C*modules and operator spaces
C*modules and operator spaces
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 twoway 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 ‘leftright 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 ‘otherhanded version’ (p.297) of the earlier result.
8.1 HILBERT C*MODULES—THE BASIC THEORY
8.1.1 (The definition)
A (right) C*module over A is a right A module Y, together with a map $<\cdot \cdot >:Y\times Y\to A$, which is linear in the second variable, and which also satisfies the following conditions;

(1)〈yy〉 ≥ 0 for all y ∈ Y,

(2)〈yy〉 =0 if and only if y =0,

(3)〈yza 〉 = 〈yz〉a for all y, z ∈ Y, a ∈ A,

(4)〈yz〉* = 〈zy〉 for all y, z ∈ Y,

(5) Y is complete in the norm $\Vert y\Vert ={\Vert \u3008yy\u3009\Vert}^{\frac{1}{2}}$.
We call $\u3008\cdot \cdot \u3009$ the Avalued inner product on Y. It follows from (3) and (4) that 〈yaz〉 = a* 〈yz〉 for all y, z ∈ Y, a ∈ A.
In (5), the fact that ‖ · ‖ is a norm follows just as for Hilbert spaces from the following Cauchy–Schwarz inequality: ‖〈yz〉‖≤ ‖y‖ ‖z‖ for y, z ∈ Y. This follows from the relation
It the linear span of the range of the inner product $\u3008\cdot \cdot \u3009$ 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 Avalued inner porduct id linear in the first variable, and condition (3) in the above is replaced by 〈ayz〉 = a〈yz〉, for y, z ∈ Y, a ∈ A. 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 ↦ 〈ηζ〉.
If Y is right C* module over A, then there is a canonical left C*module $\overline{Y}$ over A, which is simply the conjugate vector space of Y with left action $a\overline{y}=\overline{y{a}^{*}}$, and inner product $\u3008\overline{y}\overline{z}\u3009=\u3008yz\u3009$. 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 ABbimodule, 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 x 〈yz〉 = [xY]z, for al x, y, z ∈Y. Here we have written $\left[\cdot \cdot \right]$ for the Avalued product. Equivalence bimodules are also sometimes called imprimitivity bimodules, or strong Morita equivalence ABbimodules, or equivalence ABbimodules. 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 CDbimodule.
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 Amodule (see A.6.1). Indeed,
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 twosided 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 I → B(Y), taking a ∈ I to the operator y↦ya, is a linear isometry. Indeed it is clearly contractive. If Ya = 0 then we have that 〈YYa〉 = 〈YY〉a =0. This implies that I a = 0, so that a*a0. Hence a = 0. Thus the map is onetoone. It follows from A.5.9 (or from the otherhanded 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 x〈y〉, for x, y, z ∈ Y, is dense in Y.
(3) If Zis a Banach Amodule then B _{A} (Y, Z) = B _{I} (Y, Z). Indeed, suppose that u ∈ B _{I}(Y, Z), x ∈ Y and a ∈ A. By (2) and A.6.2, we may write x = ya′, with a′ ∈ I, y ∈ Y. Then u(xa) = u(ya′a) = u(y)a′a = 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 y ∈ Y, a ∈ A, and c ∈ C. The fact that this action is well declined, and that 8.1.1 (3) holds, follows from the relation 〈zy(ac)〉 = 〈zy〉ac = 〈zya〉c, for z ∈ Y.
Lemma 8.1.5
Let u: Y → Z be a bounded Amodule mop between right C*modules over A. Then 〈u(y)u(y)〉 ≤ ‖u‖^{2}〈yy〉, 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:
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: Y→Z such that there exists a map S: Z→Y with
Corollary 8.1.8
An Amodule map u: Y → Z 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 {Y_{i}: i ∈ I} is a collection of right C*modules over A, then we define the direct sum C*module by
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 ${\oplus}_{i}^{c}{Y}_{i}$ 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 Y ⊥ W (i.e. 〈y w〉 =0 for all y ∈ Y, w ∈ W). In this case, $Z\cong Y{\oplus}^{c}W$ 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: Y → Z between C* modules over B is a contractive Bmodule map, if and only if 〈u(y) u(y)〉 ≤ 〈yy〉, for all y ∈ Y. This is also equivalent to ‖(u(y), z)‖≤ ‖(y, z)‖, for all y ∈ Y and z ∈ Z, where the norms here are those of Z ⊕^{c} Z and Y ⊕^{c} 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: Y → B_{A}(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)=<yz>, for y, z ∈ y. 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 y ∈ Y as y′a, for y′ ∈ Y, a ∈ A. 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}(c_{n}(Y)). Clearly θ is onetoone. 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 if 〈Tyy〉 is selfadjoint (resp. 〈Tyy〉≥ 0) for all y ∈ Y.
Proof We sketch a proof of the difficult implication. If 〈Tyy〉 is selfadjoint for all y ∈ y, then 〈Tyy〉 = (〈TYy〉)* = 〈yTy〉. By the polarization identify (1.1), T is adjointable on Y, with T* = T. If, further, 〈Tyy〉≥0, for y ∈ Y, then T is selfadjoint, by the above. To see that T ≥ 0, suppose that A ⊂ B(H), and apply the following simple general fact about C*algebras to the collection of positive functionals $\u3008\u3008yy\u3009\zeta ,\zeta \u3009\text{on}\text{}{\mathbb{B}}_{A}\left(Y\right)\text{,}\text{}\text{}\text{}\text{for}y\in \text{Ball}\left(\text{Y}\right),\zeta \in \text{Ball}\left(H\right).$ Namely, let B be a C*algebra, and let S be a set of positive contractive functionals on B with
Corollary 8.1.13
if $a=\left[{a}_{i}\right]\in {M}_{n}\left(A\right),then{\Vert a\Vert}_{n}$ equals

(i) a is positive in M_{n} (A),

(ii) ${\sum}_{i,j}{b}_{i}^{*}{a}_{ij}{b}_{j}\ge 0$ in A, for all b _{1},…, b_{n} ∈ A,

(iii) a is a finite sum of matrices of the form $\left[{a}_{i}^{*}{a}_{j}\right],$, with a _{1}, …, a_{n} ∈ A.
Proof Define θ: M_{n}(A) → B_{A}(c_{n}(A)) by the canonical left action of M_{n}(A) on C_{n}(A). Clearly θ is a onetoone *homomorphism into the C*algebra B _{A}(C_{n}(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 $y\u3009\u3008y\ge 0,$ 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)Ibimodule. 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 $\Vert y\u3009\u3008y\Vert \le {\Vert y\Vert}^{2}.$ The reverse inequality follows from the fact that
The proof of the following result shows, conversely, that given an equivalence ABbimodule 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 Avalued 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 Amodule Y.
Lemma 8.1.15
If Y is an equivalence ABbimodule, 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 Avalued inner product, we have
For the last assertion, since λ is isometric, and by the second paragraph of 8.1.14, we have$\Vert \left[yy\right]\Vert =\Vert y\u3009\u3008y\Vert =\Vert \u3008yy\u3009\Vert .$
(p.303) Proposition 8.1.16

