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Operator Algebras and Their ModulesAn operator space approach$

David P. Blecher and Christian Le Merdy

Print publication date: 2004

Print ISBN-13: 9780198526599

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198526599.001.0001

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(p.359) Appendix

(p.359) Appendix

Source:
Operator Algebras and Their Modules
Publisher:
Oxford University Press

A.1 OPERATORS ON HILBERT SPACE

We begin by reviewing a few basic facts about operators on Hilbert space, which may be found in almost any book on functional analysis.

A.1.1

If S, T are contractive linear operators between Hilbert spaces such that ST = I, then it follows that S = T*, and T is an isometry and S a coisometry. If in addition TS = I, then T is a unitary. If P is an idempotent operator on a Hilbert space then P is a projection (i.e. P = P*) if and only if ‖ P ‖ ≤ 1.

A.1.2

If H is a Hilbert space then the space S (H) of compact operators is a norm closed (two-sided) ideal in B(H). We write S 1(H) for the usual trace class, a (two-sided) ideal in S (H) (and also in B(H)), and which is a Banach space with respect to the trace class norm ‖ T1 = tr| T |. The trace tr is a contractive functional on S 1(H), and via the dual pairing (S, T) ↦ tr(ST) it is well-known that S (H)* ≅ S 1(H) and S 1(H)* ≅ B(H) isometrically.

From this it is evident that the product on B(H), viewed as a map from B(H) × B(H) to B(H), is separately w*-continuous. That is, if StS in the w*-topology on B(H), then StTST and TStTS in the w*-topology too. The w*-topology on B(H) is also called the σ-weak topology. A linear functional on B(H) is σ-weakly continuous if and only if it is of the form k = 1 < ζ k , η k > , for ζk, ηkH with k = 1 ζ k 2 and k = 1 η k 2 finite. By such considerations, the involution * on B(H) may also be seen to be w*-continuous.

A.1.3

More generally if H, K are Hilbert spaces, we let S (H, K) denote the compact operators from H into K. For any 1 ≤ p < ∞, we let Sp(H, K) denote the Schatten p-class of compact operators T: HK such that |T|p belongs to S 1(H). This is a Banach space for the norm T p = ( t r | T | p ) 1 p . The following ideal property holds: for any TSp(H, K), V 1B(K), V 2B(H), the operator V 1 TV 2 belongs to Sp(H, K) and ‖V 1 TV 2p ≤ ‖V 1‖‖TpV2‖.

A.1.4

We write WOT for the weak operator topology. This topology makes the map T ↦ <Tζ,η> continuous on B(H), for all ζ, η ∈ H. On bounded sets the WOT and σ-weak topologies coincide. Thus a bounded net in B(H) converges in the WOT topology if and only if it converges in the w*-topology.

A.1.5

A subspace XB(H) is said to be reflexive if

X = { T B ( H ) : T ζ [ X ζ ] for all ζ H } .
(p.360) We write T for the operator TT⊕ … on H (∞), and X (∞) = {T : TX}. If W is a w*-closed subspace of B(H), then W (∞) is reflexive in B(H (∞)). To see this, suppose that Tζ ∈ [W (∞)ζ] ⊂ [B(H) (∞)ζ], for all ζ ∈ H (∞). By first setting ζ = (0, …, 0, η, 0, …), and then ζ = (η, …, η, 0, …), it is easy to argue that T = S for some SB(H). Let φ ∈ W . By A.1.2, there exist vectors ζ,η ∈ H(∞). such that φ(R) = <R ζ,η>, RB(H). Then φ(S) = <Tζ,η> = 0, since Tζ∈ [W (∞)ζ], and φ ∈ W . Thus S ∈ (W⊥) = W, so that TW (∞). (E.g. see [354,102,108] for more on this topic.)

A.2 DUALITY OF BANACH SPACES

In this section and the next, E and F are Banach spaces. We write i E: EE** for the canonical embedding. However we often suppress this map and simply consider E as a subspace of E**.

Lemma A.2.1 (Goldstine)

Ball(E) is w*-dense in Ball(E**).

Lemma A.2.2

Let u: EF* be a bounded linear map. Then there exists a unique w*-continuous ũ: E** → F* extending u. Moreoverũ‖ = ‖u‖.

