 8.2.11, which shows that Y >⊗B Hc is a right C*-module over C. That is, it is a Hilbert space with inner product ζ×η ↦ 〈η|ζ〉.

(2) In this case, Y and YBHC are left operator A-modules, by 8.2.10 and 3.4.9. Thus the first assertion follows from (1) and 3.1.7. If α(YBHc) =0 then aY = 0, by the relation 〈a(y ⊗ζ),y′⊗η〉 = 〈ζ|〈ay|y′〉η〉from (8.11). Here aA,y,y′Y, and ζ,η∈H.

8.2.14 (Avoiding the inner product)

As we have already seen, many C*-module constructions can be done, if need be, without explicit reference to the inner product. See 8.1.6, 8.2.12, and 8.4.2, for example. Here we use 8.2.13 to take this thought a little further, omitting full proofs. If Y is a right C*-module over B, and if B is a nondegenerate *-subalgebra of B(K), say, then define a B-module map Φ: YB(K,YB Kc) by Φ(y)(ζ) = y ⊗ ζ, for yY, ζ ∈K. It is easy to see that Φ(y)*Φ(z) = 〈y|z〉, for y,zY (using (8.11)). Also, Φ(Y) is a C*-module over B with the inner product (y,z)↦Φ(y)*Φ(z), and Φ is a unitary B-module map. Thus the inner product on Y is completely determined by the norm on the space YB Kc). The latter norm has reformulations avoiding use of the inner product, mentioned at the end of the Notes to Section 8.2.

We use the above to give an alternative description of the C*-module direct sum of 8.1.9. For specificity we discuss the direct sum of two right C*-modules, Y 1 and Y 2, over B. Let K be as above, and let Hi = YiB KC. We will suppress mention of the map Φ in the last paragraph, and simply write yiζ for yi⊕ ζ. Then Y 1c Y 2 may be identified with the B-submodule W of B(K,H 1H 2) consisting of the maps ζ ↦(y 1ζ,y 2ζ), for ζ∈K,y 1Y 1, y 2Y 2. Indeed, the canonical inner product on W, namely S × TS*T, takes values in B, making W into a C*-module over B which is unitarily B-isomorphic to Y 1c Y 2.

For the next result, we will need to extend the definition of K B(Y, Z) from 8.1.7, to allow Z to be any right operator B-module. Namely, we define K B(Y,Z) to be the closure in CB(Y,Z) of the span of the ‘rank one’ operators yzy′|y′ (here y, y′ ∈ Y,zZ).

Henceforth, we shall assume that all operator modules are nondegenerate.

Corollary 8.2.15

Let Y be a right C*-module over B, and let W be a right C*-module over B (or more generally let W be a right operator B-module). Then:

  • (1) W B Y ¯ K B ( Y , W ) completely isometrically.

  • (2) If Y is an equivalence A-B-bimodule, then K B ( Y , W ) = C B B ess ( Y , W ) , in the notation of the second paragraph of 3.5.2.

(p.317) Proof(1) Define ρ : W × Y ¯ K B ( Y , W ) by ρ ( w , y ¯ ) ( x ) = w y | x , for x,yYand wW. As in 8.1.19, we write 〈y|x〉 as y*x, interpreted as a product in ℒ(Y). Let [wij], [ z i j * ] and [xrs] be matrices with entries in W, Ȳ, and Y respectively. Since W is a right h-module (see 3.1.3), it is not hard to see that

[ k w i k z k j | x r s ] [ w i j ] [ z i j * x r s ] [ w i j ] [ z i j * ] [ x r s ] ,
using also the fact that ℒ(Y) is an operator algebra, and hence a matrix normed algebra (see 2.3.9). Thus ρ is completely contractive in the sense of 1.5.4. By 3.4.2, ρ induces a complete contraction from W B Y ¯ t o K B ( Y , W ) , which we will still write as ρ. Next, take a cai for K B(Y) of the form in (8.4). Given T ∈ K B(Y,W), define θ t ( T ) W B Y ¯ to be the element k T ( x k t ) x ¯ k t . From (1.40) and (8,6), one may easily check that θ t : K B ( Y , W ) W B Y ¯ is completely contractive. However, for w W , y ¯ Y ¯ ,
θ t ( ρ ( w y ¯ ) ) = k w y | x k t x ¯ k t = w k y | x k t x ¯ k t w y ¯ .
By density, θt(ρ(u)) → u, for all u, for all u W B Y ¯ This implies, by a simple modification of the principle in 1.2.7, that ρ is a complete isometry. Now it is clear that ρ maps onto K B(Y,W).

(2) If Y is an equivalence A-B-bimodule, then A ≅ K B(Y) by 8.1.15. It then follows from the argument for (8.1), that C B B ess ( Y , W ) K B ( Y , W ) . Conversely, every ‘rank one’ operator |w〉〈y| in K B(Y,W) is the limit of w e t y = w y e t which lie in C B B ess ( Y , W ) . Here (e t)t is as in 8.1.23. It follows that w y is in C B B ess ( Y , W ) , and so K B ( Y , W ) C B B ess ( Y , W ) .

8.2.16 (Hom-tensor relations)

The tensor product identity in 8.2.15 (1) is a C*-module variant of what is called a Hom-tensor relation in algebra. There are bimodule versions of this particular identity too, namely W B Y ¯ K B ( Y , W ) as bimodules, if in addition Y and W are operator bimdules. Also, there is a matching left-handed result: if Z is a right C*-module over A, and W is a left operator A-module, then Z A W A K ( Z ¯ , W ) . We leave these assertions to the interested reader.

As a sample application of the relation in 8.2.15, we show that for a right C*-module Y over B, K B(C(Y)) is *-isomorphic to the minimal tensor product K B(Y) ⊗min K , where K = K(ℓ2) again. To see this, note that the adjoint C*-module of C(Y) is R(Ȳ). Hence by 8.2.15 used twice, and by basic properties of the Haagerup tensor product from Sections 1.5 and 3.4, we have

K B ( C ( Y ) ) = ˜ C ( Y ) B R ( Y ¯ ) = ˜ C h ( Y B Y ¯ ) h R = ˜ K B ( Y ) min K .
Setting Y = B, we may deduce from this and 8.1.14, that B is strongly Morita equivalent to Bmin K∞.

(p.318) From 8.2.15 one may deduce further Hom-tensor relations such as:

K B ( Y A Z , W ) K A ( Y , K B ( Z , W ) ) completely isometrically .
Here Y is a right C*-module over A, Z is a B-rigged A-module, and W is a right operator module over B. To see the last centered relation, note that by 8.2.15, K B ( Y A Z , W ) W B ( Y A Z ) ¯ . Thus
K B ( Y A Z , W ) = ˜ W B Z ¯ A Y ¯ = ˜ K B ( Z , W ) A Y ¯ = ˜ K A ( Y , K B ( Z , W ) ) ,
using 8.2.15 twice, and 8.2.12 (2) and (4). Indeed, it is also true that for Y, Z, W as above, we have
C B B ( Y A Z , W ) = ˜ C B A ( Y , C B B ( Z , W ) )
completely isometrically. The proof of this, and other such Hom-tensor relations, may be found in [47].

One may also define the well-known ‘exterior tensor product’ of right C*-modules in operator space terms:

Theorem 8.2.17

Let Y and Z be right C*-modules over A and B respectively. Then the minimal tensor product Ymin Z (see 1.5.1) is a right C* -module over Amin B with inner product determined by

y z | y z = y | y z | z , y , y Y , z , z Z .

We omit the proof of this, which is extremely similar to that of 8.2.11. Theorem 8.2.17 also easily implies results analoguous to those in 8.2.12.

Strong Morita equivalence may be restated concisely in the language of operator modules as follows:

Theorem 8.2.18

If Y is an equivalence A-B-bimodule, then Ȳ⊗A YB completely A-A-isometrically, and YB Ȳ ≅ A completely B-B-isometrically. Conversely, if Y and X are respectively operator A-B- and B-A-bimodules, such that XhA YB completely B-B-isometrically, and also YhB XA completely A-A-isometrically, then A and B are strongly Morita equivalent, Y is an equivalence A-B-bimodule, and XȲ completely B-A-isometrically.

ProofWe will only need the first statement later, and will only prove this one here. We refer the reader to [52] Proposition 1.3 in conjunction with [65] Theorem 6.2, for a proof of the second statement.

If Y is an equivalence A-B-bimodule. then by 8.2.15 and 8.1.15, we have Y B Y ¯ K B ( Y ) A . Tracing through these identifications, one sees that the isomorphism holds as A-A-bimodules too. Similarly, Y ¯ A Y B .

8.2.19(Induced representations and Morita equivalence)

Let Y be an equivalence A-B-bimodule. If Z is a left operator B-module, then G(Y) = YB Z (p.319) is an operator A-module (see 3.4.9). If u: Z 1Z 2 is a completely contractive B-module map between left operator B-modules, then IYu: G(Z 1) → G(Z 2) is a completely contractive A-module map (see 3.4.5). That is, G(—) = YB— is a functor from the category B OMOD to the category A OMOD (see 3.5.1). Indeed the map uG(u) is linear and contractive on BCB(Z 1,Z2). It is easy, but tedious, to check that this map is actually completely contractive. Consequently, we call G a completely contractive functor. By 8.2.13, G maps the subcategory B HMOD to the category A HMOD. By 8.2.10 and 8.2.11, G also maps between the subcategories of left C*-modules over B and A respectively. Conversely, if F ( ) = Y ¯ A then F is a completely contractive functor from the category AOMOD to the category BOMOD, and F maps AHMOD to BHMOD, and also maps between the subcategories of left C*-modules. Composing these functors, and using the last theorem, and 3.4.10 and 3.4.6, we have for any left operator A-module W that

Y B ( Y ¯ A W ) = ˜ ( Y B Y ¯ ) A W = ˜ A A W = ˜ W .
Thus GF = I, and similarly FG = I. Note that from this, and from the fact that G and F are completely contractive, it is easy to see from 1.2.7 that the map uG(u) above is a complete isometry on B CB(Z 1, Z 2).

Thus if A and B are strongly Morita equivalent, then AOMOD and BOMOD are equivalent as categories (as are also AHMOD and BHMOD, and as also are the categories of left C*-modules over A and B). Of course, a similar argument gives analoguous results for the categories of right modules.

The above proves the part we will need later of the following result. Proof of the other parts may be found in [51]. We remark that this theorem is a C*-algebraic version of Morita's fundamental theorem from pure algebra (e.g. see [8,368]).

Theorem 8.2.20

Two C* -algebras A and B are strongly Morita equivalent if and only if the categories AOMOD and BOMOD are equivalent (via completely contractive functors). Moreover, if F: AOMODBOMOD is the equivalence functor, then X = F(A) is a strong Morita equivalence B-A-bimodule.

8.2.21 (The nonselfadjoint algebra case)

In the light of 8.2.18, if A and B are approximately unital nonselfadjoint operator algebras, then it is natural to define A and B to be strongly Morita equivalent if there exists an operator A-B-bimodule Y and an operator B-A-bimodule X, such that XhA Y = B completely B-B-isometrically, and Y h B X A completely A-A-isometrically. From this definition one can then proceed to develop a theory parallel to the C*-algebra case. For example, 8.2.20 generalizes: A and B are strongly Morita equivalent operator algebras if and only if the categories A OMOD and B OMOD are equivalent via completely contractive functors. Similarly, there is a generalization of C*-modules to the nonselfadjoint situation. See the Notes section for references to the literature.

(p.320) 8.2.22 (Induced representations)

Let Y be an equivalence A-B-bimodule. Thinking of modules W in BOMOD as nondegenerate representations of B on W, we saw in 8.2.19 that such representations of B induce representations of A (on YBW), and vice versa. Since ℒ(Y) is strongly Morita equivalent to B via the equivalence bimodule Yc B (see 8.1.18), we see that there are one-to-one correspondences between the ‘representations’ of B, the ‘representations’ of A, and the ‘representations’ of ℒ(Y). In particular, by A.5.8 and the discussion in 8.2.19 about H MOD, there are one-to-one correspondences between the nondegenerate *-representations of B on Hilbert space, and those of A, or of ℒ(Y). In fact we have seen part of this already in 8.2.8, although in a disguised form. There we took a nondegenerate *-representation π of ℒ(Y) on a Hilbert space H, and we saw that there were corresponding representations of A on a Hilbert space H 1, and B on H 2. At first sight the spaces H 1 and H 2 defined in 8.2.8 look different from the ones obtained via the correspondences above. We will now show that they are the same, up to a unitary. First we claim that

(8.13)
H I Y B H 2 , and H 2 Y ¯ A H 1 ,
unitarily. To see this, we will use the notation in 8.2.8, and we define a module map f: YB H 2H 1 by f(y⊗ζ) = π12(y)(ζ). Using (8.10), we have
f ( y ζ ) , f ( y η ) = π 12 ( y ) * π 12 ( y ) ζ , η = π 22 ( y | y ζ , η ) ,
for y,y′ ∈ Y, ζ, η ∈ H. By (8.11), the last number equals 〈y⊗ζ,y′⊗η〉. It follows from this that f ( ξ ) , f ( ξ ) = ξ | ξ , for any ‘finite rank’ tensor ξ ∈ YB H 2. Thus f is a well defined isometry. Also, we checked in 8.2.8 that f has dense range. Hence f is urjective. This yields the first relation in (8.13). The second is similar.