(1) If Y is an equivalence ABbimodule 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 B_{B}(Y≅LM(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 B_{B}(Y) .
Proof Suppose that Y is an equivalence ABbimodule, and that (e_{t})_{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)→B_{B}(Y)given by
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: B_{B}(Y),→LM(K _{B}(Y)) by ρ(S)(T) = ST, for S ∈ B_{B}(Y),T ∈ K B(Y). Here we are viewing LM(A) as the right Amodule 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
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:
Define a map π: ℒ(Y) → B(Y⊕^{c} B) by the obvious action (i.e. viewing an element of Y⊕^{c} 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}(Y⊕^{c} B) for each matrix m ∈ ℒ(Y), and moreover that π is a *homomorphism into B _{B}(Y⊕^{c} B). Also, one can quickly check that π is onetoone, 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}(Y⊕^{c} B) via π, thus ℒ(Y) is a C* algebra (isomorphic to the range of π. Indeed we may regard B _{B}(Y⊕^{c} B) as a 2 × 2 matrix C*algebra consisting of matrices t = [t_{ij}] whose four entries are adjointable maps. To see this, note that t_{ij} are defined in terms of t and the (adjointable) projection and inclusion maps between Y⊕^{c} B and its two summands. With this in mind, it is clear from 8.1.11 and (8.1), that K _{B}(Y⊕^{c} B) is exactly the range of π. Hence the linking C _{*}algebra may simply be thought of as K _{B}(Y⊕^{c} B).
By the last fact, and 8.1.16 (3), we see that the multiplier algebra M(ℒ(Y)) is B _{B}(Y⊕^{c} B). We define the unitized linking C*algebra ℒ^{1}(Y) of Y to be the linear span within M(ℒ(Y)) of K _{B}(Y⊕^{c} 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 ℒ^{1}(Y) is a unital C*algebra with identity 1= p  q. Clearly Y is the 12corner of both ℒ^{1}(Y and ℒ(Y). In particular,
If we take a general C*algebra A, and if Y is an equivalence ABbimodule, 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 Y⊕^{c} B is a full Bmodule. Since K _{B}(Y ⊕^{c} B)≅ℒ(Y), we have by 8.1.14 that ℒ(Y)is strongly Morita equivalent to B, via Y⊕^{c} 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 y ∈ y to the matrix in ℒ(Y) with y in the 12corner and zeroes elsewhere. If we identify B with the 22corner in a similar way, then 〈yz〉 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 x〈yz〉, or equivalently x〉y(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 ABbimodule, 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:
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
Proof By 3.1.11 Y is a right Z(M(B))module, with action yη = lim_{t} yη(e_{t}), for y∈ Y,η∈Z(M(B)). Here (e_{t})_{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 y ∈ Y. Clearly this implies that νay = ayν = aνy, if a ∈ A. 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 Amodule 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: Y → Z and β: Z → Y 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,
From 8.1.21, one may deduce a universal property of the direct sum:
Proposition 8.1.22
Suppose that {Y_{i}: 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}: Y_{i} → Y and P_{i}: Y → Y_{i} such that ${P}_{i}\circ {\epsilon}_{j}={\delta}_{i}{I}_{Y},$, and such that $\sum {\epsilon}_{i}}\circ {P}_{i$ converges strongly on Y. Then there exists an orthogonally complemented submodule W of Y such that $Y\cong \left({\oplus}_{i}^{c}{Y}_{i}\right){\oplus}^{c}$ W unitarily. If $\sum {\epsilon}_{i}\circ {P}_{i}$ converges strongly to I _{y}, then W =0.
Proof By 8.1.21, each ε_{i} and P _{i} are adjointable, with ${P}_{i}={\epsilon}_{i}^{*},$ and Q _{i} = ε_{i} o P_{i} 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 y ∈ Y. Then R is a module map on Y, and
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 (e_{t})_{t} of the form
Since K _{B} (Y) acts nondegenerately on Y (by 8.1.3 and 8.1.4), for any y ∈ Y we have that ${\sum}_{k=1}^{n\left(t\right)}}{x}_{k}^{t}\u3008{x}_{k}^{t}y\u3009\to y$. From this we deduce the following:
Corollary 8.1.24
Let Y be a right C*module over B. Then there exists a net ${\left(n\left(t\right)\right)}_{t}$ of positive integers, and contractive Bmodule maps ${\alpha}_{t}:Y\to {C}_{n\left(t\right)}\left(B\right)$ and ${\beta}_{t}:{C}_{n\left(t\right)}\left(B\right)\to Y$, such that ${\beta}_{t}\left({\alpha}_{t}\left(y\right)\right)\to y$ for every y ∈ Y. Indeed this can be done with ${\alpha}_{t}^{*}={\beta}_{t}.$
Proof We use the notation above. For y ∈ Y, define α_{t}(y) ∈ C _{n(t)}(B) to have kth entry $\u3008{x}_{k}^{t}y\u3009$. Also, define ${\beta}_{t}\left(b\right)={\displaystyle {\sum}_{k=1}^{n\left(t\right)}}{x}_{k}^{t}{b}_{k}$. Here b has kth entry b _{k} ∈ B. We have that ${\beta}_{t}\left({\alpha}_{t}\left(y\right)\right)={\displaystyle {\sum}_{k=1}^{n\left(t\right)}}{x}_{k}^{t}\u3008{x}_{k}^{t}y\u3009={e}_{t}y\to y$, as we saw immediately above the corolly. It is easily checked that α_{t}, β_{t} are adjointable, with ${\alpha}_{t}^{*}={\beta}_{t}$, so that they have same norm. We have:
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 lefthanded 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 Bmodule for which there exist nets of contractive Bmodule maps as in the lemma, then Y is a right C*module over B. This gives a characterization of C*modules among the Banach Bmodules. Indeed we have:
Theorem 8.1.26
Suppose that B is a C*algebra and that Y is a right Banach Bmodule. Suppose further that there is a net (Y_{t})t of right C*modules over B, and contractive Bmodule maps α_{t}: → Y _{t} → Y, such that β_{t}(α_{t}(y))→ y for every y ∈ Y 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
(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 Bmodule we mean a module Y for which there exists y _{1},…,y _{n} ∈ Y such that the map f: C _{n}(B)→Y given by $f\left(\left({a}_{k}\right)\right)={{\displaystyle {\sum}_{k}{y}_{k}a}}_{k}$, 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 C_{n}(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 Bmodule 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 = uf; 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 ${e}_{t}={\displaystyle {\sum}_{k}{y}_{k}\u3009\u3008{y}_{k}}$ as in 8.1.23, with $\Vert {e}_{t}{I}_{Y}\Vert <\epsilon $. Hence e _{t} is invertible with inverse S say, so that ${I}_{Y}={\sum}_{k}S(yk)><yk$, 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
Finally, any Bmodule map υ: Y → Z into a Banch Bmodule Z satisfies
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)
The formula (8.6) is also valid for nonsquare matrices. For instance, for a column y = [y _{1} … y _{n}]t ∈ C _{n}(Y) = M _{n,1}(Y), we have $\Vert y\Vert ={\Vert {\sum}_{k=1}^{n}\u3008{y}_{k}{y}_{k}\u3009\Vert}^{\frac{1}{2}}$. Viewing Y as a subspace of the linking C*algebra, we also have
Proposition 8.2.2
for C*modules Y and Z over B, every bounded Bmodule map u: Y → Z is completely bounded, with ‖u‖ = ‖ u ‖_{cb}. 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 _{n} ∈ Y,b _{1},…,b _{n} ∈ B. Set $z={\displaystyle {\sum}_{i}{x}_{i}{b}_{i}}$. Then
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 Bmodule. 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)Bbimodule. Similarly, if Y is an equivalence ABbimodule, then Y is an operator ABbimodule, and an operator M(A)M(B)bimodule.
(2) Let Y be an equivalence ABbimodule. Viewing Y as the 12corner of the lkinking C*algebra, and the ‘adjoint module’ Ȳ as the 21corner, 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 BAbimodule (see 3.1.16).
(3) If ${\oplus}_{i}^{\text{c}}{Y}_{i}$ is a direct sum of right C*modules over B, equipped with as canoncial operator space structure, then ${M}_{n}\left({\oplus}_{i}^{c}{Y}_{i}\right)\cong {\oplus}_{i}^{c}{M}_{n}\left({Y}_{i}\right)$ 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:Y → L(Y) from 8.1.19. The amplification C_{I,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
(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 Bmodule, and suppose that there exist maps α_{t} and β_{t} satisfying all the conditions in 8.1.26. If α_{t} and β_{t} 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 (α_{t})_{n} and (β_{t})_{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 Amodule, 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)
(7) The canonical operator space structure on the C*algebra B _{B}(Y) coincides with the inherited operator space structure from CB_{B}(Y). Indeed, by (6), we have M_{n}(CB_{B}(Y)) ≅ B_{B}(C_{n}(Y)). On the other hand, by 8.1.11 (3), we have M_{n}(B _{B}(Y)) ≅ B _{B}(C_{n}(Y)) ⊂ B_{B}(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(CB_{B}(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 Bmodule X is countably generated if there is a sequence (x_{n}) in X such that Span{bx_{n}: b ∈ B, n ∈ N} is dense in X. By a (countable) right quasibasis of a right C*module Y over B, we mean a row [y_{k}] ∈ R^{w}(Y) (see 1.2.26 for this notation), such that
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) = C_{I}(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 $\u3008yy\u3009={\displaystyle {\sum}_{k=1}^{\infty}}\u3008y{y}_{k}\u3009\u3008{y}_{k}y\u3009$, for all y ∈ Y. This permits us to define an isometric Bmodule map α: Y → C(B), by α(y) = (〈y_{k}y〉)_{k}. By a simple calculation analoguous to that in 1.2.27, there is a well defined contractive Bmodule map β: C(B) → Y given by β((b_{k}))= ∑_{k} y_{k}b_{k}. Clearly β ∘ α = I_{Y}. Our claim then follows from 8.1.21 (2).
A left quasibasis for Y is a column (z_{k}) ∈ C^{w}(Y) with $y={\displaystyle {\sum}_{k=1}^{\infty}}y\u3008{z}_{k}{z}_{k}\u3009$ for all y ∈ Y. If Y is full, then the latter condition is equivalent to the same condition, but for y ∈ B, since in that case B = Y * Y and Y = YB. Taking adjoints, we see the latter is also equivalent to (p.312)
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 (BrownKasparove stabilization)
Suppose that Y is a right C*module over B. Then (using the notation above):