Proof Set u ˜ = i F * u * * where iF: FF** is the canonical isometry. □

Lemma A.2.3

Let E be a closed linear subspace of a Banach space F.

  1. (1) As subsets of F** we have Ē w* = E ⊥⊥.

  2. (2) The second dual of the inclusion map EF is an isometry from E** onto E⊥⊥. Thus E** ≅ E ⊥⊥ isometrically, via this canonical isometry.

  3. (3) (F/E)** ≅ F**/E ⊥⊥ isometrically, and w*-w*-homeomorphically, via the ‘transpose’ of the canonical isomorphism (F/E)* → E . This is the same as the map obtained from q**, where q: FF/E is the canonical quotient map, by factoring out Ker (q**) = E ⊥⊥.

  4. (4) FE ⊥⊥ = E.

Proof Items (1), (2), and (3) are in the standard sources. For (4), notice that EE ⊥⊥F**. If yF but yE, choose φ ∈ F* such that φ(E) = 0, but φ(y) ≠ 0. If yE ⊥⊥ there exists by (1) a net (xt)t in E with xty in the w*-topology of F**. Hence 0 = φ(xt) → φ(y) ≠ 0, a contradiction. □

We will also (silently) use the following simple principle many times:

Lemma A.2.4

If T: EF is a w*-continuous map between dual Banach spaces, and if W is a w*-closed subspace of Ker(T), then the induced map from E/W to F is w*-continuous.

Theorem A.2.5 (Krein–Smulian)

  1. (1) Let E be a dual Banach space with predual E *, and let F be a linear subspace of E. Then F is w*-closed in E if and only if Ball(F) is closed in the w*-topology on E. In this case F is also a dual Banach space, with predual E */F , and the inclusion of F in E is w*-continuous.

  2. (p.361)
  3. (2) If uB(E, F), where E and F are dual Banach spaces, then u is w*-continuous if and only if whenever xtx is a bounded net converging in the w*-topology in E, then u(xt) → u(x) in the w*-topology.

  4. (3) Let E and F be as in (2), and u: EF a w*-continuous isometry. Then u has w*-closed range, and u is a w*-w*-homeomorphism onto Ran(u).

Proof Items (1) and (2) may be found in the standard texts; (2) is often stated for functionals φ but the result as stated here follows from this by considering φ ∘ u. For (3), note that it is easy to check using (1) that Ran(u) is w*-closed in F. Thus the restriction of u to Ball(E) takes w*-closed (and thus w*-compact) sets to w*-compact (and thus w*-closed) sets in Ran(u). Thus the inverse of u restricted to the ball is w*-continuous, so u −1 is w*-continuous by (2). □

A.3 TENSOR PRODUCTS OF BANACH SPACES

We review a few facts about tensor products of Banach spaces E and F, whose proofs may be found in many texts (see [118, 121, 324, 407], for example).

A.3.1

If (x k)k and (y k)k are finite families in E and F respectively, then one may define for z = ∑k x ky k in the algebraic tensor product EF, the quantity

k x k y k = sup { | k φ ( x k ) ψ ( y k ) | : φ Ball ( E * ) , ψ Ball ( F * ) } .
This is a norm on EF. The completion of EF in this norm is called the injective tensor product and written as E ⊗̆ F. This tensor norm gets its name from the fact that it has the injective property. Namely, if u i: E iF i are isometries for i = 1,2, then the corresponding map u 1u 2: E 1 ⊗̆ E 2F 1 ⊗̆ F 2 is an isometry too. More generally if u 1, u 2 are contractive then so is u 1u 2.

We remark that the definition and facts in the last paragraph have obvious variants for the N-fold injective product X 1 ⊗̆ … ⊗̆ X N of any N-tuple of Banach spaces. There is an associativity law: for example, if N = 3 then we have that X 1 ⊗̆ X 2 ⊗̆ X 3 = (X 1 ⊗̆ X 2) ⊗̆ X 3 = X 1 ⊗̆ (X 2 ⊗̆ X 3).

We may identify any element z = ∑k x ky k as above with a bounded operator u: F* → E, namely u(ψ) = ∑kψ(yk)x k for any ψ ∈ F*. We say that u is associated with z. Under this identification, EF coincides with the space of all finite rank and w*-to-norm continuous operators from F* into E. Clearly

(A.1)
E ̆ F B ( F * , E ) isometrically .
Of course E ⊗̆ FB(E*, F) isometrically too. Likewise if F is a dual space with predual F *, we may identify EF with finite rank operators from F * into E and we have E ⊗̆ FB(F *, E), isometrically.