By (8.12), (8.13), and 3.4.6, we have

(8.14)
( Y c B ) B H 2 ( Y B H 2 ) c ( B B H 2 ) H 1 c H 2 H .
This shows that the ℒ(Y)-module induced from the B-module H2 via the equivalence bimodule Yc B, by the procedure of 8.2.19, is unitarily isomorphic to H. Thus, conversely, the B-module induced from the ⊗(Y)-module H by the procedure of 8.2.19, is unitarily isomorphic to H 2.

The argument in and below (8.14) shows that any Hilbert B-module K induces a nondegenerate *-representation π of ℒ(Y) on (YBK)⊕K. It is easy to check that π is corner-preserving, in the sense of 2.6.15. Its ‘four corners’ consist of a representation of A on B(YBK), maps from Y to B(K, YBK) and from Ȳ to B(YBK,K,K), and the given representation of B on K. Also, by 8.2.13 (2), π is faithful if the given representation of B on K was faithful.

8.2.23 (Inducing universal representations)

Suppose that Y is an equivalence A-B-bimodule, and that K is a B-universal Hilbert module (see 3.2.7), or equivalently, a generator for BHMOD (see 3.2.8). By 8.2.19, Y induces an equivalence (p.321) of categories AHMOD ≅ BHMOD. By the simplest category theory, the induced represention of A on H = YB K is A-universal. One corollary of this: we see from 3.2.12 and the double commutant theorem, that the second commutant of A in B(H), is completely isometically isomorphic to A**.

This construction is pleasantly functorial, and is therefore quite useful (for example, it is key to the proof of the difficult implication in Theorem 8.2.20).

8.2.24 (Rieffel subequivalence)

One of the pleasant consequences of purely algebraic Morita equivalence, is that if two rings A and B are Morita equivalentvia an A-B-bimodule Y, then there are one-to-one lattice isomorphisms between the following lattices: (1) the two-sided ideals I of A, (2) the two-sided ideals J of B,(3) the A-B-submodules X of Y, and (4) the two-sided ideals D of ℒ(Y). Rieffel showed that similar correspondences hold in the C*-alogebra setting, with the word ‘norm-closed’ added. In fact this is quite easy to see: First, we replace the inner products and module actions with concrete multiplication in the linking algebra, as in 8.1.19. Define a map from lattice(2) to lattice (3) by J ↦ XJ = YJ. Conversely, define a map from lattice (3) to lattice (2) by XJ(X) = Y*X, for X in lattice (3). Using A.6.2, we have X = AX = YY*X. Thus XJ(X) = X. Given J in lattice (2), a similar argument shows that J(XJ) = J. Thus indeed the lattices (2) and (3) are lattice isomorphic. Similarly for (1) and (3). Note that if X and J are in correspondence as above, then by A.6.2,

J = J J = ( Y X ) Y X = X Y Y X = X X .
Since J = XX,X is a full right C*-module over J. By symmetry, if I is the ideal in lattice (1) associated with X, then X is a full left C*-module over I. Thus X is a equivalence I-J-bimodule, and I and J are strongly Morita equivalent. The linking C*-algebra ℒ(X) of X may be taken to be the *-subalgebra D of ℒ(Y) with corners I,X,X*, and J, by the unicity of the C*-norm on a *-algebra (see A.5.8). It is easy to check that D is an ideal of ℒ(Y).

Now we shall check that the lattices (2) and (4) are isomorphic, via the correspondence JD above. By 8.1.18, we may use the lattice isomorphisms we have already verified, but with Y and A replaced by Yc B and ℒ(Y). Thus an ideal J of B corresponds to a ℒ(Y)-B-submodule W of Yc B, namely the submodule W = (Yc B)J = Xc J. Also, we obtain a corresponding ideal I′ = WW* of ℒ(Y). However it is easily checked from facts in the previous paragraph that WW* =D.

8.2.25 (Rieffel quotient equivalence)

let X be a closed A-B-sunmodule of an equivalence A-B-bimodule Y. Via the correspondences in 8.2.24, let I be the corresponding ideal in A, and let J be the corresponding ideal in B.let D be the subset of ℒ(Y) with for corners I,X,X⋆ and J. We saw above that X is an equivalence I-J-bimodula, and D ≅ ℒ(X *-isomorphically. Moreover, D is a closed ideal in ℒ(Y). We consider the quotient map π: ℒ(Y) → ℒ(Y)/D. By 2.6.15, the canonical four couners of ℒ(Y) induce corners of ℒ(Y)/D, and π is corner-preserving. Write π = [πij], as in 2.6.15. It is straghtforward to check (p.322) that the 1-2-corner W of ℒ(Y)/D is a TRO, and also is an equivalence bimodule over π11(A) and π22 (B). For example, to see that WW = π22(B), observe that

π 22 ( z | y ) = π 21 ( z ¯ ) π 12 ( y ) W W , y , z Y .
It follows that π22(B)⊂ WW, and the converse inclusion is clear.

Next, note that the ‘1-2-corner’ π12 of π is a complete contraction from Y to W, with kernel X. Since there is a completely contractive projection from ℒ(Y) onto its 1-2-corner c(Y), it follows easily that π12 is a complete quotient map. Indeed, to see that it is a quotient map, suppose that we are given an element wW of norm < 1. Then since π is a quotient map, there exists an element w of ℒ(Y) of norm <1 which π maps to w. The 1-2-corner of W is in Y, has norm <1,and is mapped by π12 to w.

Thus W is completely isometrically isomorphic to Y/X, the latter in the sense of 1.2.14. Similarly the 1-1-corner of ℒ(Y)/D is *-isomorphic to A/I, and the 2-2-corner of ℒ(Y)/D is π-isomorphic to B/J.Thus we see that A/I and B/J are stongly Morita equivalent. The equivalence (A/I)-(B/J)-bimodule may be taken to be Y/X (or equivalently,W). One easily sees that with respect to these indentifications, the B/J-valued inner produet on Y/X, for example, is simply:

y + X | y + X = y | y + J , y , y Y .
Thus the quotient Y/X has (a) a canonical C*-module structure (discussed above), and (b) a quotient operator space structure (from 1.2.14); and very fortunately these two structures are compatible.That is, the cononical operator space structure on the C*-module Y/X equals its quotient operator space structure.

8.3 TRIPLES, AND THE NONCOMMUTIVE SHILOV BOUNDARY

8.3.1 (Triple sustems)

In the literature, the word ternary is often used in place of our usage of the word triple, to aviod confusion with the use of that word in the JB*-triple literature. Since we will not discuss. JB*-triples, we will use the word ‘triple’ consistently, with apologies to those for whom it has different connotations. In 4.4.1 we discussed the ‘triple product’[x,y,z] = xy*on a TRO. It is convenient for us definea triple system to be an operator space Y possessing a map[·,·,·]: Y ×Y×YY (again called a triple product), such that Y is complete isometric to a TRO Z via a linear map θ: YZ which is a triple morphism, that is,θ ([x,y,z)] = [θ(x), θ(y),θ(z)] for all x,y,zY. In fact by 4.4.6, it is clear that an operator space Y can have at most one such triple product, and thus this triple product is uniquely determined by the norms on M n(Y), for n≥1. We will therefore write xy*z for this unique triple product on Y, without there being too much danger of confusion. We are not aware of a simple formula for the triple product in terms of the matrix norms; however we mention in passing that there is a remarkable intrinsic charcterize of triple systems in terms of these norms due to Neal and Russo [289].

(p.323) In any case, by the above, we may simply define a triple system to be an operator space which is linearly completely isometric to a TRO. By a triple epimorhism we mean a surjective triple morphism, whereas a triple isomorphism is a one-to-one surjective triple morphism. By a subtriple of a triple system, we mean a closed subspace which is closed under the triple product. Thus TROs may be defined to be the subtriples of B(K,H), or of a C*-algebra. Any C*-module Y, with its canonical operator space structure, is a triple system. If Y is an equivalence bimodule, then this triple product is just xy|z〉 = [x|y]z; which we simply write xy*z (see also 8.1.19).

From the perspective of this section, TROs, C*-modules, equivalence bimodules, and triple systems, are essentially the same thing. That is, we may use these words interchangeably in the statements of most results below. Thus although we often restrict our attention to TROs, such results will immediately imply corresponding results for triple systems or C*-medules.

It is clear that if Y is a triple system, then so is m n(Y). In particular, if Y = B(K, H), then the triple product onM n(Y) simply corresponds to the obvious triple product on B (K (n),H (n)).

Lemma 8.3.2 (Harris-Kaup)

letθ:YW be a triple morphism between TROs. Then:

  • (1) θ is contractive, and indeed completely contractive.

  • (2) θis completely isometric if and only if it is one-to-one.

Proof (1) The amplification θn: M n(Y) →m n(W) is also a triple morphism between TROs (see 8.3.1). Thus it is enough to prove the first statement, or equivalently that ‖θ(y)*θ(y)‖ ≤ ‖y*y)‖ for any yY, or equivalently the following containment of spectra: σ (θ(y)*θ(y)) ∪ σ(y*y) ∪{0}. The latter follows immediately from the following claim: A nonzero scalar λ; is in σ(y*y) if and only if there does not exist a zY such that

(8.15)
( λ I y y * ) z ( λ I y * y ) = y .
To prove this claim, note that if λ ∉σ (y*y) then z = yIy*y) −2 satisfices (8.15). However zY, since (λIy*y)−2 is, by spectral theory, a limit of polynomials in y*y. Conversely, suppose that λ∈σ(y*y). By basic spectral theory, we can find a sepuence b nY*Y with (λIy*y)b n →0, but b n itself does not converge to 0. Hence y*yb n, and therefore also yb n, cannot converge to 0. From this, it is easy to see that there can exist no z satisfying (8.15).

(2) Again, it suffices to prove that if θ in (1) is one-to-one, then it is isometic. This will follow if σ(θ(y)*θ(y)) ∪{0} = σ(y*y) ∪{0}, in the notation of (1). However if λ;∈ σ(y*y)\σ (θ(y)), with λ≠0, then by Urysohn's lemma there exists a continuous nonnetgative function f on σ(y*y), which is zero on σ(θ(y)*θ(y)) and at 0, but is nonzero at λ. Clearly f (y*y)≠ 0. Since f may be approximated uniformly by polymials p n with no constant term, we have that f(y*y) ∈ YY.

For any zY and any polynomial p, θ(zp(y*y)) = θ(z) p (θ(y)*θ(y)). Replacing (p.324) p by p n and letting n → ∞,we have θ(zf(y*y)) = 0. Thus zf(θ(y)*θ θ(zf(θ(y)*θ(y)) = 0. Thus zf (y*y) = 0. Since this is true all zY, by 8.1.4 (1) we have f(y*y) = 0. This is a contradiction.  □

We can rephrase part of 8.2.25 in the language of triple systems as follows:

Proposition 8.3.3

Suppose that Y is an equivalence A-B-bimodule, and that X is a closed A-B-submodule of Y. Then the quotient operator space Y/X is a triplw system. Indeed if q:Y → X is the cononical quotient map, then the triple product on Y/X is given by [q (x), q(y), q(w)] = q(xy*w), for x, y, z∈ Y. Thus q is a triple epimorhism.

If X is a closed linear subspace of a triple system Y, such that xy*z and zy*x are in X for all xX and y,zY, then we say that X is a triple ideal of Y. In this case, the quotient Y/X is a triple system, by 8.3.3.

Corollary 8.3.4

letθ:YZ be a triple morphism between TROs. Then:

  • (1) Ker(θ) is a triple ideal in Y.

  • (2) Ran(θ) is closed, and is a subtriple of Z.

  • (3) θ is a complete quotient map onto its range.

  • (4) The induced map θ̃:Y/Ker (θ) → Z is a completely isometric triple morphism onto Ran(θ).