(1) C(B)⊕_{c}Y ≅C(B) unitarily, if Y has a right quasibasis.

(2) C(B)⊕_{c}C(Y) ≅C(B) unitarily, if Y has a right quasibasis.

(3) C(B)⊕_{c}C(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 ABbimodule 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)⊕_{c}W for a submodule W of C(B). By the ‘associativity’ of the C*module sum, we may employ the ‘Eilenberg Swindle’:
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
(p.313) 8.2.7 (The BrownGreenRieffel 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 ABbimodule 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 bimodule 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, $q=\widehat{\pi}\left(p\right)$ is a projection in B(H), and we may decompose B(H) as a 2 × 2 matrix operator algebra. Indeed the ijcorner of B(H) is simply B(H_{j},H_{i}), where Ran(q) = H _{1} and Ker(q) = H _{2}. By 2.6.15, π is cornerpreserving, and we may decompose π as [π_{ij}]. The maps π_{ij} are complete contractions, which are complete isometries if π is faithful. Also π_{11} and π_{22} are *representations of A and B on H _{1} and H _{2} respectively. In fact π_{11} and π_{22} 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, π_{11} is nondegenerate. A similar argument applies to π_{22}. Note also that [π_{12}(Y)H _{2}] = H _{1}, since we have H _{1} = [π_{11}(A)H _{1}] = [π_{2}(Y)π_{21}(Ȳ)H _{1}] ⊂[π_{12}(Y)H _{2}]. A similar argument shows that [π_{12}(Y)⋆H _{1}] = H _{2}.
If c is the corner map mentioned in 8.1.19, then we have
8.2.9 (Rigged C*modules)
In C*module theory, the most important tensor product is the socalled 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 socalled Brigged Amodule 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 Amodule 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 Amodule. Then Z is a Brigged Amodule 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 Amodule.
Proof If Z is a Brigged Amodule, then by the observations above, Z is certainly a nondegenerate left Bmodule. By 8.2.3 (1), Z is a left operator module over B _{B}(Z). By 3.1.12, Z is an operator Amodule.
If Z is a left operator Amodule, 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 ABbimodule. Define θ: A → CB _{B}(Y) by θ(a)(Y) = ay. Then θ is a contractive homomorphism. Indeed θ is a *homomorphism into Δ(CB_{B}(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 Brigged Amodule.
Theorem 8.2.11
Suppose that Y is a right C*module over A, and that Z is a Brigged Amodule. Then the module Haagerup tensor product Y ⊗_{hA} Z is a right C*module over B, with Bvalued inner product determined by the formula
Proof By 8.1.24 and 8.2.2, there exist completely contractive Amodule maps α_{t}: Y → C_{nt} (A) and β_{t}: C_{nt} (A) → Y such that β_{t} ∘ α_{t} → I_{Y} strongly on Y. By the functoriality of the module Haagerup tensor product (see 3.4.5), we obtain contractive Bmodule maps α_{t} ⊗ I_{Z}: Y ⊗_{hA} Z → C_{nt}(A) ⊗_{hA} Z and β_{t} ⊗ I_{Z}: C_{nt}(A) ⊗_{hA} Z → Y ⊗_{hA} Z. By density of the elementary tensors, the net of maps (β_{t} ⊗ I_{Z}) ∘ (α_{t} ⊗ I_{Z}) converges strongly to the identity map on Y ⊗_{hA} Z. By 3.4.11, we have C_{nt}(A) ⊗_{hA} Z ≅ C_{nt}(Z), and the latter is a right C*module over B. Via this isomorphism, it is easily checked that the induced inner product on C_{n} (A) ⊗ _{hA} Z is given by the formula
(p.315) In the remainder of this section, we will simply write Y ⊗_{A} Z for Y ⊗_{hA} Z.
8.2.12 (Properties of the tensor product)
One may view the last result as the assertion that the wellknown 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: Y → Y′ is a bounded right Amodule map between right C*modules over A, and if υ: Z → Z→ Z′ is a bounded ABbimodule map between Brigged Amodules, then u⊗υ extends to a bounded right Bmodule map between the interior tensor products: Y ⊗_{A} Z → Y ⊗_{A} 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 (Y ⊗ Z) ⊗_{B} W = Y ⊗_{A} (Z ⊗_{B} W) unitarily, if Y is a right C*module over A, if Z is a Brigged Amodule, and if W is a Crigged Bmodule. This follows immediately from the associativity of the Haagerup tensor product (see 3.4.10).
(3) (Commutation with the direct sum) We have
The second centered relation above is almost identical, however one first should check that ${\oplus}_{i}^{c}{Z}_{i}$ is indeed a Brigged Amodule. Observe that the canonical left action of $A\text{}\text{}\text{on}{\oplus}_{\text{i}}^{\text{c}}{Z}_{i}$ is well defined since by 8.1.5,
(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 $\text{'}\text{adjointC*module'}\text{}\text{of}Y{\otimes}_{A}Z\text{is}\overline{Z}{\otimes}_{A}\overline{Y}$, 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) Y ⊗_{B} H^{c} is a H^{c} is a Hilbert space.