A.3.2

Let Ω be a compact space. We let C(Ω; E) denote the Banach space of all continuous functions f: Ω → E. Equip C(Ω; E) with the supremum norm, that is, ‖f‖ = sup{‖f(t)‖E: t ∈ Ω}. We simply write C(Ω) for C(Ωℂ). We may (p.362) identify C(Ω) ⊗ E with a subspace of CE) by regarding any f = ∑k gkxk (with gkC(Ω) and xkE) as a function, f(t) = ∑k gk(t)xk. Then the norm of f in CE) is equal to sup{|∑k gk(t)φ(xk)|: t ∈ Ω, φ ∈ Ball(E*), hence is equal to its injective tensor norm by (A.1). Moreover C(Ω) ⊗ E is dense in C(Ω; E) (e.g. see [407, IV; 7.3] for a proof) hence we have

(A.2)
C ( Ω ) ̆ E C ( Ω ; E ) isometrically .
Similarly if Ω is a locally compact space, we let C 0(Ω; E) denote the Banach space of all continuous functions from Ω to X vanishing at ∞, equipped with the supremum norm. Then (A.2) extends to the relation C 0(Ω)⊗̆EC 0(Ω; E).

A.3.3

A bounded bilinear map T: E × FZ is a bilinear map for which there is a constant C such that ‖T(x, y)‖ ≤ Cx‖‖y‖, for all xE, yF. The least such C is written as ‖T‖. We say that T is contractive if ‖T‖ ≤ 1.

The Banach space projective tensor product E ⊗̂ F is the completion of the algebraic tensor product EF in a certain norm. We do not need to explicitly write down this norm, instead we will simply state the universal property of E ⊗̂ F, namely that it linearizes bounded bilinear maps. More precisely, the canonical map ⊗: E × FE ⊗̂ F is a contractive bilinear map, and for any bounded bilinear T: E × FZ, the associated linear map EFZ is continuous with respect to the just mentioned norm, and extends to a bounded linear map T̃ E ⊗̂ FZ with ‖T̃‖ = ‖T‖. From this it is easy to see that

(A.3)
B ( E ̂ F , Z ) B ( E , B ( F , Z ) ) B ( F , B ( E , Z ) ) isometrically .
In particular, via the obvious isomorphisms,
(A.4)
( E > ^ F ) * B ( E , F * ) B ( F , E * ) isometrically .

A.3.4

A bounded operator u: EF is said to be 2-summing if I l2u extends to a bounded operator from l 2⊗̆E into l 2(F). We set

π 2 ( u ) = I 2 u : 2 ̆ E 2 ( F ) .

It is not hard to check that π2(·) is a (complete) norm on the space Π2(E, F) of 2-summing operators from E into F. We also note that π2(u) is the supremum of the norms of the mappings I n 2 u : n 2 ̆ E n 2 ( F ) , for n ∈ ℕ.

A.3.5

We review three tensor products related to Hilbert space factorization. Let (ek)k denote the canonical basis of l 2. For zEF define

(A.5)
γ 2 ( z ) = inf { k = 1 n e k x k 2 ̆ E k = 1 n e k y k 2 ̆ F }
where the infimum is over all finite families ( x k ) k n in E and ( y k ) k n in F such that z = k = 1 n x k y k . The quantity γ2 is a norm on EF, and we let Eγ2 F denote the resulting completion.

(p.363) Consider k = 1 n e k x k for x 1,…,xnE as above, and let u: E* → ℓ 2 be the associated linear map. According to (A.1), k = 1 n e k x k 2 ̌ E is equal to the usual operator norm of u. Analoguously, we define k = 1 n e k x k 2 ( E * , 2 ) to be π2(u). Then for zEF we define

(A.6)
g 2 ( z ) = inf { k = 1 n e k x k 2 ( E * , 2 ) k = 1 n e k y k 2 ̌ F }
where the infimum is over ways to write z = k = 1 n x k y k , with xkE, ykF. As above, g 2 is norm, and we let Eg2 F denote the resulting completion.