Proof Item (1) is obvious. The induced map θ̃: Y/Ker(θ) → z is, by 8.3.3, a one-to-one triple morphism onto Ran (θ). By Lemma 8.3.2,θ̃ is completely isometric, which gives (2),(3), and (4).  □

Corollarly 8.3.5 (Hamana)

A linear map θ: YW between full c*-modules or TROs, is a triple morphism if and only ifθ is the 1-2-corner of a cornerpreserving *-homomorphismπ: ℒ(Y) → ℒ(W) between the Morita linking C*- algebras. In this caseis one-to-one(resp. Surjective)if and only if πis one-to-one (resp. surjective).

Proof The first ‘if’ is clear. For the converse, We may suppose that θ is a triple morphism between TROs. Define ρ ( k = 1 n x k y k * ) = k = 1 n θ ( x k ) θ ( y k ) * , for x 1,…,x n,y n,…,y nY. Now θ(Y) is a TRO in W, and so by 8.1.4(1) used twice, and by 8.3.4(3), we have

k θ ( x k ) θ ( y k ) * = sup { k θ ( x k ) θ ( y k ) * w   :   w Ball ( θ ( Y ) ) } = sup { k θ ( x k ) θ ( y k ) * θ ( z )   :   z   Ball ( Y ) } sup { k x k y k * z   :   z   Ball ( Y ) } = k x k y k * .
Hence ρ is well-defined, and extends to a contraction, which we still call ρ, from YY⋆ to WW⋆. It is easy to see that ρ is a *-homomorphism. Similarly one gets a (p.325) canonical *-homomorphism σ: Y*YW*W. It is now clear how to define the corner-preserving *-homomorphism π: ℒ(Y) → ℒ(W), whose 1-2-corner is θ.

It is clear that if π is one-to-one(resp. surjective) then so is θ. If θ is one-to-one, then it is isometric. In this case, the one inequality in the centered equations above is also an equality. It follows that ρ is isometric, hence one-to-one. Similarly, π is one-to-one. Since π is corner-preserving, it is now easy to see that π is also one-to-one. If θ is surjective, then ρ is surjective, since it has dense range in WW. Similarly σ, and hence also π, is surjective.

8.3.6 (Inner products on triple systems)

By 4.4.6, a completely, isometric linear isomorphism between two equivalence bimodules is a triple morphism, Thus, it is the 1-2-corner of a *-isomorphism π between the Morita linking C*-algebras, as in 8.3.5. The 1-1-and 2-2-corners of π are *-isomorphisms between the algebras acting on the given bimodules.

If Y is a triple system, then Y has a canonical equivalence bimodule structure. Namely, let A be the subspace of B(Y) densely spanned by the maps z ↦[x,y,z], and let B be the subspace densely spanned by the maps z↦[z,y,x], for x,yY.If Y is a TRO, then it is clear A is just the copy of YY⋆ in B(Y). That is,A = K YY(Y); and by 8.1.14, Y is a left c*-module over A. Hence it follows that if Y is a triple system, then Ais a C*-algebra in the product of B(Y), and Y is a left C*-module over A in a canonical fashion. Similarly, B is a C*-algebra with the reversed product of B(Y), and Y is a right C*-module over B. Clearly, Y is an equivalence A-B-bimodule. We will see in 8.4.2 that A is a *-subalgebra of A l(Y) in the notation of 4.5.7, and BA r(Y).

Next we charcterize the possible right C*l-module actions on a triple system which are compatible with the underlying operator space structure (via equation (8.6)). First we note that if Y is a full right C*-module over a C*-algebra C, and if σ is faithful *-homomorphism from C onto an ideal of a C*-algebra D, then we may make Yinto a right C*-module over D as follows. We define yd = yσ−1(d) and 〈〈y|z〉〉 = σ(〈y|z〉), for d∈σ(C), and y,zY. It is easy to check that Y is a right c*-module over σ(C), and hence by 8.1.4(4), over D. We denote this C*-module as Y σ.Morever, the oprator space struture on Y σ given by (8.6), is easily seen to coincide with the former one. We claim that every right C*-module action on Y over any C*-algebra D, which is compatible with the given operator space structure, arises in precisely this way, To see this, write Y′ for Y viewed as a C*-module over D. Thus the identity map I from Yto Y′ is a complete isometry, and a triple isomorphism. Hence by the first paragraph of 8.3.6, there is associated a*-isomorphism σ from C onto the ideal Iof D. It is easy algebra to check that Y′ = Y σ (using the fact that σ and I are corners of a *-isomorphism between the Morita linking C*-algebras (see 8.3.5)).

By the last two paragraphs, it follows that if a triple system Y is a right C*-module over a C*-algebra D, such that the norms from (8.6) equal the given matrix norms on Y, then this C*-module equals Y σ for a faithful *-homomorphism σ from B onto an ideal of D. Here B is as in the second last paragraph. This gives another ‘picture’ of the C*-modules over a C*-algebra D, as the triple systems (p.326) Y, together with a faithful *-homomorphism σ: BD as above.

We may now extend the notation introduced in 8.1.19 to triple systems. Thus products such as y 1 y 2 * , y 1 y 2 * y 3 y 4 * y 5 , and so on, make sense for elements (y i) in a triple system. Indeed y 1 y 2 * represents the operator z ↦ [y 1,y 2,z] in the C*-algebra A above, whereas y 1 y 2 * y 3 y 4 * y 5 = [ [ y 1 , y 2 , y 3 ] y 4 , y 5 ] Y . As in 8.1.19, any such expressions may be intepreted as products in the Morita linking C*-algebra of Y, where Y is regarded as an equivalence A-B-bimodule as above.

8.3.7 (Nondegenerate triple morphisms)

If θ:YB(K,H) is a triple morphism defined on a TROY Y, then we say that θ is nondegenerate if (i) θ(Y)K is dense in H, and (ii) θ(Y)*H is dense in K. Using the fact that Y is the norm closure of YYY, it is easy to see that (i) is equivalent to saying that the corresponding *-homomorphism from YY⋆ to B (H) given by 8.3.5,if nondegenerate. Also, (ii) is equivalent to saying that the corresponding *-homomorphism from YY to B(K) is nondegenerate. Also, (ii)is equivalent to saying that ∩yεY Ker(θ(y)) = {0}.

Thus (using 8.2.8 if necessary), we see that θ is nondegenerate if and only if the correspoding *-homomorphism from ℒ(Y) to B (HK) from 8.3.5 is nondegenerate. Note that this implies that if, further, θ is one-to-one, then we may view M(ℒ(Y)) ⊂B(HK), as Section 2.6. If θ is not nondegenerate then we may ‘cut it down’ to be nondegenerate, by replacing H with H′ = [θ(Y )K], and K with K′ = [θ(Y)*H].

8.3.8 (The noncommutative Shilov boundary)

We next discuss the noncommutative Shilov boundary, or ‘triple envelope’, of a nonunital opertor space. To a certaion extent this will parallel the development in Section 4.3, to which the reader may want to refer back to periodically.

Suppose that X is an operator space.If i: XY is a linear complete isometry into a triple system Y, such that Y is the smallest subtriple of Y containing i(x), then we say that (Y,i) is a triple extension of X. Notice that in this case, by the argument for (4.7),

Y = Span ¯ { i ( x 1 ) i ( x 2 ) * i ( x 3 ) i ( x 2 n ) * i ( x 2 n + 1 ) : n , x 1 , , x n X }
(notation as in 8.3.6). If Y is a full C*-madule or equivalence bimodule, then it is easy to see that this the same as saying that the copy of i(X) supported in the 1-2-corner of the Morita linking algebra ℒ(Y), generates ℒ(Y) as a C*-algebra. We say that two triple extensions (Y,i) and (Y′,i′) are X -equivalent if there exists a triple isomorphism θ:YY′ such that θ ∘ i =i .

We define a triple envelope or noncommutative Shilov boundary of X to be any triple extension (Y,i) with the universal property of the next theorem.

Theorem 8.3.9 (Hamana)

If X is an operator space, then there exists a triple extension(Y,i) of X with following universal property: Given any triple extension (Z,j) of X there exists (necessarily unique and surjective) triple morphism θ:ZY, such thatθ ∘ j =i.

(p.327) 8.3.10 (Remarks on the universal property)

Before we begin the proof, we make a series of important but simple remarks stemming from the universal property of the justification for the use of the term ‘noncommutative Shilov boundary’. We omit the proofs, which are almost identical to 4.3.2.

First, suppose that (Y,i) is a triple extension with the universal property of the theorem. Then there exists no triple ideal W of Y such that qwi is a completely isometry on X, where qw: YY/W is the canonical quotient triple morphism. This follows by applying the universal ptoperty with j = qwi. The second remark is that the set of triple extenstions (Y,i) satisfying the universal property of the theorem, is one entire equivalence class of the relation we called X-equivalence defined above 8.3.9. Third, if (Z,j) is any triple extension of X, and if θ:ZY is the triple epimorphism provided by the universal property, and if W = Ker (θ), then (Z/W,qwj) is clearly X-equivalent to (Y,i). Thus by the second remark,(Z,W,qwj) may be taken to be a triple envelope of X. Here again, qw: ZZ/W is the quotient morphism. Putting these remarks together we obtain our fourth remark, namely that the triple envelope of X may be taken to be any triple extension (Y,j) of X for which there exists no closed triple ideal W of Y such that q wj is completely isometric on X.

8.3.11 (Proof of Theorem 8.3.9)

For an operator space X we define 𝒯 (X) and J as in 4.4.7. Note that 𝒯(X) is simply the subtriple of I (X) generated by J(X).

suppose that (Z,j) is a triple extenstion of X. We may suppose, by 8.3.6, that Z is an equivalence C-D-bimodule. Let C 1 and D 1 be the unitizations of C and D respectively, and let ℒ1 (Z) be the ‘unitized linking C*-algebra’ of 8.1.17:

1 ( Z ) = [ C 1 Z Z D 1 ] .
Inside ℒ1(Z) there is a canonical copy 𝒮1 of the Paulsem 𝒮(X). Moreover, it is easy to see that 𝒮1 generates ℒ1(Z) as a C*-algrbra. Similary, consider the C*-algebra C e * ( S ( X ) ) generated by 𝒮(X) in I(𝒮(X)), using the notation of 4.4.2. This is a C*-envelope of 𝒮(X) (see the proof of 4.3.1). By 4.3.1, there exists a *-homomorphism π : 1 ( Z ) C e * ( S ( X ) ) , such that π extends the canonical map from 𝒮1 to 𝒮(X). By 2.6.15, we see that π takes each of the four corners in ℒ1(Z) to the matching corner in C e * ( S ( X ) ) . Let θ be the 1-2-corner of π. As in 8.3.5, θ is a triple morphism, and it is surjective since π is surjective.

We will henceforth write (𝒯(X),J) for any triple envelope of X.

8.3.12 (Propeties of the triple envelope)

(1) As one would expect, if Z is a triple system, then Z is a triple envelope of itself. Indeed, applying the universal property of the theorem to the identity map j: ZZ, we obtain a triple epimorphism θ: Z → 𝒯 (Z) with θ ∘ j = i. Thus θ = i, and so θ is a triple isomorphism.

(2) If u:X 1X 2 is a surjective linear comopletely isometry between operator spaces, then one may ‘extend’ u to a triple isomorphism between any triple (p.328) envelopes (𝒯(X 1), J 1) and (𝒯(X 2),J 2). Indeed a routine ‘diagram chase’ shows that (𝒯(X 2),J 2u) is a triple envelope for X 1.

(3) The triple envelope shares many of the properties of the injective envelope that we met in Chapter 4. For example, any triple envelope of X is both a rigid and an eseential extension of X, in the sense of 4.2.3. The proof is the same as that of 4.3.6.

(4) Another useful fact is that 𝒯(M mn(X)) ≅ M mn(𝒯(X)) completely isometrically isomorphically (or equivalently, by 4.4.6 and 8.3.2, triple isomorphically) for any operator space X, and for m, n ∈ N. More generally it is true that T(K I,J(X)) ≅ K IJ(T(X)) for arbitrary cardinals I, J. One may deduce such relations from the analoguous assertion for the injective envelope (see 4.2.10 or 4.6.12), and the definition of the triple envelope given in the proof of 8.3.9. Similarly, although we shall not need this, T ( i X i ) i T ( X i ) triple isomorphically. Another way to prove such relations is to use fourth remark in 8.3.10 (see Appendix A in [53] for details).

(5) If A is a unital operator space, or an approximately unital operator algebra, then one can show that any C*-envelope of A is a triple envelope of A. Most of this was observed at the end of 4.4.7. For the rest, see [53].