(2) If Y is a Brigged Amodule, then Y ⊗_{B} H^{c} is a Hilbert Amodule. If θ and the canonical map from A into B(Y) are both onetoone, then so is the canonical map from A into B(Y ⊗_{B} H^{c}).
Proof(1) This follows from 8.2.11, which shows that Y >⊗_{B} H^{c} is a right C*module over C. That is, it is a Hilbert space with inner product ζ×η ↦ 〈ηζ〉.
(2) In this case, Y and Y ⊗_{B}H^{C} are left operator Amodules, by 8.2.10 and 3.4.9. Thus the first assertion follows from (1) and 3.1.7. If α(Y ⊗_{B}H^{c}) =0 then aY = 0, by the relation 〈a(y ⊗ζ),y′⊗η〉 = 〈ζ〈ayy′〉η〉from (8.11). Here a∈A,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 Bmodule map Φ: Y → B(K,Y ⊗_{B} K^{c}) by Φ(y)(ζ) = y ⊗ ζ, for y ⊗ Y, ζ ∈K. It is easy to see that Φ(y)*Φ(z) = 〈yz〉, for y,z ∈ Y (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 Bmodule map. Thus the inner product on Y is completely determined by the norm on the space Y ⊗_{B} K^{c}). 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 H_{i} = Y_{i} ⊗_{B} K^{C}. We will suppress mention of the map Φ in the last paragraph, and simply write yiζ for yi⊕ ζ. Then Y _{1}⊕^{c} Y _{2} may be identified with the Bsubmodule W of B(K,H _{1} ⊕ H _{2}) consisting of the maps ζ ↦(y _{1}ζ,y _{2}ζ), for ζ∈K,y _{1} ∈ Y _{1}, y _{2}∈Y _{2}. Indeed, the canonical inner product on W, namely S × T ↦ S*T, takes values in B, making W into a C*module over B which is unitarily Bisomorphic to Y _{1} ⊕^{c} 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 Bmodule. Namely, we define K _{B}(Y,Z) to be the closure in CB(Y,Z) of the span of the ‘rank one’ operators y↦z〈y′y′ (here y, y′ ∈ Y,z ∈ Z).
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 Bmodule). Then:

(1) $W{\otimes}_{B}\overline{Y}\cong {\mathbb{K}}_{B}\left(Y,W\right)$ completely isometrically.