Next we let

(A.7)
γ 2 * = inf { k = 1 n e k x k Π 2 ( E * , ) k = 1 n e k y k Π 2 ( F * , 2 ) } ,
where again the infimum over all ways to write z = k = 1 n x k y k , in EF. Again, γ 2 * is a norm and we let E γ 2 * F denote the resulting completion.

It is not hard to see that γ 2 ( ) g 2 ( ) γ 2 * ( ) on EF.

A.3.6

Let H, K be hilbert spaces. Then the Space KH̅ of finite rank operators from H to K is dense both in S (H, K) and in S 1 (H, K). It is clear from (A.1) that S (H, K) ≅ K ⊗̌ H̅, and it turns out that

(A.8)
S 1 ( H , K ) K ^ H ¯ K γ 2 * H ¯ .

A.4 BANACH ALGEBRAS

We refer the reader in this section to [74, 106, 297] for any omitted proofs if assertions below, or for more background.

A.4.1

A C-Banach algebra is a Banach space A which is also an algebra such that ‖ab‖ ≤ Ca‖‖b‖ for all a, bA. If C = 1 then we simply say Banach algebra. We say that a Banach Algebra A is unital if it has a unit (i.e. identity) of norm 1. A bounded approximate identity is a bounded net (et)t with etaa and aeta. This is a contractive approximate identity (cai) if moreover ‖et‖ ≤ 1 for all t. If A Possesses a Cai we say that A is approximately unital.

A.4.2

Let A be a unital Banach algebra. A state on A is a contractive unital functional on A. An element hA is said to be Hermitian if φ(h) ∈ ℝ for every state φ on A. Note that by the Hahn–Banach theorem we may replace states on A here by contractive unital functionals on Span {1,h}. Equivalently, h is Hermitian if ‖exp(ith)‖ ≤ 1 for all t ∈ ℝ. We write Her (A) for the set of Hermitian elements of A. By the first definition of Hermitians above, it is evident that if u: AB is a unital contractive linear map between unital Banach algebras, then u(Her(A)) ⊂ Her(B).

(p.364) It is well-known that if ϕ(h) = 0 for all states ϕ on A, then h = 0. Also, if A is a unital C*-algebra, then Her(A) is exactly the set Asa of selfadjoint elements.

A nonzero homomorphism on a unital Banach algebra is a state, and is called a character. The maximal ideal space MA of a commutative unital Banach algebra A is the set of characters of A, together with the w*-topology inherited from A*.

A.4.3

Suppose that A is an approximately unital Banach algebra. We may define a unitization of A by considering the canonical ‘left regular representation’ λ: AB(A), and identifying A + C1 with the span of λ(A) + C IA, which is easy to see is a unital Banach subalgebra of B(A). Thus if aA and α ∈ C then

(A.9)
a + α 1 = sup { a c + α c : c A , c 1 } .

We write this unitization as A 1 if A is nonunital.

It is occasionally useful that there are some other equivalent expressions for the quantity above. For example, if (et)t is a cai for A then

(A.10)
a + α 1 = lim t a e t + α e t = sup t a e t + α e t .
To see that this limit exists and that these quantities are the same, let β be the quantity on the right-hand side of (A.9), let ε > 0 be given, and choose c ∈ Ball(A) with ‖ac + αc‖ > β − ε. Then ‖aetc + αetc‖ → ‖ac + αc‖, so that there is a t 0 with ‖aetc + αetc‖ > β − ε for tt 0. Then
β < a e t c + α e t c a e t + α e t β .
This proves what we asserted. One can see that one does not change the quantities in (A.9) and (A.10) by considering expressions ca + αc or eta + αet.

As a consequence of (A. 10), if π: AB is a contractive homomorphism between Banach algebras, such that (π(et))t is a cai for B for some cai (et)t for A, then π extends uniquely to a contractive unital homomorphism π̃ between the unitizations. To see this define π̃(a + α1) = π(a) + α1, for aA and α ∈ C, and appeal to formula (A.10) twice to see that π̃ is contractive too. If, further, π is isometric, then by (A. 10) it follows that π̃ is isometric too.

A.5 C*- ALGEBRAS

We refer the reader in this section to any book on C*-algebras for any omitted proofs of assertions below, or for more background.