As a sample application, we give another proof of an earlier result (see 4.5.13):

Corollary 8.3.13 (A Banach-Stone theorem)

If A and B are unital operator algebras, and if v: AB is a linear surjective complete isometry, then there exists a unital completely isometric isomorophism π from A onto B, and a unitary U ∈ Δ (B), such that v = Uπ(·).

Proof By (5) above, we may take T ( A ) = C e * ( A ) , and similarly for B. By 8.3.12 (2), one may ‘extend’ v to a triple isomorophism from C e * ( A ) to C e * ( B ) , which we still write as v. If V(1) =U and v(a) =1, then

U * U = ( a ) υ ( 1 ) * υ ( 1 ) = υ ( a ) υ ( 1 ) * υ ( 1 ) = υ ( a ) = 1.
Similarly UU* =1, so that U is a unitary in C e * ( B ) . Then π U 2212v(·) satosfies the hypotheses of 1.3.10, and hence it is a *-isomorphism. Thus
U 1 υ ( a 2 ) = π ( a 2 ) = π ( a ) 2 = U 1 υ ( a ) U 1 υ ( a ) = U 2 .
Hence U* =U 2212 = v (a 2) ∈ B, and so U ∈ Δ (B). Thus the restriction of π to A is a unital completely isometric homomorphism onto B. □

8.3.14 (The Shilov inner product)

It is often convenient to take the triple envelope 𝒯(X) of X to be an equivalence bimodule. In this case we call the restriction to X of the associatecd C*-module inner products on 𝒯(X), the Shilov inner products on X. We have already met this concept in 4.4.8.

8.4 C*-MODULE MAPS AND OPERATOR SPACE MULTIPLIERS

In the previous section, we saw that the noncommutative Shilov boundary of an operator space may be viewed an a C*-module. This is pleasant, since then we (p.329) may hope to apply C*-module methods directly to the study of operator spaces. For such applications, the multiplier algebras of an operator space considered in Chapter 4, are often useful, since they are intimately connected with C*-module theory, in several ways. The reader might turn to Section 4.5 for the definitions of ℳl(X) and 𝒜l(X). We now describe these algebras in C*-module terms.

8.4.1 (Multipliers and the triple envelope)

For any operator sapce X, let (𝒯(X),J) be a triple envelope of X, which we may take to be a right C*-module over a C*-algebra ℱ. The space of bounded right module maps on 𝒯(X) is an operator space, as we mentioned after 8.2.2. We temporarily write LM(X) for the subspace consisting of those module maps leaving J(X) invariant. We will now show that ℳl(X) ≅LM(X) as operator algebras. That is,

(8.16)
l ( X ) { a C B F ( T ( X ) ) : a J ( X ) J ( X ) }
completely isometrically isomorphically. If aLM(X), then since (iv) implies (i) in 4.5.2,a |X is a left multiplier of X. (A direct proof of this may also be given, using the last paragraph in 8.1.19.) Indeed by 4.5.2, the map aa |X gives the isometric isomorphism in (8.16). Now M n(CB (𝒯(X))≅B (C n(𝒯(X))), by 8.2.3 (6). Thus if a = [a ij] ∈ M n(CB (𝒯(X))), then a may be regared as a module map on C n(𝒯(X)). By 8.3.12 (4), C n is a triple envelope of C n(X). We deduce from the isometric case of (8.16), that
l ( C n ( X ) ) { a C B F ( C n ( T ( X ) ) ) : a J n , 1 ( X ) J n , 1 ( X ) } M n ( L M ( X ) )
isometrically, By (4.11), we now have (8.16) completely isometrically.

We next show that

(8.17)
A l ( X ) { a B ( T ( X ) ) : a J ( X J ( X ) J ( X ) , a * J ( X ) J ( X ) ) }
*-isomorphically. This follows from (8.16), and the fact that each side of (8.17) is just the diagonal algebra of the matching sides of (8.16) (see 4.5.7 and 8.2.4). Indeed, the diagonal algebra may be defined to be the span of the Hermitian elements. The Hermitian (in this case, selfadjoint) elements of the algebra LM(X) above, clearly are in the set on the right side of (8.17). Conversely, if a is in the latter set, then writing a = a + a * 2 + i a a * 2 i , we see that a∈ Δ (LM(X)).

By 8.2.4 and the last paragraph of 8.1.19, we also may regrad (as subalgebras)

A l ( X ) M l ( X ) K ( T ( X ) ) * * ( T ( X ) ) * * .

Corollary 8.4.2

Suppose that Y is a right C*-module over a C*-algebra B.Then ℳ l (Y) = CB B(Y) and 𝒜 l (Y) =B B(Y). Similar assertions hold for the right multiplier algebras, indeed ℳ r(Y) ≅RM(I) and 𝒜r(Y) ≅ M(ℐ), where I is as in 8.1.4.

Proof The first two assertions are immediate from the proofs of (8.16) and (8.17) above, and using 8.3.12(1). The second two also 8.2.4 and the ‘otherhanded version’ of facts from 8.1.14–8.1.16.  □

(p.330) Corollary 8.4.3

Suppose that A is a C*-algebra. Any nondegenerate left operator A-module X is a closed A-submodule, of a B-rigged Z (see A-submodule of a B-rigged A-module Z is a left A-module.

Proof If X is a left operator A-module, then by 4.6.2 (2) there is an associated *-homomorphism θ: A → 𝒜l(X). Using (8.17), θ may be viewed as a *-homomorphism into B (𝒯(X)). By (4.7) we have θ(e t)(Z) → Z, for ZZ. That is, 𝒯(X) is an ℱ-rigged A-module. Clearly X is an A-submodule of 𝒯(X).

Conversely, by 8.2.10 any B-rigged A-module Z is a nondegenerate left operator A-module; and therefore so is any A-submodule.

The following adds to the picture of adjointable multipliers on operator spaces that we began in 4.5.8:

Theorem 8.4.4

Let X be an operator space. For a map T: XX, the following are equivalent:

  • (i) 𝒯 ∈ 𝒜l(X).

  • (ii) There exists a linear complete isometry σ from X into a C*-algebra, and a map R: XX, such that σ(T(x))* σ(y) = σ(x)* σ(R(y)), for x,yX.

  • (iii) There exists a map R: XX, such that

T ( x ) | y = x | R ( y ) , x , y X .
The inner product in (iii) is a Shilov inner product for X (see 8.3.14).

Proof That (i) implies (ii) follows from 4.5.8(1), taking R= σ2212 (S*σ(·)).

Suppose that T satisfies (ii), and that (𝒯(X),J) is a triple envelope of X, with 𝒯(X) a full C*-module over a C*-algebra F. The subtriple Y generated by σ(X) is a triple extension of X in the sense of 8.3.8. By 8.3.9, there exists a triple epimorphism θ: Y → 𝒯(X) such the θ ∘ σ = J. By the proof of 8.3.5 there is an associated *-homomorphism from Y*Y to F, taking z 1 * z 2 to 〈θ(z 1) θ (z 2)〉, for z 1, z 2Y. Applying this *-homomorphism to the equation in (ii), we obtain the equation in (iii).

To see that (iii) implies (i), we define B l(X) to be the set of maps 𝒯 on X satisfying (iii). Set T#x002A; = R, where R is as in (iii). It is easy to cheek, just as in 8.1.7, that B l(X) is a closed subalgebra of B(X), and that * is an isometric involution on B l(X) satisfying the C*-identity. Thuy B l(X) is a C*-algebra.

Next we note that 𝒜l(X) is the range in B(X) of the isomorphism in (8.17). Explicitly, this map takes an a in the set on the right of (8.17), to the operator T a = J −1(aJ(·)) on X. We have T a∈ B l(X). Indeed a proof similar to the proof that (i) implies (ii) above, shows that aT a is a *-homomorphism into B l(X). We therefore will be done if we can show that the range of this faithful *-homomorphism equals B l(X). To do this, it suffices to show that if U is a unitary in B l(X), then U is the range of the map above (since any unital C*-algebra is spanned by its unitary elements). For such U, we have (p.331)

U x | U y = U * U x | y = x | y , x , y X .
Now 𝒯(X) = Xℱ, by the centered equation in 8.3.8. Define a map taking the element y = k = 1 n x k b k , to k = 1 n U ( x k ) b k , for x 1 , , x n X , b 1 , , b n . To see that this map is well defined and bounded, we compute:
k = 1 n U ( x k ) b k | k = 1 n U ( x k ) b k = i , j = 1 n b i * x i | x j b j = y | y .
Hence the map above extends to a map a ∈ Bℱ(𝒯(X)). Clearly Ta = U. □

The C*-algebra B l(X) in the last proof is another useful description of Al(X). It also shows why the name adjoinatable is approprite for these maps.

8.4.5 (Comparisons with C*-module maps)

There are very many quite striking parallels between multipliers (resp. adjointable multipliers) on operator spaces, and bounded module maps (resp. adjointable maps) on C*-modules. We have seen some of these already. We mention a few more: for example, from 4.7.4 and 4.7.1 we know that for a dual opreator space X,𝒜l(X) is a W* -algebra, and any u ∈ 𝒜l(X) is w* -continuous. In the next section we shall see that such results play an important roal for W* -mouldes. We shall also briefly mention there some conncetions with the one-sided M-ideals of Section 4.8. Indeed, a good deal of the results in the noncommutative M-ideal theory follow by applying C* module techniques. Many more such parallels may be found discussed in [56,73].

In 8.1.10 we remarked that a linear map u on a right C*-module Y is a contractive module map if and only if the map uIY on C 2(Y) is contractive. This of course is the analogue for C* -modules of condition 4.5.2 (ii), which characterizes opreator space multipliers. Beginning from this fact from 8.1.10, and using basic facts about C* -modules met early in this chapter, one may give another development of the theory of operator space multipliers that we saw in Chapter 4, but avoiding many of the technical details about the injective envelope used in Section 4.5. See [56] for details. such an approach is close to the orginal development of the operator space multiplier theory (see[53]). For example, we give a quick proof that (ii) implies (iv) in Theorem 4.5.2. Condition (ii) there says that uIX: C 2(X) → C 2(X) is compleetely contractive. By 1.2.11, I(X) is ‘completely complemented’ in B(H for some H. Thus there is completely contractive 2 -module map projection C 2(B(H))≅B(H, H (2))→ C 2(I(X). Hence, by 3.6.2, we can extend uIx to a comletely contractive 2 -module map c 2(I(X))→c 2(I(x)) ⊂ C 2(B(H)). Such a map is necessarily of the form ũ⊕υ. By the ‘rigidity’ property of Ix),υ = IX. Since I(Xis a C*-module,by the reult at start of this paragraph, ũ is a contractive module map on I(X). BY(4.7),it restricts to a contractive module map on 𝒯(X).

8.5 W*-MODULES

The theory of W*-modules may be thought of as a ‘dual varint‘ of the theory of C*-modules. Indeed our development in this section is parallel, to some ex (p.332) tent, to the pattern of Sections 8.1 and 8.2 above. However, we shall see that W*-modules are quite a bet simpler than C*-modules. For example, W*-modules behave much more like Hilbert spaces, and there is a very powerful ‘stabel isomorphism theorem’ (8.5.28 below) W*-modules, which is very useful for operator space applications.

Throughout this section, M and N are W*-algebras.

8.5.1 (The definitions)

We say that a right C*-module Y over a C*-algebra A is selfdual if every bounded A-moduoe map u: YA is of the form u(·) = 〈z|·〉, for some zy. We say that y is a right W*-module if y is a selfdual right C*-module over a W*-algebra.

If Y is a right C*-module over a W*-algebra. M, then we will consistently write ℐw for the w*-closure in M of the span of the range of the M-valued inner product on y (recall from 8.1.4 that ℐ is the norm closure of this span). It is easy to see, using simple w*-closed ideal in M. We say that y is w*-full if ℐw = M.

In the following remarks, y is a selfdual right c*-module over A.

  • (1) It follows for exmple form 8.1.11 (2) and 8.2.15,that

    (8.18)
    C B A ( Y , A ) Y ¯ completely isometrically .

  • (2) If Z is another right c*-module over A, then

    (8.19)
    B A ( Y , Z ) = B A ( Y , Z ) , and B A ( Y ) = B A ( Y )
    . Indeed, the fact that any uBA(Y, z) is adjointable follows by considering the A-valued map 〈z|u(·)〉, for fixed zZ.

  • (3) The adjoint c*-module Ȳ (see 8.1.1) is a selfdual left C*-module over A. We leave the details as an exercdise.