(2) If Y is an equivalence ABbimodule, then $\mathbb{K}B\left(Y,W\right)=C{B}_{B}^{\text{ess}}\left(Y,W\right)$, in the notation of the second paragraph of 3.5.2.
(p.317) Proof(1) Define $\rho :W\times \overline{Y}\to {\mathbb{K}}_{B}\left(Y,W\right)$ by $\rho \left(w,\overline{y}\right)\left(x\right)=w\u3008yx\u3009$, for x,y ∈ Yand w ∈ W. As in 8.1.19, we write 〈yx〉 as y*x, interpreted as a product in ℒ(Y). Let [w_{ij}], $\left[{z}_{ij}^{*}\right]$ and [x_{rs}] be matrices with entries in W, Ȳ, and Y respectively. Since W is a right hmodule (see 3.1.3), it is not hard to see that
(2) If Y is an equivalence ABbimodule, then A ≅ K _{B}(Y) by 8.1.15. It then follows from the argument for (8.1), that $C{B}_{B}^{\text{ess}}\left(Y,W\right)\subset {\mathbb{K}}_{B}\left(Y,W\right)$. Conversely, every ‘rank one’ operator w〉〈y in K _{B}(Y,W) is the limit of $w\u3009\u3008{e}_{t}y=w\u3009\u3008y{e}_{t}$ which lie in $C{B}_{B}^{\text{ess}}\left(Y,W\right)$. Here (e _{t})_{t} is as in 8.1.23. It follows that $w\u3009\u3008y\text{}\text{is}\text{}\text{}\text{in}\text{}C{B}_{B}^{\text{ess}}\left(Y,W\right)$, and so ${\mathbb{K}}_{B}\left(Y,W\right)\subset C{B}_{B}^{\text{ess}}\left(Y,W\right)$.
8.2.16 (Homtensor relations)
The tensor product identity in 8.2.15 (1) is a C*module variant of what is called a Homtensor relation in algebra. There are bimodule versions of this particular identity too, namely $W{\otimes}_{B}\overline{Y}\cong \mathbb{K}B\left(Y,W\right)$ as bimodules, if in addition Y and W are operator bimdules. Also, there is a matching lefthanded result: if Z is a right C*module over A, and W is a left operator Amodule, then $Z{\otimes}_{A}W{\cong}_{A}\mathbb{K}\left(\overline{Z},W\right)$. 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(ℓ^{2}) 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
(p.318) From 8.2.15 one may deduce further Homtensor relations such as:
One may also define the wellknown ‘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 Y ⊗_{min} Z (see 1.5.1) is a right C* module over A ⊗_{min} B with inner product determined by
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 ABbimodule, then Ȳ⊗_{A} Y ≅ B completely AAisometrically, and Y⊗_{B} Ȳ ≅ A completely BBisometrically. Conversely, if Y and X are respectively operator AB and BAbimodules, such that X ⊗_{hA} Y ≅ B completely BBisometrically, and also Y ⊗_{hB} X ≅ A completely AAisometrically, then A and B are strongly Morita equivalent, Y is an equivalence ABbimodule, and X ≅ Ȳ completely BAisometrically.
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 ABbimodule. then by 8.2.15 and 8.1.15, we have $Y{\otimes}_{B}\overline{Y}\cong {\mathbb{K}}_{B}\left(Y\right)\cong A$. Tracing through these identifications, one sees that the isomorphism holds as AAbimodules too. Similarly, $\overline{Y}{\otimes}_{A}Y\cong B$.
8.2.19(Induced representations and Morita equivalence)
Let Y be an equivalence ABbimodule. If Z is a left operator Bmodule, then G(Y) = Y ⊗_{B} Z (p.319) is an operator Amodule (see 3.4.9). If u: Z _{1} →Z _{2} is a completely contractive Bmodule map between left operator Bmodules, then I_{Y} ⊗ u: G(Z _{1}) → G(Z _{2}) is a completely contractive Amodule map (see 3.4.5). That is, G(—) = Y⊗_{B}— is a functor from the category B ^{OMOD} to the category A ^{OMOD} (see 3.5.1). Indeed the map u ↦ G(u) is linear and contractive on B^{CB}(Z _{1},Z_{2}). 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 $F()=\overline{Y}{\otimes}_{A}$ then F is a completely contractive functor from the category _{A}OMOD to the category _{B}OMOD, and F maps _{A}HMOD to _{B}HMOD, 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 Amodule W that
Thus if A and B are strongly Morita equivalent, then A^{OMOD} and B^{OMOD} are equivalent as categories (as are also A^{HMOD} and B^{HMOD}, 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 A^{OMOD} and B^{OMOD} are equivalent (via completely contractive functors). Moreover, if F: A^{OMOD} → B^{OMOD} is the equivalence functor, then X = F(A) is a strong Morita equivalence BAbimodule.
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 ABbimodule Y and an operator BAbimodule X, such that X ⊗_{hA} Y = B completely BBisometrically, and $Y{\otimes}_{hB}X\cong A$ completely AAisometrically. 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 ABbimodule. Thinking of modules W in B^{OMOD} as nondegenerate representations of B on W, we saw in 8.2.19 that such representations of B induce representations of A (on Y⊗_{B}W), and vice versa. Since ℒ(Y) is strongly Morita equivalent to B via the equivalence bimodule Y ⊕^{c} B (see 8.1.18), we see that there are onetoone 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 onetoone 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
By (8.12), (8.13), and 3.4.6, we have
The argument in and below (8.14) shows that any Hilbert Bmodule K induces a nondegenerate *representation π of ℒ(Y) on (Y ⊗_{B}K)⊕K. It is easy to check that π is cornerpreserving, in the sense of 2.6.15. Its ‘four corners’ consist of a representation of A on B(Y⊗_{B}K), maps from Y to B(K, Y⊗_{B}K) and from Ȳ to B(Y ⊗_{B}K,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 ABbimodule, and that K is a Buniversal Hilbert module (see 3.2.7), or equivalently, a generator for B^{HMOD} (see 3.2.8). By 8.2.19, Y induces an equivalence (p.321) of categories _{A}HMOD ≅ _{B}HMOD. By the simplest category theory, the induced represention of A on H = Y ⊗_{B} K is Auniversal. 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 ABbimodule Y, then there are onetoone lattice isomorphisms between the following lattices: (1) the twosided ideals I of A, (2) the twosided ideals J of B,(3) the ABsubmodules X of Y, and (4) the twosided ideals D of ℒ(Y). Rieffel showed that similar correspondences hold in the C*alogebra setting, with the word ‘normclosed’ 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 ↦ X_{J} = YJ. Conversely, define a map from lattice (3) to lattice (2) by X ↦J(X) = Y*X, for X in lattice (3). Using A.6.2, we have X = AX = YY*X. Thus X_{J(X)} = X. Given J in lattice (2), a similar argument shows that J(X_{J}) = 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,
Now we shall check that the lattices (2) and (4) are isomorphic, via the correspondence J ↦ D above. By 8.1.18, we may use the lattice isomorphisms we have already verified, but with Y and A replaced by Y ⊕^{c} B and ℒ(Y). Thus an ideal J of B corresponds to a ℒ(Y)Bsubmodule W of Y ⊕^{c} B, namely the submodule W = (Y⊕^{c} B)J = X ⊕^{c} 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 ABsunmodule of an equivalence ABbimodule 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 IJbimodula, 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 cornerpreserving. Write π = [π_{ij}], as in 2.6.15. It is straghtforward to check (p.322) that the 12corner W of ℒ(Y)/D is a TRO, and also is an equivalence bimodule over π_{11}(A) and π_{22} (B). For example, to see that W⋆W = π_{22}(B), observe that
Next, note that the ‘12corner’ π_{12} 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 12corner c(Y), it follows easily that π_{12} is a complete quotient map. Indeed, to see that it is a quotient map, suppose that we are given an element w ∈ W 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 12corner of W ^{′} is in Y, has norm <1,and is mapped by π_{12} to w.
Thus W is completely isometrically isomorphic to Y/X, the latter in the sense of 1.2.14. Similarly the 11corner of ℒ(Y)/D is *isomorphic to A/I, and the 22corner of ℒ(Y)/D is πisomorphic 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/Jvalued inner produet on Y/X, for example, is simply:
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×Y →Y (again called a triple product), such that Y is complete isometric to a TRO Z via a linear map θ: Y →Z which is a triple morphism, that is,θ ([x,y,z)] = [θ(x), θ(y),θ(z)] for all x,y,z ∈Y. 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 onetoone 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 x〈yz〉 = [xy]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 (HarrisKaup)
letθ:Y→ W 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 onetoone.
Proof (1) The amplification θ_{n}: 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 y ∈ Y, 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 z ∈Y such that
(2) Again, it suffices to prove that if θ in (1) is onetoone, 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) ∈ Y⋆Y.
For any z∈Y 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 z ∈ Y, 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 ABbimodule, and that X is a closed ABsubmodule 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 x ∈X and y,z ∈Y, 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:Y →Z 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 onetoone 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: Y→W between full c*modules or TROs, is a triple morphism if and only ifθ is the 12corner of a cornerpreserving *homomorphismπ: ℒ(Y) → ℒ(W) between the Morita linking C* algebras. In this case,θis onetoone(resp. Surjective)if and only if πis onetoone (resp. surjective).
Proof The first ‘if’ is clear. For the converse, We may suppose that θ is a triple morphism between TROs. Define $\rho \left({\sum}_{k=1}^{n}{x}_{k}{y}_{k}^{*}\right)={\displaystyle {\sum}_{k=1}^{n}\theta \left({x}_{k}\right)}\theta \left({y}_{k}\right)*$, for x _{1},…,x _{n},y _{n},…,y _{n} ∈Y. 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
It is clear that if π is onetoone(resp. surjective) then so is θ. If θ is onetoone, 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 onetoone. Similarly, π is onetoone. Since π is cornerpreserving, it is now easy to see that π is also onetoone. If θ is surjective, then ρ is surjective, since it has dense range in W⋆W. 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 12corner of a *isomorphism π between the Morita linking C*algebras, as in 8.3.5. The 11and 22corners 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,y∈Y.If Y is a TRO, then it is clear A is just the copy of YY⋆ in B(Y). That is,A = K _{Y⋆Y}(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 ABbimodule. We will see in 8.4.2 that A is a *subalgebra of A _{l}(Y) in the notation of 4.5.7, and B⊂A _{r}(Y).
Next we charcterize the possible right C*lmodule 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 〈〈yz〉〉 = σ(〈yz〉), for d∈σ(C), and y,z ∈ Y. 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: B → D as above.
We may now extend the notation introduced in 8.1.19 to triple systems. Thus products such as ${y}_{1}{y}_{2}^{*},y1{y}_{2}^{*}{y}_{\text{3}}{y}_{4}^{*}{y}_{5}$, 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 ${y}_{1}{y}_{2}^{*}{y}_{3}{y}_{4}^{*}{y}_{5}=\left[\left[{y}_{1},{y}_{2},{y}_{3}\right]{y}_{4},{y}_{5}\right]\in Y$. 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 ABbimodule as above.
8.3.7 (Nondegenerate triple morphisms)
If θ:Y → B(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 YY⋆Y, 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 Y⋆Y 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 (H⊕K) from 8.3.5 is nondegenerate. Note that this implies that if, further, θ is onetoone, then we may view M(ℒ(Y)) ⊂B(H⊕K), 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: X → Y 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),
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:Z→ Y, 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 qw ∘ i is a completely isometry on X, where qw: Y → Y/W is the canonical quotient triple morphism. This follows by applying the universal ptoperty with j = qw ∘ i. 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 Xequivalence defined above 8.3.9. Third, if (Z,j) is any triple extension of X, and if θ:Z → Y is the triple epimorphism provided by the universal property, and if W = Ker (θ), then (Z/W,qw ∘ j) is clearly Xequivalent to (Y,i). Thus by the second remark,(Z,W,qw ∘ j) may be taken to be a triple envelope of X. Here again, qw: Z → Z/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 _{w} ∘ j 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 CDbimodule. Let C ^{1} and D ^{1} be the unitizations of C and D respectively, and let ℒ^{1} (Z) be the ‘unitized linking C*algebra’ of 8.1.17:
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: Z → Z, 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 _{1} → X _{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 _{2} ∘ u) 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, $\mathcal{T}\left({\oplus}_{i}^{\infty}{X}_{i}\right)\cong {\oplus}_{i}^{\infty}\mathcal{T}\left({X}_{i}\right)$ 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 BanachStone theorem)
If A and B are unital operator algebras, and if v: A → B 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 $\mathcal{T}\left(A\right)={C}_{e}^{*}\left(A\right)$, and similarly for B. By 8.3.12 (2), one may ‘extend’ v to a triple isomorophism from ${C}_{e}^{*}\left(A\right)$ to ${C}_{e}^{*}\left(B\right)$, which we still write as v. If V(1) =U and v(a) =1, then
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 ℳ_{l}(X) and 𝒜_{l}(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 ℳ_{l}(X) ≅LM(X) as operator algebras. That is,
We next show that
By 8.2.4 and the last paragraph of 8.1.19, we also may regrad (as subalgebras)
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 𝒜_{r}(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 Amodule X is a closed Asubmodule, of a Brigged Z (see Asubmodule of a Brigged Amodule Z is a left Amodule.
Proof If X is a left operator Amodule, then by 4.6.2 (2) there is an associated *homomorphism θ: A → 𝒜_{l}(X). Using (8.17), θ may be viewed as a *homomorphism into B _{ℱ}(𝒯(X)). By (4.7) we have θ(e _{t})(Z) → Z, for Z ∈ Z. That is, 𝒯(X) is an ℱrigged Amodule. Clearly X is an Asubmodule of 𝒯(X).
Conversely, by 8.2.10 any Brigged Amodule Z is a nondegenerate left operator Amodule; and therefore so is any Asubmodule.
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: X → X, the following are equivalent:

(i) 𝒯 ∈ 𝒜_{l}(X).