A.5.1

A concrete C*-algebra is a closed *-subalgebra of B(H) for a Hilbert space H. A von Neumann algebra is a concrete C*-algebra which is closed in the w*-topology on B(H), and which contains IH. By A.1.2 it follows that the product on a von Neumann algebra is separately w*-continuous in each variable, and that the involution is also w*-continuous. An (abstract) C*-algebra is a Banach algebra A with a conjugate linear involution *: AA such that (a*)* = a and (ab)* = b*a*, for all a, bA, which also satisfies the C*-identity: ‖a*a‖ = ‖a2, for aA. A C*-subalgebra of a C*-algebra is a closed selfadjoint subalgebra.

(p.365) A.5.2

An element a in a C*-algebra A is positive if a = b*b for some bA. We write A + for the set of such elements, and write ab if baA +, and if a and b are selfadjoint. We will assume familiarity with the basic properties of this ordering and the continuous functional calculus for normal operators.

A C*-algebra is approximately unital, and indeed has a positive increasing cai. The unitization A 1 in A.4.3 of a C*-algebra A is a C*-algebra.

A functional ϕ ∈ A * is called a state if it is ‘positive’ (i.e. ϕ(a) ≤ 0 if a ≤ 0) and has norm 1. This is equivalent to other definitions of states elsewhere in this book. Indeed there is a host of equivalent definitions of states, or indeed of elements in A +. For example,

(A.11)
a 0 ϕ ( a ) 0 for all states ϕ of A ( or of A 1 ) .

A *-homomorphism (resp. *-isomorphism) is a homomorphism (resp. isomorphism) satisfying π(a)* = π(a)* for all a. Such maps are automatically positive.

A.5.3

A primary result in the subject of operator algebras is the Gelfand–Naimark theorem, which states that every abstract C*-algebra A is *-isomorphic to a concrete C*-algebra. A major part of the proof of this result is the Gelfand–Naimark–Segal (GNS) construction, which shows that the positive functionals ϕ on a C*-algebra A are the functions of the form <π(·)ζ,ζ>, for a *-homomorphism π: AB(H), and a vector ζ ∈ H, such that [π(A)ζ] = H. If ϕ is a state then we can take ‖ζ‖ = 1.

A W*-algebra is an abstract C*-algebra with a Banach space predual, such that there exists a w*-continuous *-isomorphism from A onto a von Neumann algebra. By a theorem of Sakai, the last part of this definition is redundant, but we will avoid using this deeper fact here.

A.5.4

The commutative C*-algebras, are by a theorem of Gelfand, exactly the spaces C 0(Ω) mentioned in A.3.2. We will assume that the reader is familiar with the basic correspondences between constructions in the category of compact spaces K, and the category of commutative unital C*-algebras. For example, the correspondences between closed subsets of K, and ideals of C*-algebras and their quotients. Or, more generally, the correspondence between continuous (resp. and one-to-one, and surjective) functions between compact spaces, and unital (resp. and surjective, and one-to-one ) *-homomorphisms between commutative unital C*-algebras. These correspondences follow easily from Gelfand's theory. Recall that algebraic isomorphisms between C(K)-spaces (resp. between closed subalgebras of C(K)-spaces containing constant functions), are *-isomorphisms (resp. isometric).

A.5.5

The universal representation πu: AB(Hu) of a C*-algebra A is constructed by taking a direct sum of *-homomorphisms associated with all the states on A by the GNS construction (see A.5.3 above). Thus πu has the property that for any state ϕ of A, there exists a unit vector ξ ∈ Hu such that ϕ = <πu(·)ξ,ξ>.

(p.366) Every nondegenerate *-homomorphism from A to B(H), for any Hilbert space H, is unitarily equivalent to the restriction to an invariant subspace of a direct sum of sufficiently many copies of πu.

Theorem A.5.6

The second dual of a C*-algebra A is a w*-algebra. Indeed A** is linearly isometric, via a w*-continuous map, to a von Neumann algebra.