  • (4) Selfduality is an operator space invariant. That is, if Z is another C*-module over a possibly different C*-algebra B, say, and if Y and Z are linearly completely isometric, then Z is selfdual as B-module. To see this, one first uses the next result to see that we can assume that Y and Zare both full. By 4.4.6, Y and Z are triple isomorphic. The rest is a pleasant algebraic exercise.

Lemma 8.5.2

Let be a right C*-module over a C*-algebra A, and letbe as in 8.1.4. Then BA(Y, A) = BC(Y, D) as sets, for any C ∈{A, A 1,M(A,ℐ} and D ∈{A, A 1, M(A),ℐ,M(ℐ)}. Hence y is selfdual as an A-module if and only if y is selfdual as a D-module, for any D ∈{A 1, M(A),ℐ,M(ℐ)}.

Proof Suppose that uB(Y, D)By cohen's theorem A.6.2, we may write any yY as y = y 1 a 1 a 2 for a 1, a 2 ∈ ℐ. Hence u(y = u(y 1)a 1 a 2Aℐ⊂ ℐ. Moreover, if cC then u(yc = u(y 1 a 1(a 2 c)) = u(y 1)a 1(a 2 c) = u(y)c. Thus

B I ( Y , D ) B C ( Y , I ) B I ( Y , I ) B I ( Y , D ) .

Thus BC(Y, D) = B (Y, D) = B Y,ℐ), for any C, D as above.

The last assertion follwos immeditely from the first. □

(p.333) Proposition 8.5.3

Suppose that Y is a right C*-module over a W*-algebra M. Thenw is the multipulier algebra of ℐ.

Proof Clearly ℐ a w*-dense ideal in ℐw. Fix a faithful unital w*-continuous *-repersentation π of ℐw on H, say. This is easily seen, using 2.1.9, to be a nondegenerate *-repersentation of ℐ. By section 2.6, we may identify M(ℐ) with {TB(H):Tπ(ℐ) ⊂ π(ℐ),π(ℐ)T ⊂ π(ℐ)}. In fact the latter algebra clearly cotains π(ℐw), and is a subalgebra of π(ℐ)″ by 2.6.5. On the other hand,

π ( I ) = π ( I ) ¯ w * π ( I w )
by the double comutant theroem. Thus π(ℐw) = M(ℐ). □

Lemma 8.5.4

Let Y be a right C*-module over a W*-algebra M. Then:

  • (1) Y is a W*-module over M if and only if Y has a Banach space predual with respect to which the inner product on Y is separately w*-continuous.

If Y is a W*-module, then:

  • (2) Y has a unique Banach space predual with respect to which the inner product on Y is separately w*-continuous.

  • (3) With respect to the w*-topology induced by the predule in (2), a bounded net (xt)t converges to x in Y if and only ify|x t〉 → 〈y|xin the w*-topology of M, for all yY.

  • (4) Let W = M * ⊗̑M Ȳ (see section 3.4). Then W is an operator space predual of Y inducing the w*-topology in (2) and (3) above.

Proof We reamrk that (4) and its proof, and the proof of (2), are simpler if one replaces M * ⊗̑M Ȳ by the (quite analoguous, but much simpler) Banach module version of the projective tensor product. To understand the argument better, the reader may want to consider this variant first.

If Y is a W*-module then by 3.5.10 and the other-handed version of (8.18),

( M * ^ M Y ¯ ) * M C B ( Y ¯ , M ) Y
completely isometrically. Unraveling these isomorphisms yields a completely isometric surjection ρ: YW*, given by
ρ ( x ) ( ψ y ¯ ) = ψ ( y | x ) , ψ M · , x , y Y .
By the norm density of the finite rank tensore from M *Ȳ in W, it is clear that with respect to the w*-topology on y induced by W, the ‘if and only if’ condition in (3) holds. Thus by A.2.5, the inner product is separately w*-continuous with respect to this topology.

Define θ: WY* by θ(ψ ⊗ ȳ)(x) = ψ(〈y|x〉), for ψ ∈ M * and x, yY. It is easy to check that θ = (ρ*)|W. Suppose that y *is a predual of Y, regarded as a subspace of Y*, for which the inner product on Y is separately w*-continuous. (p.334) Then θ maps into Y *. Viewing θas map into Y *, θ* corresponds to the map ρ above. Hence θ is a (completely isometric surjection onto y *, and ρ is a homeomorphism for the w*-topologies. We have proved (2)–(4).

The ‘only if ’ in (1) follows from the proof of (4). On the other hand,suppose that y has a Banach space predual, and that the M-valued inner product is separately w*-continuous. Let u:YM be a bounded M-module map. By 8.1.23, we may choose a cai (et)t for K M(Y), with terms of the form k = 1 n | x k x k | for some xkX. For xX, we have

(8.20)
u ( e t ( x ) ) = k = 1 n u ( x k ) x k | x = k = 1 n x k u ( x k ) * | x = w t | x ,
where w t = k = 1 n x k u ( x k ) * (Which depends on t). It follows from 8.1.11 (2) that ‖wt‖ = ‖uet‖≤ ‖u‖. Thus (wt)t is a bouded net with Y, and so it has a w*-convergent subnet, with limit w say. Replace the net with the subnet. By the hypothesis, 〈w|x〉 → 〈w|x〉. Since u(et(x))→u(x)in norm, by (8.20)we have u(x = 〈w|x, for all xY. Thus Y is selfdual over M.

We will henceforth use the phrase the w*-topologyof a W*-module Y, for the (unique) topology in (2)–(4) above.

Corllary 8.5.5

Suppose that Y is a right W* -module over M. Then:

  1. (1) BM(Y) = B M(y), and this is a W*-algebra.

    1. (2) A bounded net (Ti)i in B M(Y) connverges in the w* -topology to TBM(Y) if and only if Ti(y) → T(y) in the w*-topology of Y, for all yY. Indeed, Y ⊗̑M W is a predual for BM(Y), where W is as in 8.5.4(4).

Proof BY 3.5.10, 8.5.4(4), and (8.19), we have

( Y M W ) * = ˜ C B M ( Y , W * ) = C B M ( Y ) = B M ( Y ) .
The latter space is a C* -algebra, and it is therefore a W*-algebra by the theorem of Sakai mentioned at the start of Section 2.7. The assertion involving net convergence is proved similarly to the analoguous statements in 8.5.4(4).

There is a stronger variant of the following, which assumes only a Banach space predule. the version here will suffice for most operator space applications. Our proof showcases the mulitipliers from Chapeter 4, and has the advantage of generalizing to the nonselfadjoint algebra sitution. see the Notes for details.

Theorem 8.5.6 (Zettel, Effros–Ozawa–Ruan)

Let Y be a full right C*-module over a C*-algebra A, and suppose that Y has as operator sapce predual. If M is M(A) then M and B A(Y) are W*-algebras, and y is a w*-full W*-module over M. Moreover, Y has a unique operator sapce predual, the space in 8.5.4(4).

Proof Let Y * be a fixed operator space predual of Y. We will use the fact from 8.4.2 that 𝒜l(Y) = B M(Y), and 𝒜r(Y) = M(A) = M. By 4.7.4 we know that (p.335) 𝒜l(Y) and 𝒜r(Y) are W*-algebras, and hence so are B M(Y) and M. By 4.7.5, the canonical trilinear map B M(Y) × Y × MY is separately w*-continuous.

We will now check that the inner product on Y is separately w*-continuous, with respect to the w*-topology determined by Y *. To this end, suppose that we have a bounded net (yt)t converging to y in the w*-topology of Y. Fix x, w in Y. By the above, the ‘rank one’ operator |w〉〈x| is w*-continuous on Y. Hence wx|yt〉 → wx|y〉 in the w*-topology. Let ( x y t μ ) be a w*-convergent subnet of the bounded net (〈x|yt〉), converging to bM say. By the last paragraph, we have that w x y t μ w b in the w*-topology. Hence wb = wx|y〉. Since this is true for all wY, it follows from 8.1.4 (1) that b = 〈x|y〉. Hence 〈x|y t〉 → 〈x|y〉 in the w*-topology. By A.2.5(2), the inner product is separately w*-continuous. By 8.5.4 and 8.5.3, Y is a w*-full W*-module over M and we have the other consequences stated in 8.5.4 and 8.5.5. The uniqueness follows easily from 8.5.4 (2), and from basic oprator space dulity principles (see Section 1.4). □

Corollary 8.5.7

Let Y be a right C*-module over a W*-algebra M. Then Y is a W*-module if and only Y has an oprator space predual. In this case the oprator sapce predual is unique.

Proof The ‘only if’ follows from 8.5.4 (4). For the other direction, by 8.5.3, M(ℐ) = ℐw. Thus if Y has as operator space predual, then, by 8.5.6, Y is a W* -module over ℐw. By 8.5.2, Y is also selfdual over M. The uniqueness was proved in 8.5.6.

Corollary 8.5.8

A bounded module map u: YZ between W*-modules over M, is w*-continuous.

Proof BY (8.19), u is adjointable. We will apply A.2.5 (2). If (yt)t is a bounded net converging to yY in the w*-topology of Y then

u ( y t ) | z = y t | u * ( z ) y t | u * ( z ) = u ( y ) | z , z Y .
By 8.5.4 (3), u(yt) → u(y) in the w*-topology. Thus u is w*-continuouse.

We separate one other intersting fact, which follows esily from 8.2.3 (1), and the first paragraph of the proof of 8.5.6, for example.

Corolly 8.5.9

A right W*-module Y over a W*-algebra M is a normal dual operator B M(Y)-M-bimodule in the sense of 3.8.2.

8.5.10 (The linking W*-algebra)

If Y is a right W*-module over M, then we define the linking W*-algebra of Y to be ℒw(Y) = BM(Yc M). This equals B M(Yc M) by (8.19), since Yc M is a selfdual M-module, as is easily verified. As in 8.1.17, by considering the adjointable inclusion and projection maps between Yc M and its two summands, it is clear that ℒw(Y) may be viewed as a 2×2 matrix algebra with corners B M(Y), Y, Ȳ and M. Thus any W*-module is a corner eN(1–e), for a W*-algebra N and a projection eN. The converse is also true, namely that if e is a projection in a W*-algebra N, then Y = eN(1‐e) (p.336) is a right W*-module over (1‐e)N(1‐e). This may be seen by using 8.5.4 if necessary. Thus we obtain another ‘plcture’ of W*-module as the corners of W*-algebra. This should be compared with 8.1.19.

The linking W*-algebra of a W*-module is very useful when it comes to calculations, because the inner product and module action have been replaced by multiplication in the W*-algebra, just as we saw in 8.1.19. As one illustration of this princepal, we invite the reader to ckeck that the proofs in 8.2.24 and 8.2.25 may be adapted in an obvious way to give the analoguouse results for w*-closed ideals and quotients. In fact the W*-module version of these results are simpler, due to the well-known correspondence between two-sided w*-closed ideals in a W*-algebra and central projections.

8.5.11 (W*-closed TROs)

Via the linking W*-algrbra, one may now view W*-closed subtriples of B(K,H), or of a W*-algebra. Indeed, as we just saw, we may write any W*-module Y over M, as a corner eN(1‐e) of a W*-algebra N with M≅(1‐e)N(1‐e). Suppose that N has been respresented faithfully as a von Neumann subalgebra of B(H) say. Then the projection e determiners a splitting H = H 1H 2 say, and it is evdent that M corresponds to a von Neumann algebra in B(H 2), and Y corresponds to a w*-closed subtriple, and an M-submodule, of B(H 2,H 1). Another useful way to represent Y, is to suppose that M is a von Neumann algebra in B(K), and to consider the isometry Φ:YB(k,YM k)from 8.2.14 satisfying Φ(Y)*Φ(Z)=〈Y|〉, for y,ZY;with Y unitarily isomorphic to the TRO Φ (Y). We leave it as an exercise that Φ is a w*-homeomorphism on to Φ(Y),which is w*-closed. (Hint: use A 2.5 and 8.5.4(3),and simple net arguments of the kind found in 8.5.36,for example.)

Conversely,using 8.5.4, for example, it is easy to check that any w*-closed subtriple Z of b(K,H),or of a W*-algebra, is a w*-equivalence bimodule in the sense described next:

8.5.12 (Weak Morita equivalence)

W*-equivalence M-N-bimodules aredefined analoguously to the equivalence bimodules in 8.1.2,with the words ‘C*-module’ replaced by ‘W*-module’, and ‘full’ by ‘w*-full’. If there exists such a bimodule ’over M and N, then we say that M and N are weakly Morita equivalent. It is not hard to see that weak Morita equivalence is an equivalence relation coarser than *-isomorphism of W*-algebras (see the Notes to this to section).