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

(iii) There exists a map R: X → X, such that
Proof That (i) implies (ii) follows from 4.5.8(1), taking R= σ^{2212} (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 _{2} ∈ Y. 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 𝒜_{l}(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 a ↦ T _{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)
The C*algebra B _{l}(X) in the last proof is another useful description of A_{l}(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,𝒜_{l}(X) is a W* algebra, and any u ∈ 𝒜_{l}(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 onesided Mideals of Section 4.8. Indeed, a good deal of the results in the noncommutative Mideal 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 u ⊕ I_{Y} 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 u ⊕I_{X}: 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 ${\ell}_{2}^{\infty}$ module map projection C _{2}(B(H))≅B(H, H ^{(2)})→ C _{2}(I(X). Hence, by 3.6.2, we can extend u⊕I_{x} to a comletely contractive ${\ell}_{2}^{\infty}$ 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 = I_{X}. 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 Amoduoe map u: Y→ A is of the form u(·) = 〈z·〉, for some z ∈ y. 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 ℐ^{w} for the w*closure in M of the span of the range of the Mvalued 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 ℐ^{w} = 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)$$\begin{array}{cc}C{B}_{A}\left(Y,A\right)\cong \overline{Y}& \text{completely isometrically}\end{array}\mathrm{.}$$ 
(2) If Z is another right c*module over A, then
(8.19). Indeed, the fact that any u ∈ B_{A}(Y, z) is adjointable follows by considering the Avalued map 〈zu(·)〉, for fixed z ∈ Z.$${B}_{A}\left(Y,Z\right)={\mathbb{B}}_{A}\left(Y,Z\right),\text{}\text{}\text{}\text{}\text{and}\text{}\text{}\text{}{\mathbb{B}}_{A}\left(Y\right)={B}_{A}\left(Y\right)$$ 
(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 Bmodule. 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 let ℐ be as in 8.1.4. Then B_{A}(Y, A) = B_{C}(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 Amodule if and only if y is selfdual as a Dmodule, for any D ∈{A ^{1}, M(A),ℐ,M(ℐ)}.
Proof Suppose that u ∈ B_{ℐ}(Y, D)By cohen's theorem A.6.2, we may write any y ∈ Y as y = y _{1} a _{1} a _{2} for a _{1}, a _{2} ∈ ℐ. Hence u(y = u(y _{1})a _{1} a _{2} ∈ Aℐ⊂ ℐ. Moreover, if c ∈ C then u(yc = u(y _{1} a _{1}(a _{2} c)) = u(y _{1})a _{1}(a _{2} c) = u(y)c. Thus
Thus B_{C}(Y, D) = B _{ℐ}(Y, D) = B _{ℐ} 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. Then ℐ^{w} is the multipulier algebra of ℐ.
Proof Clearly ℐ a w*dense ideal in ℐ^{w}. Fix a faithful unital w*continuous *repersentation π of ℐ^{w} 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 {T ∈ B(H):Tπ(ℐ) ⊂ π(ℐ),π(ℐ)T ⊂ π(ℐ)}. In fact the latter algebra clearly cotains π(ℐ^{w}), and is a subalgebra of π(ℐ)″ by 2.6.5. On the other hand,
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.

(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 (x_{t})t converges to x in Y if and only if〈yx _{t}〉 → 〈yx〉 in the w*topology of M, for all y ∈ Y.

(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 otherhanded version of (8.18),
Define θ: W → Y* by θ(ψ ⊗ ȳ)(x) = ψ(〈yx〉), for ψ ∈ M _{*} and x, y ∈ Y. 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 Mvalued inner product is separately w*continuous. Let u:Y→ M be a bounded Mmodule map. By 8.1.23, we may choose a cai (e_{t})t for K _{M}(Y), with terms of the form ${\sum}_{k=1}^{n}{x}_{k}\u3009\u3008{x}_{k}$ for some x_{k} ∈ X. For x ∈ X, we have
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) B_{M}(Y) = B _{M}(y), and this is a W*algebra.

(2) A bounded net (T_{i})_{i} in B _{M}(Y) connverges in the w* topology to T∈ B_{M}(Y) if and only if T_{i}(y) → T(y) in the w*topology of Y, for all y∈Y. Indeed, Y ⊗̑_{M} W is a predual for B_{M}(Y), where W is as in 8.5.4(4).

Proof BY 3.5.10, 8.5.4(4), and (8.19), we have
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 𝒜_{l}(Y) = B _{M}(Y), and 𝒜_{r}(Y) = M(A) = M. By 4.7.4 we know that (p.335) 𝒜_{l}(Y) and 𝒜_{r}(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 × M → Y 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 w〈xy_{t}〉 → w〈xy〉 in the w*topology. Let $\left(\u3008x{y}_{{t}_{\mu}}\u3009\right)$ be a w*convergent subnet of the bounded net (〈xy_{t}〉), converging to b ∈ M say. By the last paragraph, we have that $w\u3008x{y}_{{t}_{\mu}}\u3009\to wb$ in the w*topology. Hence wb = w〈xy〉. Since this is true for all w ∈ Y, it follows from 8.1.4 (1) that b = 〈xy〉. Hence 〈xy _{t}〉 → 〈xy〉 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) = ℐ^{w}. Thus if Y has as operator space predual, then, by 8.5.6, Y is a W* module over ℐ^{w}. 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: Y → Z between W*modules over M, is w*continuous.
Proof BY (8.19), u is adjointable. We will apply A.2.5 (2). If (y_{t})_{t} is a bounded net converging to y ∈ Y in the w*topology of Y then
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)Mbimodule 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 ℒ^{w}(Y) = B_{M}(Y ⊕^{c} M). This equals B _{M}(Y ⊕^{c} M) by (8.19), since Y ⊕^{c} M is a selfdual Mmodule, as is easily verified. As in 8.1.17, by considering the adjointable inclusion and projection maps between Y ⊕^{c} M and its two summands, it is clear that ℒ^{w}(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 e ∈ N. 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 wellknown correspondence between twosided 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 _{1}⊕H _{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 Msubmodule, 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:Y→B(k,Y⊗_{M} k)from 8.2.14 satisfying Φ(Y)*Φ(Z)=〈Y〉, for y,Z∈Y;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 MNbimodules 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 MNbimodule, thaen M≅B_{N}(Y)*isomorphically.
Proof By the ‘lefthanded’ version of 8.5.3, and by 8.1.15,M≅M(K _{N}(Y)).
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, z ∈ Y, we have 0 = (1–p)y〉 〈(x) = (1–p)y〈zx〉. 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 B_{N}(Y)Nbimodule. More generally, if Y is a right W*module over Nthen Yis a W*equivalence B_{N}(y)ℐ^{w}bimodule, where ℐ^{w}is as above.
Conversely, if Y is a W*equivalence MNbimodule, then by 8.5.3 and the proof of 8.5.13, we have M≅M(K _{N}(Y)) and N≅M(ℐ). Since Yis an equivalence K _{N}(Y)ℐbimodule (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 ℒ^{w}(Y) are weakly Morita equivalent.
8.5.15 ( W*summands )
If Y is a right W*module over M then by 3.8.11, ${C}_{I}^{w}\left(Y\right)$ is a dual operator Mmodule for any cardinal I. It is easy to see that ${C}_{I}^{w}\left(Y\right)$ is also a W*module. For example, if Y is represrented, as in 8.5.11, both as an Msubmodule, 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 3.8.10 – 3.8.12) ${C}_{I}^{w}\left(Y\right)$ may be identified with an Msubmodule, 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×T↦S*T, It is then easy to see that the corresponding inner product on ${C}_{I}^{w}\left(Y\right)$ is givem by
Similar assertions hold for ${R}_{I}^{w}\left(X\right)$, 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 Msubmodule of Z,