Proof Let πu: AB(Hu) be the universal representation of A (see A.5.5). Let π u ˜ : A * * B ( H u ) be the unique (contractive) w*-continuous extension of πu as in A.2.2 (thus π u ˜ = i * π u * * , where i: S 1(Hu) = B(Hu)*B(Hu)* is the canonical injection). Let ψ be a linear functional on A of norm 1. It is well-known that we may write ψ = <π(·)ξ,ξ′>, for unit vectors ξ,ξ′ and a *-homomorphism π (see 1.2.8 for an easy proof of this fact; also Zsido has recently shown us a beautiful simple proof that will be in [401]). For aA we have that

| ψ ( a ) 2 | π ( a ) ξ 2 = < π ( a * a ) ξ , ξ > ,
from which we see that the positive map ϕ = <π(·)ξ,ξ> has norm 1, and is therefore a state on A. Thus by A.5.5, ϕ = <πu(·)η,η> for a unit vector η ∈ Hu. We now have |ψ(a)|2 ≤ <πu(a * a)η,η> = πu(a)η‖2. Thus the functional πu(a)η ↦ ψ(a) on [πu(A)η] is well defined and contractive. By the Riesz representation theorem, there exists a vector ζ ∈ Hu such that ψ = <πu(·)η,ζ>. Hence, since A is w*-dense in A**, we have <ν,ψ> = <π˜u(ν)η, ζ> for ν ∈ A**. We deduce that |<ν,ψ>| ≤ ‖π˜u(ν)‖, which implies that π˜u is an isometry. Since A is w*-dense in A**, we have π u ˜ ( A * * ) π u ( A ) ¯ w * . By the Krein–Smulian theorem A.2.5, the range of π u ˜ is w*-closed. Hence π u ( A ) ¯ w * π u ˜ ( A * * ) , and so π u ( A ) ¯ w * = π u ˜ ( A * * ) . Thus A** is linearly isometric to the von Neumann algebra π u ( A ) ¯ w * . □

A.5.7

By the last result together with an observation in A.5.1, we see that if A is a C*-algebra then A** possesses a product extending that of A, which is separately w*-continuous in each variable, and with respect to which A** is a w*-algebra. By Goldstine's lemma A.2.1 such a product on A** must be unique. We call this the canonical w*-algebra structure on A**.

Proposition A.5.8

Let π: AB be a homomorphism between C*-algebras. Then π is contractive if and only if π is a *-homomorphism. If these hold then first, π has closed range, and induces a *-isomorphism between the C*-algebras A/Ker(π) and π(A). Second, π is isometric if and only if π is one-to-one.

Proof Most of these may be found in any book on C*-algebras (and will be generalized in 8.3.2). We merely sketch the part that cannot be so easily found, namely that a contractive homomorphism π is a *-homomorphism. If A is unital then we can assume that B is unital and that π(1) = 1 (otherwise replace B by π(1)Bπ(1)). If aA+ and if φ is a state on B, then φ o π is a state on A. Using (A.11) twice shows that ϕ(π(a))) ≤ 0, and that π(a) ≤ 0. Thus π is positive, and is therefore also a *-homomorphism. In the nonunital case extend π (p.367) to a contractive homomorphism between the unitization C*-algebras (or simply consider π**: A** → B**, and use the results on second duals above), and then apply the ‘unital case’ above to obtain the result.

Theorem A.5.9

[409,39] Let A be a C*-algebra, B a Banach algebra, and π: AB a contractive homomorphism. Then π(A) is norm closed, and it possesses an involution with respect to which it is a C*-algebra. Moreover, π is then a*-homomorphism into this C*-algebra. If π is one-to-one then π is an isometry.

Proof We may assume that B is the closure of π(A). If A is unital then B is unital, and π(1A) = 1B. Since π(Her(A)) π Her(B) (see A.4.2),

π ( A ) = π ( Her ( A ) ) + i π ( Her ( A ) ) Her ( B ) + i Her ( B ) B .
Hence Her(B) + iHer(B) is dense in B. By the Vidav–Palmer theorem [74,298] B is a C*-algebra (in fact Her(B) + iHer(B) is always norm closed [297, Theorem 2.6.7]). Thus we may use A.5.8 if necessary to obtain the stated conclusions. If A is not unital, we may conclude as in the proof of A.5.8 (here one may use A.4.3, for example, to extend π to suitable unitizations).