Corollary 8.5.13

If Y is a W*-equivalence M-N-bimodule, thaen MBN(Y)*isomorphically.

Proof By the ‘left-handed’ version of 8.5.3, and by 8.1.15,MM(K N(Y)).

Now the result follows by 8.1.16(3), and (8.19)   □

8.5.14 (W*-modules are W**equivalence bimodules)

Analoguously to 8.1.14, any right W*-module Y over a W*-algebra Nis a w*-full left W*-module over (B N y). That it is selfdual as a left module follows, for example, from 8.5.7.To see that it is w*-full suppose that p is tha support projection in (B N(y)for the (p.337) w*-closed ideal generated by the ‘rank one’ operators. For all x, y, zY, we have 0 = (1–p)|y〉 〈|(x) = (1–p)yz|x〉. By 8.1.4 (2), 1–p = 0, so that p = 1.

Consequently, if Y is a w*-full right W*-module over N, then Y is a W*-equivalence BN(Y)-N-bimodule. More generally, if Y is a right W*-module over Nthen Yis a W*-equivalence BN(y)-ℐw-bimodule, where ℐwis as above.

Conversely, if Y is a W*-equivalence M-N-bimodule, then by 8.5.3 and the proof of 8.5.13, we have MM(K N(Y)) and NM(ℐ). Since Yis an equivalence K N(Y)-ℐ-bimodule (see 8.1.18),it follows frows from 8.1.20 that Mand N have isomorphic centers. Also, as in 8.1.18, we have that if Yis a w*-full right W*-module over N, then N and ℒw(Y) are weakly Morita equivalent.

8.5.15 ( W*-summands )

If Y is a right W*-module over M then by 3.8.11, C I w ( Y ) is a dual operator M-module for any cardinal I. It is easy to see that C I w ( Y ) is also a W*-module. For example, if Y is represrented, as in 8.5.11, both as an M-submodule, and as a w*-closed subtriple, of B(k,H), with M acting as a von Neumann subalgebra of B(K), then (if necessary, by simple arguments of the kind in 3.8.103.8.12) C I w ( Y ) may be identified with an M-submodule, and w*-closed subtriple, W of B(K,H (I)). By 8.5.4 (1), W is a W*-module over M, with inner product S×TS*T, It is then easy to see that the corresponding inner product on C I w ( Y ) is givem by

( y i ) | ( z i ) = i I y i | z i , ( y i ) , ( z i ) C I w ( Y ) ,
where the convergence of the sum is in the w*-topology of M. The columns with a finite number of nonzero entries are w*-dense in C I w ( Y ) . Indeed if PJ is the pronection from C I w ( Y ) onto the set of ‘columns supported on CJ(Y’, for a finite subset J of I, then (PJ)J converges in the w*-topology of B M ( C I w ( Y ) ) to the identity map on C I w ( Y ) , by 8.5.5 (2)and 1.6.3 From this it is not hard to show that B M ( C I w ( Y ) ) M I ( B M ( Y ) ) as W*-algebras. Since we shall not use this fact, we omit the proof. In [73], the interested reader will find the prorf of an operator space generalization of this fact. Namely, A l ( C I w ( Y ) ) M I ( A l ( Y ) ) as W*-algebras, for any dual operator space X.

Similar assertions hold for R I w ( X ) , for a left W*module X over M

We say that a submodule Y of a W*-module Z is w*-orthogonally complemented in Z, if Y is orthogonally complemented in the sense of 8.1.9,and if Y is a w*-closed subspace of Z In this case Y is a W*module too, by 8.5.4.

Lemma 8.5.16

Suppose that Z is a right W*-module over a W*algbra M,and that Y is a subspace of Z. The following are equivalent:

  • (i) Y is an orthogonally complemented M-submodule of Z,

  • (ii) Y is a w*orthogonally coplemented M-submodule of Z,

  • (iii) Y is a w*-closed M-submodule of Z,

  • (iv) Y is a right M-summand of Z, in the sense of 4.8.1.

(p.338) Proof Clearly (ii) implies (i) and (iii). If p is a projection in B M(Y), then p is w*-contious by 8.5.8. Thus Ran (p) is w*-closed. Therefore (i) implies (ii). By definition, and by 4.5.15 (iii) and 8.4.2, the right M-summands of Z are precisely the ranges of the adjointable projections on Z. Thus (iv) is equivalent to (i). Finally, given (iii), it follows from 8.5.4 (1) that Y is selfdual as an M-module. Hence the inclusion map form Y into Z is adjointable by (8.19). Its adjoint is easily checked to be an orhogomal projection onto Y, giving (i).

Proposition 8.5.17

If Y is a right C*-module over C*-algebra B, then Y** Y** with its cannical B**-module action (see 3.8.9), is a W*-module over B**, IF Y is an equvalence A-B-bimodule, then Y** is a W*-equivalence bimodule over A** and B**.

proof First suppose that Y is full over B. We recall that Ypℒ(1−p), as in (8.3). Here ℒ is the linking C*-algebra of Y, and p is a projection in M(ℒ). BY replacing Y with pℒ(1–p), we may assume that Y is a subtriple of ℒ, and that B = Y*Y ⊂ ℒ. Let Z be the w*-colsure of Y in pℒ**(1–P). Clearly, Z = pℒ**(1–p), a w*-closed subtriple of ℒ**. By A.2.3, Y** ≅ Z completely isometrically, and w*-homeomorphically. Similarly, if N = (1–p) ℒ**(1–p). then B** ≅ N, as W*-algebras. We have B = Y*YZ*ZN. Thus the w*-closure of Z*Z contians the w*-closure of B in ℒ**, namely N. Thus the w*-closure of Z*Z equals N. By 8.5.11, Z is a w*-full right W*-module over N. We may transfer these structures, to make Y**. a w*-full right W*-module over B**. This B** -module action on Y** coincides with the canoncal second dula action from 3.8.9. This is because the product map Z×NZ is separately w*-continuous, and extends the canoical map Y×BB. A similar argument yields the last assertion of Proposition.

If Y not full over B then we consider the ideal ℐ from 8.1.4. BY the above, Y** is a righy W*-module over ℐ**. However ℐ** is an ideal in B**, and so y** is a W*-module over B**, by 8.5.2 and 8.1.4 (4). □

8.5.18 (Dual triple systems)

Approproate weak*versions of the theory of TROs and triple systems presented in the first half of Section 8.3, also go through without difficulty. In fact the theory becomes simpler, due to the correspondence between w*-closed two-sided ideals in a W*-algebra and central projections.

By 8.5.6 and the remarks above it, a TRO Ywhich has a predual, is a w*-full W*-module over M(Y*Y). Putting this together with 8.5.11 and 8.5.14, we see that such ‘dual TROs’; the triple systems which have a predual; W*-modules;amd W*-equivalence bmodules, are essentially the thing, in a sense similar to the discussion in 8.3.1.

A w*-continuous triple morphism u: YZ between w*-closed TROs, has range which is a w*-closed TRO. Indeed, let W = ker(u), which is a w*-closed triple ideal in Y. BY the last several lines of 8.5.10, and the argumet for 8.3.3, there is a central projecction e such that W = eY. Let W′ = (1–e)Y, then uW,is a one-to one w*-continuous triple morphism with Ran (u) = Ram(W). By 8.3.2, uW is isometric, so that by A.2.5 its rage os w*-closed.

(p.339) If Y is a TRO, then Y** is a TRO and a W*-equivalence bimodule, as in the proof of the last Proposition, for example. Note that if u: YB(K,H) is a triple morphism, then by routine w*-approximation arguments, the canonical w*-continuouse extension ñ:Y**→B(K,H) (see A.2.2) is a triple morphism too. As one application of this, we deduce that any subtriple Y of B(K,H) satisfies a ‘Kaplansky density theorem’, namely that Ball (Y) is w*-dence in the unit ball of the w*-closure of Y. This follows from the above, 8.3.4 (3), and A.5.10.

A completely isometric surjective linear map (or equivalently, a surjective triple isomorhism) between dule TROs, is automatically W*-cotinuous. This follows from the uniqueness of the predual of a W* -module (see 8.5.7).

Theroem 8.5.19 The right M -ideals (see 4.8.1) in a right Hilbert C*-module are exactly closed right submodules.

Proof Let Z be a right C*-module over a C*-algebra B. If Y is a right M-ideal of Z, then the w*-clsure W of Y in Z** is a right M-summand of Z**. Viewed as subsets of Z**, we have

Y B ( W B * * ) Z Y Z = Y ,
using A.2.3(4). Thus Y is a B-submodule of Z.

Conversely, if Y is a B-submodule of Z, then its w*-closure W in Z**, is a B**-submodule. Indeed this follows from the fact that the B**-module action on Z** is separately w*-continuous, and A.2.1. By 8.5.16, W is a right M-simmand of Z**, so that Y is a right M-ideal of Z.

8.5.20 (C*-module and M-ideals)

In fact one may view the theory of one-sided M -ideals in oprator sapce, introdiced briefly in Section 4.8, as a genaralization of the behaviour of submodules of C* -modules. See [56,57,73] for details.

In connection with the last result, we note that the calssical M-ideals (see 4.8.1) in an equivalence A-B-bimodule are exactly the A-B-submodules. One direction of this is not hard. For example, if Y is an A-B-submodule of an equivalence A-B-bimodule Z, then by 8.5.19, Y is both a left and a right M-ideal of Z. By 4.8.4, Y is a complete M-ideal, and heance an Mideal. The reverse direction seems to be harder. Suppose that Y is an M-ideal in Z. If aA sa, consider the map TZ = az, for zZ. It is easy to see that T ∈ Her(B(Z)) (see A.4.2), and so by [195,Corollary I.1.25], aY < Y. Thus Y is a left A-submodule, and similarly it is a right B-submodule. Hence the M-idelas in a TRO Y are the (YY*)-(Y*Y)-submodules.

8.5.21 (Partial isometries in C*-module)

We say that an an element u in a right C*-module Y over M is a partial isometry if p = 〈u|u〉 is an orthogonal projection in M. This element p is called the initial projection of u. Note that it follws that up = u (since〈u-up|u-up〉 = 0). Thus |u〉〈u| is an orthogonal projection in the W*-algebra BM(Y).

(p.340) We say that two partial isometries u and υ in Y are orthogonal if 〉u|υ〉 = 0. In this case the orthogonal projection |u〉〈u| and |υ〉〈υ| are mutually orthogonal.

Lemma 8.5.22

Suppose that Y is a right W*-module over M, and that yY. Then y = u|y|, where | y | = y | y 1 2 M , and u is a partial isometry in Y whose initial projection is the range projection of |y| in the von Neumann algebra sense (see 2.2.7 in [320])).

Proof We view y as an element of the linking W*-algebra of Y (see 8.5.10), and take its polar decomposition there, as in 2.2.9 of [320]. It is easy to see from the formula given there for u, that uY, that u*uM, and that the latter is the range projection of |y| in M.

Lemma 8.5.23 (Paschke)

Let Y be a right W*-module. Then Y has an orthonormal basis. That is, there exists a set {xi}iI in Y consisting of mutually orthogonal nonzero partial isometries, such that x = ∑i xixi|xin the w*-topology of Y, for all xY. In particular, ∑iI |xi〉 〈xi| comverges in the w*-topology of BM(Y), to Iy.

Proof We consider the subsets B of Y consisting of mutually orthogonal nonzero partial isometries in Y, ordered by inclusion. At least one such set exists by 8.5.22, and Zorn's lemma we may choose a maximal such set, {xi: iI} say. We first claim that if 〈 xi|x〉 = 0 for all iI, then x = 0. To see this, write x = u|x| for a partial isometry uY as in 8.5.22. Then 〈xi|u〉|x| = 0. if p is the initial projection of u, then u = up, and so 〈xi|up|x| = 0. Since p is the range projection for |x|, We see that 〈xi|up = 0 = 〈xi|u〉. This contradicts the maximality above, if u ≠ 0.

By the remarks before 8.5.22, TJ = ∑iJ |xi〉 〈xi| is an orthogonal projection in BM(Y), for any finite subset JI. We therefore have

T J x | x = i J x | x i x i | x x | x , x X .
The increasing net (TJ)J converges in the w*-topology of BM(Y) to an operator T say, with 0 ≥ TI. By 8.5.5 (2), for any xX the sum ∑i xixi|x〉 converges in the w*-topology of Y, to T(x). Taking the inner product with xj, for a fixed jI, we have 〈xj|T(x)〉 = 〈xj|x〉, by the w*-continuity of the inner product. Hence T(x) = x the first part of the proof.