(ii) Y is a w*orthogonally coplemented Msubmodule of Z,

(iii) Y is a w*closed Msubmodule of Z,

(iv) Y is a right Msummand 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 Msummands 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 Mmodule. 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 ABbimodule, then Y** is a W*equivalence bimodule over A** and B**.
proof First suppose that Y is full over B. We recall that Y≅pℒ(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*Y ⊂ Z*Z⊂N. 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×N→Z is separately w*continuous, and extends the canoical map Y×B→B. 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 twosided 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: Y→Z 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 u_{W}′,is a oneto one w*continuous triple morphism with Ran (u) = Ram(_{W}′). By 8.3.2, u_{W}′ 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: Y→B(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 Mideal of Z, then the w*clsure W of Y in Z** is a right Msummand of Z**. Viewed as subsets of Z**, we have
Conversely, if Y is a Bsubmodule 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 Msimmand of Z**, so that Y is a right Mideal of Z.
8.5.20 (C*module and Mideals)
In fact one may view the theory of onesided 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 Mideals (see 4.8.1) in an equivalence ABbimodule are exactly the ABsubmodules. One direction of this is not hard. For example, if Y is an ABsubmodule of an equivalence ABbimodule Z, then by 8.5.19, Y is both a left and a right Mideal of Z. By 4.8.4, Y is a complete Mideal, and heance an Mideal. The reverse direction seems to be harder. Suppose that Y is an Mideal in Z. If a∈A _{sa}, consider the map T_{Z} = az, for z∈Z. 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 Asubmodule, and similarly it is a right Bsubmodule. Hence the Midelas 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 = 〈uu〉 is an orthogonal projection in M. This element p is called the initial projection of u. Note that it follws that up = u (since〈uupuup〉 = 0). Thus u〉〈u is an orthogonal projection in the W*algebra B_{M}(Y).
(p.340) We say that two partial isometries u and υ in Y are orthogonal if 〉uυ〉 = 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 y ∈ Y. Then y = uy, where $\lefty\right={\u3008yy\u3009}^{\frac{1}{2}}\in M$, 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 u ∈ Y, that u*u ∈ M, 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 {x_{i}}_{i∈I} in Y consisting of mutually orthogonal nonzero partial isometries, such that x = ∑_{i} x_{i} 〈x_{i}x〉 in the w*topology of Y, for all x ∈ Y. In particular, ∑_{i∈I} x_{i}〉 〈x_{i} comverges in the w*topology of B_{M}(Y), to I_{y}.
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, {x_{i}: i ∈ I} say. We first claim that if 〈 x_{i}x〉 = 0 for all i ∈ I, then x = 0. To see this, write x = ux for a partial isometry u ∈ Y as in 8.5.22. Then 〈x_{i}u〉x = 0. if p is the initial projection of u, then u = up, and so 〈x_{i}u〉px = 0. Since p is the range projection for x, We see that 〈x_{i}u〉p = 0 = 〈x_{i}u〉. This contradicts the maximality above, if u ≠ 0.
By the remarks before 8.5.22, T_{J} = ∑_{i∈J} x_{i}〉 〈x_{i} is an orthogonal projection in B_{M}(Y), for any finite subset J ⊂ I. We therefore have
8.5.24 (The Parseval identity)
Suppose that {x_{i}: i ∈ I} is an orthonormal basis for Y, as in 8.5.23. It follows from the proof above, and 8.5.4 (1), that
As a consequence, we obtain another characterzation of W*modules, as exactly the w*closed submodules of ${C}_{I}^{w}\left(Y\right):$
(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 Misomorphis to an orthogonally complemented submodule of ${C}_{I}^{w}\left(M\right)$, 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 {x_{i}: i ∈ I} be an otthonormal basis for Y as in 8.5.23. Define α: Y → ${C}_{I}^{w}\left(M\right)$ by α(y) = (〈x_{i}y〉)_{i}, for y ∈ Y. 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 Mmodule 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 ${\oplus}_{i\in I}^{wc}{Y}_{i}$ of a family {Y_{i}: i ∈ I} of right W*modules over a W*algebra M, to be the set of (y_{i})_{i∈I} ∈ ∏_{i∈I} Y_{i}, such that the finite paratial sums of the series ∑_{i∈I}〈y_{i}y_{i}〉 are uniformly bounded above. Equivalently, it is the set of (y_{i})_{i}∈I such that ∑_{i∈I}〈y_{i}y_{i}〉 converges in the w*topology of M. It easy to check, using the polarization identity (1.1), that for (y_{i}) and (z_{i}) in ${\oplus}_{i\in I}^{wc}{Y}_{i}$, the finite partial sums of ∑_{i∈I}〈y_{i}z_{i}〉 converge in the w*topology of M. We write 〈(y_{i})(z_{i})〉 for the w*limit. Most of the conditions in the difinition fo a C*module are easy to check for ${\oplus}_{i\in I}^{wc}{Y}_{i}$, and all will follow from considerations in the next paragraph. Note that if Y is a right W*module, then the W*module ${C}_{I}^{w}\left(Y\right)$ 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 $Y={\oplus}_{i\in I}^{wc}{Y}_{i}$ in a slightly different way. We begin with a faithful normal repesentation of M on a Hilbert space K. We suppose that for each i ∈ I, Y_{i} is represented (as in 8.5.11 say) as a w*closed M submodule, and a subtriple, of B(K,H_{i}), for a Hilbert space H_{i}. Set H = ⊕_{i} H_{i}, and let P_{i} be the projection from H onto H_{i}. We also set W = {T ∈ B(K,H): P_{i}T ∈ Y_{i} for all i ∈ I}, and equip W with its canonical inner product S × T ↦ S*T. For S,T ∈ W we have S*T = ∑_{i} S*P_{i}P_{i}T, 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 y_{i} = P_{i}T, for T ∈ W it becomes evident that W corresponds precisely to the difinition of ${\oplus}_{i\in I}^{wc}{Y}_{i}$ above. In other words, W is unitrarily Misomorphic to this sum. It follows that ${\oplus}_{i\in I}^{wc}{Y}_{i}$ 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, ${\oplus}_{i\in I}^{wc}\left({\oplus}_{i\in J}^{wc}{Y}_{ij}\right)\cong {\oplus}_{i\in J}^{wc}\left({\oplus}_{i\in J}^{wc}{Y}_{ij}\right)\cong {\oplus}_{i,j}^{wc}{Y}_{ij}$ unitarily, for right W*modules Y_{ij} 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 (Y_{i}) is a family of C*modules over a C*algebra B, then ${\left({\oplus}_{i}^{c}{Y}_{i}\right)}^{**}\cong {\oplus}_{i}^{wc}{Y}_{i}^{**}$ unitarily as B**modules. We merely sketch the proof. Let $Y={\oplus}_{i}^{c}{Y}_{i},\text{}\text{let}\text{}Z={\oplus}_{i}^{wc}{Y}_{i}^{**}$, and let ℒ be the ‘augmented linking algebra’ K _{B}(Y⊕^{c} B). Let p _{0} be the projection of Y ⊗^{c} B onto 0 ⊕ B, and similarly let p_{i} be the projection of Y ⊕^{c} B onto the copy of Y_{i}. We regard p_{i} ∈ M (ℒ) ⊂ ℒ**. In the latter W*algebra one can show that ∑_{i}p_{i} = (1−p _{0}). As in the proof of 8.5.17, we have Y_{i} ≅ p_{i}ℒp_{0}, and Y ≅ (1−p _{0})ℒp _{0}, unitarily as right Bmodules. Also as in the proof of 8.5.17, we have ${Y}_{i}^{**}\cong {p}_{i}{\mathcal{L}}^{**}{p}_{0}$ 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