A.5.10

The following simple principle is useful in proving the Kaplansky density theorem, and variants of it. Namely, suppose that E is a closed subspace of a dual space F*, and that we wish to prove that Ball(E) is w*-dense in the unit ball of its w*-closure. By A.2.2, there is a w*-continuous contraction ũ: E** → F* extending the inclusion map u: EF*. Suppose that ũ takes the open unit ball of E** onto the open unit ball of Ran(ũ). This is often automatic, as in the case that E is a *-subalgebra of a von Neumann algebra F* (this follows by A.5.8; in this case ũ is a homomorphism by a standard w*-density argument using the separate w*-continuity in A.5.7 and A.5.1). It is easy to check, by A.2.5 (1), that Ran(ũ) is w*-closed. Thus E ¯ w * = Ran ( u ˜ ) . Indeed, if z ∈ Ran(ũ), with ‖z‖ < 1, then there exists η ∈ E** with ũ(η) = z and ‖η‖ < 1. By A.2.1, there is a net (et)t in Ball(E), converging in the w*-topology to η. Thus et = u(et) → ũ(η) = z in the w*-topology of F*. It is quite obvious that this implies that Ball(E) is w*-dense in the unit ball of E ¯ w * .

A.6 MODULES AND COHEN's FACTORIZATION THEOREM

Again, see [106,297,321] for omitted proofs, complementary results, and history.

A.6.1

Reflecting on the difference between algebras and rings, it is natural that this difference should be reflected in the modules over each. For us, a (left) module over an algebra A will always be a vector space X over C, which is a (left) module in the traditional sense, but we insist also that

( α a ) x = a ( a α ) = α ( a x ) , a A , α , x X .
There is then a one-to-one correspondence between left A-modules, and representations of A on vector spaces (that is, homomorphisms π from A into the (p.368) algebra of linear maps on X, for a vector space X). This correspondence is given by the formula
π ( a ) ( x ) = a x , a A , x X .
We call π the canonical homomorphism associated with the module action.

Now suppose that X is a (left, say) A-module over a Banach algebra A. We say that X is a normed (resp. Banach) A-module if X is a normed (resp. Banach) space, and the module action A × XX is a contractive bilinear map. This is the same as saying that the associated homomorphism π: AB(X) in the last paragraph, is contractive. We shall not do so here, but for many results (such as those in Chapter 5) one may want to weaken the last condition to allow bounded module actions.

An A-B-bimodule is a left A-module which is also a right B-module, such that the two actions commute. That is, a(xb) = (ax)b for aA, bB, xX. A Banach A-B-bimodule is a Banach space and a bimodule, which is both a left and a right Banach module.

We say that a (left) Banach A-module X is a nondegenerate A-module, or that A acts nondegenerately on X, if X equals the norm closure of the linear span of the products ax for aA and xX. If A has a bounded approximate identity (et)t then this is equivalent to saying that etxx for all xX, and we will see some other equivalent conditions in A.6.4 below. A bimodule will be called nondegenerate if it is nondegenerate both as a left and a right module.

Theorem A.6.2 (Cohen's factorization)

Suppose that A is a Banach algebra with a bounded approximate identity, and that X is a left Banach A-module. Then X is a nondegenerate A-module if and only if any xX may be written in the form x = ay for some aA,yX. In this case, if further A has a cai, and if x has norm < 1, then we may also choose a and y with norm < 1.

Corollary A.6.3

Let A, X be as in A.6.2. Then every A-module map f: AX is bounded.

Proof Take a sequence an → 0 in A. Let Y = C 0(A), the set of sequences converging to 0 with terms in A. This is a right Banach A-module, which is not hard to see is nondegenerate. Applying Cohen's theorem we may write an = bnb, for b, bnA with bn → 0. Thus f(an) = bnf(b) → 0.

A.6.4

If X is a (not necessarily nondegenerate) left Banach A-module over a Banach algebra A then we define the essential part of X to be the norm closure of the linear span of the products ax for aA and xX. If A has a bounded approximate identity (et)t then this set is clearly exactly the set of xX such that etxx. Clearly in this case the essential part is a nondegenerate A-module. Thus in this case (i.e. if A has a bounded approximate identity), it follows from Cohen's theorem applied to the essential part, that the essential part of X equals {ax: aA,xX}, and also equals { k = 1 n a k x k : n , a k A , x k X } . Thus our use of the notation AX in this book (see Section 1.1), at least in this case, is less ambiguous than it may seem.