8.5.24 (The Parseval identity)

Suppose that {xi: iI} is an orthonormal basis for Y, as in 8.5.23. It follows from the proof above, and 8.5.4 (1), that

i I y | x i x i | y = y | y , y Y ,
the convergence in the w*-topology. Since ∑i|xi〉〈xi| =I, we see using (8.7) if necessary, that (xi) is an element of R I w ( Y ) , and has norm there equal to 1.

As a consequence, we obtain another characterzation of W*-modules, as exactly the w*-closed submodules of C I w ( Y ) :

(p.341) Corollary 8.5.25

A Banach module Y over a W*-algebra M is a righr W*-module over M if and only if Y is isometrically M-isomorphis to an orthogonally complemented submodule of C I w ( M ) , for some cardinL I.

Proof The ‘only if’ follows from 8.5.16 and the remark above it. Conversely, if Y is a right W*-module over M, let {xi: iI} be an otthonormal basis for Y as in 8.5.23. Define α: Y C I w ( M ) by α(y) = (〈xi|y〉)i, for yY. By 8.5.24 α is an isometry. Also, α is w*-continuos with respect to the w*-topology of Y, by 1.6.3 (2) and 8.5.4 (1), and is an M-module map. By A.2.5, the range of α is w*-colsed. Now apply 8.5.16.

8.5.26 (The ultraweak direct sum)

We define the ultraweak direct sum i I w c Y i of a family {Yi: iI} of right W*-modules over a W*-algebra M, to be the set of (yi)iI ∈ ∏iI Yi, such that the finite paratial sums of the series ∑iIyi|yi〉 are uniformly bounded above. Equivalently, it is the set of (yi)iI such that ∑iIyi|yi〉 converges in the w*-topology of M. It easy to check, using the polarization identity (1.1), that for (yi) and (zi) in i I w c Y i , the finite partial sums of ∑iIyi|zi〉 converge in the w*-topology of M. We write 〈(yi)|(zi)〉 for the w*-limit. Most of the conditions in the difinition fo a C*-module are easy to check for i I w c Y i , and all will follow from considerations in the next paragraph. Note that if Y is a right W*-module, then the W*-module C I w ( Y ) met in 8.5.15 equals the ultaweak direct sum of I copies of Y.

Although we shall not use the general ultraweak direct sum much, we mention some of its properties. For these, it is helpful to view Y = i I w c Y i in a slightly different way. We begin with a faithful normal repesentation of M on a Hilbert space K. We suppose that for each iI, Yi is represented (as in 8.5.11 say) as a w*-closed M -submodule, and a subtriple, of B(K,Hi), for a Hilbert space Hi. Set H = ⊕i Hi, and let Pi be the projection from H onto Hi. We also set W = {TB(K,H): PiTYi for all iI}, and equip W with its canonical inner product S × TS*T. For S,TW we have S*T = ∑i S*PiPiT, which is a w*-convergent sum in M. Hence this inner product is valued in M. It is easy to see that W is a right W*-module over M. Writing yi = PiT, for TW it becomes evident that W corresponds precisely to the difinition of i I w c Y i above. In other words, W is unitrarily M-isomorphic to this sum. It follows that i I w c Y i is a right W*-mofule over M.

One may deduce from the above above description, and the associativity property for Hilbert space sums, that the ultraweak direct sum is associative. Thus, for example, i I w c ( i J w c Y i j ) i J w c ( i J w c Y i j ) i , j w c Y i j unitarily, for right W*-modules Yij over M. Also, one can easily see that the set of tuples in an ultraweak direct sum which are zero except in a finite number of entrices, is w*-dense.

We shall not use this, but it can be deduced from 8.5.23 that the right W*-modules over a W*-algebra M, are exactly the ultraweak direct sums of w*-closed right ideals pM. See [302,421].

(p.342) 8.5.27 (Second duals of C*-modules sums)

If (Yi) is a family of C*-modules over a C*-algebra B, then ( i c Y i ) * * i w c Y i * * unitarily as B**-modules. We merely sketch the proof. Let Y = i c Y i , let Z = i w c Y i * * , and let ℒ be the ‘augmented linking algebra’ K B(Yc B). Let p 0 be the projection of Yc B onto 0 ⊕ B, and similarly let pi be the projection of Yc B onto the copy of Yi. We regard piM (ℒ) ⊂ ℒ**. In the latter W*-algebra one can show that ∑ipi = (1−p 0). As in the proof of 8.5.17, we have Yipip0, and Y ≅ (1−p 0)ℒp 0, unitarily as right B-modules. Also as in the proof of 8.5.17, we have Y i * * p i * * p 0 and Y** ≅ (1−p 0)ℒ**p 0, unitarily as right B**-modules. Hoowever it is not hard to check, using basic facts about w*-limits of increasing nets of projections in a W*-algebra, that

i w c Y i * * i w c p i * * p 0 ( 1 p 0 ) * * p 0 ,
unitarily as right B**-modules. This proves the result.

A powerful tool assocated with W*-modules, and inded operator spaces, is the following weak* variant of the stabilization result in 8.2.6.

8.5.28

Let T be a w*-full right W*-algebra N. Then there exists a cardinal I such that C I w ( Y ) C I w ( N ) unitarily (as right N-modules). Also, M I (Y) is linearly completely isometrically isomorphic (via a right N-module map) to the W*-algebra M I (N).

Proof We prove this very similarly to our arguments for 8.2.6, which the reader should follow along with (another argument is sketched in the Notes). By 8.5.25 and 8.5.16, there exists a cardinal I and a w*-closed N-submodule W of C I w ( N ) , such that Y c W C I w ( N ) unitarily (as right N-modules). By set theory we may assume that I 2 = I. It follows from this, and (1.59) for example, that C I w ( C I w ( N ) ) C I w ( N ) . Using the latter fact, and using ‘associativity’ of the ultraweak sum (see 8.5.26), the ‘Eilenberg swindle’ works, similarty to the proof of 8.2.6 (1), to give:

C I w ( N ) C I w ( Y ) c C I w ( W ) C I w ( Y ) c C I w ( Y ) c C I w ( W ) C I w ( Y ) c C I w ( N ) ,
unitarily as N-modules. We may assume by 8.5.14 that Y is a W*-equivalence M-N-bimodule. By (the ‘other-handed variant’ of) Lemma 8.5.23, and since N is isomorphic to the algebra of left M-modules maps on Y (see 8.5.14), there exists a cardinal J and a subset {zj: jJ} of Y, such that 1N = ∑jzj|zj〉 in the w*-topology on N. Thus ( z j ) C J w ( Y ) , which permits us, exactly as in the proof of 8.2.6 (2) (and in fact a little more easily), to define maps showing via 8.1.21 that N is unitarily isomorphic to an orthogonally complemented N-submodule of C J w ( Y ) . Thus by 8.5.16, there exists a w*-closed submodule p of C J w ( Y ) such that N c P C J w ( Y ) , , By the atrument in the the first pargraph of this proof, we obtain that C J w ( Y ) C J w ( Y ) c C J w ( Y ) unitarily. By set theory, we may suppose that I = j. Thus C J w ( Y ) C J w ( Y ) .

(p.343) The last assertion follows since

M I ( Y ) R I w ( C I w ( Y ) ) R I w ( C I w ( N ) ) M I ( N ) ,
using (1.20).   □

The following follows at once from 8.5.17:

Corollary 8.5.29

If Y is any full right C*-algebra B,then there is a cardinal I such that C I w ( Y * * ) C I w ( B * * ) completly B**-isometrically. Also, M I(Y**) is completely B**-isometric to the W*-algbra M I(B**).

8.5.30 (W*-algebra ‘covers’of an operator space)

In orrator space applications one sometimes applies the preceding result, with Y an injective or triple envelope of an operator space X (see Sections 4.4 and 8.3).These are triple systems, and may be taken to be C*-modules, as discussed in those sections. By 8.5.59 we see that for some cardinal I, M I(y**) is a W*-algebra.

Thus for any operator space x, there is a useful and essentially canonical W*-algebra ‘contaning’ M,I(X).

Corollary 8.5.31 (The stable isomorphism therem for W*-algebras)

Two W*-algebras M and N are weakly Morita equivalent if and only if there existsa cardona I such that M I(M) ≅ M I(N) *-isomorphically.

Proof The ‘only if’ may be proved similarly to 8.2.7, but replacing the use of 8.2.6 (5) with the last assertion of 8.5.28 (and the ‘other-handed’ variant of that assertion). Another argument is sketched towards the end of the Notes for 8.5. The other direction is easier. For example, one may appeal to the later result 8.5.38, and the fact that the commutant of M ⊗̄ M I is M′ ⊗ 1.   □

8.5.32 (A basic construction)

Suppose that Y is a full right C*-module over a C*-algebra B. We may suppose that Y is an equivalence A-B-bimodule. Consider the Morita linking C*-algebra ℒ (Y), and identify Y with a corner pℒ(Y)(1–p) as usual (see (8.3)). Fix a faithful nondegenerate representation ℒ(Y) on a Hilbert space. As we saw in 8.2.8, this Hilbert space may to be HK, where H and K are two Hilbert space on which respectively A and B are faithfully and nondegenerately represented. Since A = YY*, we have [YK] = H. Similarly, [H*H] = K. Using these facts it is easy to explicitly compute the commutant ℒ(Y)′ in B(HK). Indeed, a simple calulation shows the following facts. First, ℒ(Y)′ is in set of diagonal matrices, whose 1–1 entry R is in A′, and whose 2–2 entry S is in B′ Second, these entries are mutually dependent, and this dependencs is given by the eqution Ry = yS for all yY. This last equation provides a map π: B′ → A′, defined by π(S) = R. That is,

(8.21)
π ( S ) y = y S , y Y , S B .
In fact π is a *-isomorphism from B′ onto A′. One (perhaps too slick) way to see this proceeds as follows. Recall from 8.2.19 that the strong Morita equivalence of A and B gives a category isomorphism between the categories of modules over A and B respectively. This isomorphism is implemented by a pair (p.344) of functors F and G, given by F(W) = YB W, and G(V) = Ȳ ⊗A V. By basic algebra, the map TF(T) = IYT yields a surjective isomorphism BB(W) ≅ AB (F(W)). Indeed this map is clearly contractive (by 8.2.12 (1)), but one may exhibit a contractive inverse via the other functor G, just as in pure algebra (e.g. see [8, Proposition 21.2]). Thus the last isomorphism is also isometric. By 8.2.22, F(K) = YBK is unitarily A-isomorphic to H above. Setting W = K and appealing to the above facts, we have BB(K) ≅ AB(H) isometrically isomorphically. This says precisely that B′ ≅ A′; and it may be verfied that this isomorphism is exactly our map π above. We have by 1.2.4 that π is a *-isomorphism. We remark in passing that this proves that strongly Morita equivalent C*-algebras naturally have Hilbert space representations in which their commutantas are *-ismorphic.

Summarizing, we saw that the commutant of ℒ(Y) is the set of matrices

[ π ( S ) 0 0 S ] , S B .
It is now easy to see that the second commutant ℒ(Y)′′ is the set of matrices
[ c e f * d ] , c A ; d B ; e , f E ,
where E = {TB(K,H): TS = π(S)T for all SB′}. In other words, E =B B(K,H), where we are viewing H as a B′-module via π. Also, we have E = pℒ(Y)′′(1–p). Thus E is a subtriple of B(K,H), and E has a B′′-valued inner product defined by T 1 | T 2 = T 1 * T 2 , for T 1, T 2B(K,H). Indeed we have
B = Y * Y E * E B ′′ ,
the last inclusion following from the definition of E and a direct calculation. Taking w*-closures in B(K) in the last eqution, and using the double commutant theorem, we deduce that E*E is w*-dense in B′′. From 8.5.4 (1), we deduce that E is a w*-full right W*-module over B′′. Hence, by symmetry, E is a W*-equivalence A′′-B′′-bimodule.

By the double commutant theorem, ℒ(Y) is w*-dense in ℒ(Y)′′. It is clear from this, and from the fact that E = pℒ (Y)″(1–p), that Y is a w*-dense A-B-submodule of E. We will see some applications of this momentarily, and in the Notes section.

8.5.33 (Universal representations)

We mention in passing that if we begin the construction in 8.5.32 by choosing a ‘universal representation’ (see 3.2.7) of ℒ(Y) (or equivalently, by 8.2.23 and 8.1.18, a representatoin of ℒ(Y) induced from a universal represemtatoin of B, say), then that construction allows one to recover 8.5.17. In fact 8.5.32 yields much more information in this case, such as the fact that Y** ≅ B′B(K,H), for appropriate Hilbert modules H and K. It also allows us to treat representations of B**, or y**, in a functorial way that is often important in applications.