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 ${C}_{I}^{w}\left(Y\right)\cong {C}_{I}^{w}\left(N\right)$ unitarily (as right Nmodules). Also, M _{I} (Y) is linearly completely isometrically isomorphic (via a right Nmodule 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 Nsubmodule W of ${C}_{I}^{w}\left(N\right)$, such that $Y{\oplus}^{c}W\cong {C}_{I}^{w}\left(N\right)$ unitarily (as right Nmodules). By set theory we may assume that I ^{2} = I. It follows from this, and (1.59) for example, that ${C}_{I}^{w}\left({C}_{I}^{w}\left(N\right)\right)\cong {C}_{I}^{w}\left(N\right)$. 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:
(p.343) The last assertion follows since
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 ${C}_{I}^{w}\left({Y}^{**}\right)\cong {C}_{I}^{w}\left({B}^{**}\right)$ 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 ‘otherhanded’ 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 ABbimodule. 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 H ⊗ K, 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(H ⊗ K). 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 y ∈ Y. This last equation provides a map π: B′ → A′, defined by π(S) = R. That is,
Summarizing, we saw that the commutant of ℒ(Y) is the set of matrices
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 ABsubmodule 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 Nrigged Mmodule 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 MNcorrespondences.
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 Nrigged Mmodule if and only if it is a normal dual operator MNbimodule in the senes of 3.8.2.
Proof If Z is a normal Nrigged Mmodule, then by 8.5.9 it is a normal dual operator B _{N}(Z)Nbimodule. Since the left action comes from a normal *homomorphism θ:M → B _{N}(Z), Z is a normal dual MNbimodule.
Conversely, if Z is a normal dual operator MNbimodule, then as in the proof of 4.7.6, there is a normal *homomorphism θ: M → 𝒜_{l}(Z). However 𝒜_{l}(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 Y ⊗_{M}H is a normal Hilbert Nmodule (in the sense of 3.8.5). To see this, suppose that (a_{t})_{t} is a bounded net in N converging in the w*topology to an a ∈ N. Then by 8.5.35, a_{t}y → ay in the w*topology of Y, for any Y ∈ Y. By 8.5.4 (3), if ζ, η ∈ H and y, z ∈ ∈ Y, then
By 8.2.13, if, further, M is faithfully represented on H, and of the canonical map from N into B _{M}(Y) is onetoone, then N is faithfully represented on the Hilbert space Y ⊗_{M} H. We shall not use this, but it follows easily from A.6.2 that of ℐ is as in 8.1.4, then Y ⊗_{M} H = Y ⊗_{ℒ}H.
Corollary 8.5.37 (Rieffel)
Suppose that π: M → B(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 y ⊗_{M} K), such that Y ≅ _{R}B(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 _{R}B(K,H) of B(K,H), with its canonical B(K)valued inner product, is a right Wmodule over π(M).
In fact the isomorphism in (1) is a unitary Mmodule 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 = Y ⊗_{M} K. Similarly, using also the definition of ℒ_{W}(Y) in 8.5.10, we have a faithful normal *representation of ℒ_{W}(Y) = B _{M}(Y ⊕^{c} M) on the Hilbert space(Y ⊕^{c} M) ⊗_{M} K. The latter space, as in (8.14), is unitarily equivalent to (Y ⊗_{M} K) ⊕ K = H ⊕ K. Just as in the last paragraph of 8.2.22, the associated normal representation of ℒ^{w} (Y) on H ⊕ K is onetoone (and hence completely isometic, by 1.2.4) and cornerpreserving, and its ‘12corner’ is a map from Y to B(K,H). Clearly the latter map is completely isometric and wclosed 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 Msubmodule 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 ℐ^{w} = pM for such a projection p. Let K′ = π(p)K, and apply the previous case to the canonical representation of ℐ^{w} on K′. Write θ for this last representation, and let N = θ(ℐ^{w})′. We obtain a Hilbert space H on wich N is normally represented, such that Y ≅ _{N}B(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 Rmodules, and it is easily checked that _{R}B(K,H) ≅ _{N}B(K′,H) unitarily as Mmodules via the map T ↦ T _{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: M → B(K) and ρ: N → B(H), with π(M)′ ≅ ρ(N)′ *isomorphically. Moreover, in this case, writing R for π(M)′ and for ρ(N)′, the TRO _{R}B(K,H) in 8.5.37 (2) is a W*equivalence NM bimodule.
Proof If Y is a W* equivalence NMbimodule, 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 Y ⊗_{M}K, 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 = _{R}B(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 ℐ^{w} of Y⋆Y is a w*closed ideal in π(M). Thus there is a central projection e in π(M) with ℐ^{w} = eπ(M). Since Y = Yℐ^{w} (e.g. see 8.1.4 (2)), we have Y = Y e. Since e ∈ π(M), (1 −e)K is an Rmodule. Let P be 1 − e, viewed as a map from K onto (1 − e)K. Since e ∈ R′, p is an Rmodule map. Since H is a cogenerator of _{R}NHMOD (see 3.8.6), if (p.347) (1 − e)K ≠ (0) then there exists a nonzero Rmodule map T: (1 − e)K → H. Thus TP ∈ Y, 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 Mmodule 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 = Y ⊗_{M} 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 Y ≅ B_{M}(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 B_{M}(L ^{2}(M),H). One may show that B_{M}(L ^{2}(M),H) ⊗_{M} L ^{2}(M) ≅ H unitarily. Indeed it is easy to see that the canonical map from B_{M}(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 Mrigged Nmodules (see 8.5.34). A correspondence between N and M is a Hilbert space H which is a normal NMbimodule as above. Thus, such Hilbert space are in a bijective relation with the normal Mrigged Nmodules. 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 Nrigged Mmodule 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 Σ_{i∈I} x_{i} ⊗ y_{i}, 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)
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 8.2.11– 8.2.19.
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 ‘OSnuclwarity’, ‘OSsemidiscreteness, 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 OSnuclear oprator space X has the WEP, or the fact from 7.1.9 that an OSsemidiscrete dual operator space is injective. In fact, for a finitedimensional opertor spaceX, all of the following properties are equivalent: injectivity, OSnuclearity, OSsemidiscreteness, 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 M_{n} (that is, there exists a completely isometry from X onto a subspace W of M_{n}, and a completely contractive projection from M_{n} 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 finitedimensional TRO X, the linking C*algebra is ℐ(X) is finitedimensional. By (8.3), X is completely contractively complemented in ℐ(X). However any finitedimensional TRO X, the linking C*algebra ℒ(X) is finitedimensional. By (8.3), X is completely contractively complemented in ℒ(X). However any finitedimensional C*algebra is completely contractively complemented in some M_{n}, so that X is completely contractively complemented in M_{n} 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 OSsemidiscrete,

(iii) X has the WEP,

(iv) X is completely isometrically isomorphic and w*homeomorphic to a ‘corner’ pM(1 − p), for an injective W*algebra M, and a projection P ∈ M.
(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 OSsemidiscrete (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 MNbimodule, over W*algebras M and N say. By the proof of