(p.345) 8.5.34 (Normal rigged W*-modules)

We define a normal N-rigged M-module to be a right W*-module Z over N for which there exists a (unital) normal *-homomorphism θ: M → B N(Z). This name is due to Rieffel, who does not however insist on all of the conditions above. In the literature, they are often called M-N-correspondences.

The following shows again that the ‘interesting objects’ in the theory fall within the operator module framework:

Proposition 8.5.35

A W*-module over N is a normal N-rigged M-module if and only if it is a normal dual operator M-N-bimodule in the senes of 3.8.2.

Proof If Z is a normal N-rigged M-module, then by 8.5.9 it is a normal dual operator B N(Z)-N-bimodule. Since the left action comes from a normal *-homomorphism θ:M → B N(Z), Z is a normal dual M-N-bimodule.

Conversely, if Z is a normal dual operator M-N-bimodule, then as in the proof of 4.7.6, there is a normal *-homomorphism θ: M → 𝒜l(Z). However 𝒜l(Z) = B N(Z) by 8.4.2.

8.5.36 (Inducing normal representations)

Suppose that H is a Hilbert space on which M is normally represented on. if Y is a right W*-module over M, then it follos from 8.2.13 that YMH is a normal Hilbert N-module (in the sense of 3.8.5). To see this, suppose that (at)t is a bounded net in N converging in the w*-topology to an aN. Then by 8.5.35, atyay in the w*-topology of Y, for any YY. By 8.5.4 (3), if ζ, η ∈ H and y, z ∈ ∈ Y, then

a t y η , z ζ = z | a t y η , ζ z | a y η , ζ = a y η , z ζ .
Since finite sums of ‘rank one tensors’ are norm dense in YM H, it is evident that aty⊗η → ay⊗η weakly in the Hilbert space YhM H. For the same reason, we may now deduce that atξ → aξ weakly, for any ξ ∈ YM H. Thus by A.2.5(2), YM H is a normal Hilbert N-module.

By 8.2.13, if, further, M is faithfully represented on H, and of the canonical map from N into B M(Y) is one-to-one, then N is faithfully represented on the Hilbert space YM H. We shall not use this, but it follows easily from A.6.2 that of ℐ is as in 8.1.4, then YM H = YH.

Corollary 8.5.37 (Rieffel)

Suppose that π: MB(K) is a faithful normal representation, and let R = π(M)′, the commutant in B(K).

(1) If Y is a W*-module over M, then there exists a Hilbert space H on which R is normally represented (namely yM K), such that YRB(K,H) completely isometrically, and w*-homeomorphically.

(2) Conversely, if H is a Hilbert space on which R is normally represented, then the w*-closed right π(M)- submodule RB(K,H) of B(K,H), with its canonical B(K)-valued inner product, is a right W-module over π(M).

In fact the isomorphism in (1) is a unitary M-module map.

(p.346) Proof (2) This is clear by a direct computation, using the double commutant theorem, and 8.5.4 (1).

(1) We will use 8.5.32 (Rieffel' s proof is sketched in the Notes section).

First assume that Y is w*-full over M. Since Y is a W*-module, the normal representation of M on K induces, by 8.5.36, a faithful normal representation of B M(Y) on H = YM K. Similarly, using also the definition of ℒW(Y) in 8.5.10, we have a faithful normal *-representation of ℒW(Y) = B M(Yc M) on the Hilbert space(Yc M) ⊗M K. The latter space, as in (8.14), is unitarily equivalent to (YM K) ⊕ K = HK. Just as in the last paragraph of 8.2.22, the associated normal representation of ℒw (Y) on HK is one-to-one (and hence completely isometic, by 1.2.4) and corner-preserving, and its ‘1-2-corner’ is a map from Y to B(K,H). Clearly the latter map is completely isometric and w-closed subspace of B(K,H). If ℐ is as in 8.1.4 then, as we saw in the proof of 8.5.3, ℐ acts nondegenerately on K, and M(ℐ) = π(ℐ)″ = π(M). By 8.2.13 (2), A = K M(Y) acts nondegenerately on H. Now we are in a position to apply the arguments in 8.5.32. By the last facts in 8.5.32, we have E = Y. Since Y is an M-submodule of E, the last assertion the Corollary is also clear in this case.

In the general case, we use the fact that a W*-closed ideal in a W*-algebra is of the form pM for a central projection p. Thus ℐw = pM for such a projection p. Let K′ = π(p)K, and apply the previous case to the canonical representation of ℐw on K′. Write θ for this last representation, and let N = θ(ℐw)′. We obtain a Hilbert space H on wich N is normally represented, such that YNB(K′,H). However, there is a canonical normal *-homomorphism r ↦ π(p)r, from R onto N. Thus K′ and H may be viewed as R-modules, and it is easily checked that RB(K,H) ≅ NB(K′,H) unitarily as M-modules via the map TT |K.□

Corollary 8.5.38 (Connes)

W*-algebras M and N are weakly Morita equivalent if and only if there exist faithful normal representions π: MB(K) and ρ: NB(H), with π(M)′ ≅ ρ(N)′ *-isomorphically. Moreover, in this case, writing R for π(M)′ and for ρ(N)′, the TRO RB(K,H) in 8.5.37 (2) is a W*-equivalence N-M -bimodule.

Proof If Y is a W* -equivalence N-M-bimodule, and if π is as in 8.5.37, then the proof of (1) of that result, together with the paragraph after (8.21), shows that if ρ is the induced representation of N on YMK, then π(M)′ ≅ ρ(N)′. A quick proof the converse is given in the Notes, however it does not yield the final assertion. So instead, suppose that π,ρ,R are as in the statement of 8.5.38. IfY = RB(k,H) then, as in 8.5.37 (2), it is easy to see that Y is a TRO, which is a right and a left W*-module over π(M) and ρ(N) respectively. We need to show that these W*-modules are full. The w*-closure ℐw of YY is a w*-closed ideal in π(M). Thus there is a central projection e in π(M) with ℐw = eπ(M). Since Y = Yw (e.g. see 8.1.4 (2)), we have Y = Y e. Since e ∈ π(M), (1 −e)K is an R-module. Let P be 1 − e, viewed as a map from K onto (1 − e)K. Since eR′, p is an R-module map. Since H is a cogenerator of RNHMOD (see 3.8.6), if (p.347) (1 − e)K ≠ (0) then there exists a nonzero R-module map T: (1 − e)KH. Thus TPY, and so TP = TPe, which is absurd. Thus e = 1, is w*-full on the right. A similar arument shows that Y is w*-full on the left.  □

8.5.39 (Correspondences)

Another important ‘picture’: of W*-modules is related to the standard form L 2 (M) of a W*-algebra M (see 3.8.5, or for full details, see, for example, [175] or [408, Chapters VIII and IX]). One reason why the standard form is of importance here, is that it is a normal faithful Hilbert space representation of M such that M′ ≅ M op. Thus in Corollary 8.5.37 we may replace R = M′ by M op. We viwe a left M op-module action on a Hilbert space, as a right M-module action of M. In particular, by 8.5.37 (1), if Y is a right w*-module over M, then there exists a Hilbert space H = YM L 2(M) on which M op is normally *-represented, or equivalently on which M is normally represented on the right of H, such that YBM(L 2(M), H). Conversely, by 8.5.37 (2), any Hilbert space H on which M is normally represented as tight action (that is, any normal *-representation of M op), gives rise to a right W*-module over M, namely BM(L 2(M),H). One may show that BM(L 2(M),H) ⊗M L 2(M) ≅ H unitarily. Indeed it is easy to see that the canonical map from BM(L 2(M),H) ⊗M L 2(M) to H is isometric. That it has dense range, and is thus surjective, follows from modular theory (see [27, Theorem 2.2], and references therin). Thus we see that there is a bijective correspondence between such Hilbert spaces H, and right W*-modules over M.

It is easily see that the bijection above restricts to a bijective corresponce between the Hillbert spaces H on which M is normally represented on the right, and on which another W*-algebra N is normally represented on the left; and the class of normal M-rigged N-modules (see 8.5.34). A correspondence between N and M is a Hilbert space H which is a normal N-M-bimodule as above. Thus, such Hilbert space are in a bijective relation with the normal M-rigged N-modules. We do not have spaces to even touch on this extremely imortant topic in detail here, but refer the reader to [101, Section V.B], [408, Chapters VIII and IX], and [346,6,27,119], for example, and references therein.

8.5.40 (The W*-module tensor product)

This is a ‘W*-module version’ of the interior tensor product discussed in Section 8.2. We write it as Y⊗̄M Z, for a right W*-module Y over M, and a normal N-rigged M-module Z. Just as the C*-module interior tensor product is just the module Haagerup tensor product, the W*-module tensor product ⊗̄M coincides with the module weak* Haagerup tensor product ⊗w*hM which we discussed briefly in 3.8.14. Some of the benefits of knowing that these tensor products coincide are that, first, we get useful expressions for elements in this tensor product as w*-concergent sums ΣiI xiyi, with a convenient description of the norm of such a sum. This facilitates easy computations. Second, as in 8.2.12, we can appeal to the useful properties of this tensor product to show that this product is functorial, associative, commutes with the ultraweak sum, and so on. One may deduce that, for example, analoguosly to 8.2.15, we have (p.348)

W ¯ M Z C B M ( Z ¯ , W ) completely isometrically .

We omit the proof of this result, which is similar to the proof of 8.2.15, and is a variant of a result originally from [119]. See [48] for a complete discussion of this tensor product, which parallels our earlier development of the C*-module tensor product. Indeed there are precisely analoguous W*-module versions of results 8.2.11–8.2.19. We refer the reader to [361,48] for a development of this theory which parallels our earlier discussion. To avoid this already lengthy chapter becoming completely unwieldy we have omitted these results. However, the reader who has followed the discussion till now, should at least have no difficulty stating appropriate W*-module versions of 8.2.118.2.19.

8.6 A SAMPLE APPLICATION TO OPERATOR SPACES

Because of space limitations, we will only list one of the very many applications of the preceding theory to operator spaces. See the Notes for references to the literature for other applications.

8.6.1 (Injectivity and semidiscreteness)

We return to the notions of ‘OS-nuclwarity’, ‘OS-semidiscreteness, and the ‘WEP’, introduced in Section 7.1. For a general operator space, the relationships between these concepts, and other properties such as ‘injectivity’, are quite interesting. Some of these are not difficult, such as the result from 7.1.5 that any OS-nuclear oprator space X has the WEP, or the fact from 7.1.9 that an OS-semidiscrete dual operator space is injective. In fact, for a finite-dimensional opertor spaceX, all of the following properties are equivalent: injectivity, OS-nuclearity, OS-semidiscreteness, and the WEP. These are also equivalent to X being a triple system, and also equivalent to saying that for some n ∈ N, X is a completely contractively complemented subspace of Mn (that is, there exists a completely isometry from X onto a subspace W of Mn, and a completely contractive projection from Mn onto W). Most of these equivalences are quite trivial to see. Indeed, if X is injective then it is a TRO by 4.4.2. For any finite-dimensional TRO X, the linking C*-algebra is ℐ(X) is finite-dimensional. By (8.3), X is completely contractively complemented in ℐ(X). However any finite-dimensional TRO X, the linking C*-algebra ℒ(X) is finite-dimensional. By (8.3), X is completely contractively complemented in ℒ(X). However any finite-dimensional C*-algebra is completely contractively complemented in some Mn, so that X is completely contractively complemented in Mn too. We leave the remaining implications to the reader.

Theorem 8.6.2 (Effros, Ozawa, and Ruan)

If X is a dual operator space then the following are equivalent:

  • (i) X is injective,

  • (ii) X is OS-semidiscrete,

  • (iii) X has the WEP,

  • (iv) X is completely isometrically isomorphic and w*-homeomorphic to acornerpM(1 − p), for an injective W*-algebra M, and a projection PM.

(p.349) Proof We saw in 7.1.3 that (i) is equivalent to (iii). We saw in 7.1.9 that (ii) implies (i). Using the fact that an injective W*-algebra is OS-semidiscrete (see[99,419]), it is clear that (iv) implies (ii). Finally, if X is an injective dual operator space then by 4.4.2, X may be regarded as an TRO, and hence an equivalence bimodule. By 8.5.6, X is a W*-equivalence M-N-bimodule, over W*-algebras M and N say. By the proof of