(p.359) Appendix
(p.359) Appendix
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 (twosided) ideal in B(H). We write S ^{1}(H) for the usual trace class, a (twosided) ideal in S ^{∞}(H) (and also in B(H)), and which is a Banach space with respect to the trace class norm ‖ T ‖_{1} = tr T . The trace tr is a contractive functional on S ^{1}(H), and via the dual pairing (S, T) ↦ tr(ST) it is wellknown 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 S_{t} → S in the w*topology on B(H), then S_{t}T → ST and TS_{t} → TS in the w*topology too. The w*topology on B(H) is also called the σweak topology. A linear functional on B(H) is σweakly continuous if and only if it is of the form ${\sum}_{k=1}^{\infty}<\cdot {\zeta}_{k},{\eta}_{k}>$, for ζ_{k}, η_{k} ∈ H with ${{\displaystyle {\sum}_{k=1}^{\infty}\Vert {\zeta}_{k}\Vert}}^{2}$ and ${{\displaystyle {\sum}_{k=1}^{\infty}\Vert {\eta}_{k}\Vert}}^{2}$ 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 S^{p}(H, K) denote the Schatten pclass of compact operators T: H → K such that T^{p} belongs to S ^{1}(H). This is a Banach space for the norm ${\Vert T\Vert}_{p}={\left(tr{\leftT\right}^{p}\right)}^{\frac{1}{\text{p}}}$. The following ideal property holds: for any T ∈ S^{p}(H, K), V _{1} ∈ B(K), V _{2} ∈ B(H), the operator V _{1} TV _{2} belongs to S^{p}(H, K) and ‖V _{1} TV _{2}‖_{p} ≤ ‖V _{1}‖‖T‖_{p}‖V_{2}‖.
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 σweak 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 X ⊂ B(H) is said to be reflexive if
A.2 DUALITY OF BANACH SPACES
In this section and the next, E and F are Banach spaces. We write i _{E}: E → E** for the canonical embedding. However we often suppress this map and simply consider E as a subspace of E**.
Lemma A.2.2
Let u: E → F* be a bounded linear map. Then there exists a unique w*continuous ũ: E** → F* extending u. Moreover ‖ũ‖ = ‖u‖.
Proof Set $\tilde{u}={i}_{F}^{*}\circ {u}^{**}$ where i_{F}: F → F** is the canonical isometry. □
Lemma A.2.3
Let E be a closed linear subspace of a Banach space F.

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

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

(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: F → F/E is the canonical quotient map, by factoring out Ker (q**) = E ^{⊥⊥}.

(4) F ∩ E ^{⊥⊥} = E.
We will also (silently) use the following simple principle many times:
Lemma A.2.4
If T: E → F 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) 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.
(p.361)

(2) If u ∈ B(E, F), where E and F are dual Banach spaces, then u is w*continuous if and only if whenever x_{t} → x is a bounded net converging in the w*topology in E, then u(x_{t}) → u(x) in the w*topology.

(3) Let E and F be as in (2), and u: E → F a w*continuous isometry. Then u has w*closed range, and u is a w*w*homeomorphism onto Ran(u).
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 _{k}⊗y _{k} in the algebraic tensor product E⊗F, the quantity
We remark that the definition and facts in the last paragraph have obvious variants for the Nfold injective product X _{1} ⊗̆ … ⊗̆ X _{N} of any Ntuple 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 _{k} ⊗ y _{k} as above with a bounded operator u: F* → E, namely u(ψ) = ∑_{k}ψ(yk)x _{k} for any ψ ∈ F*. We say that u is associated with z. Under this identification, E ⊗ F coincides with the space of all finite rank and w*tonorm continuous operators from F* into E. Clearly
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 C(Ω E) by regarding any f = ∑_{k} g_{k} ⊗ x_{k} (with g_{k} ∈ C(Ω) and x_{k} ∈ E) as a function, f(t) = ∑_{k} g_{k}(t)x_{k}. Then the norm of f in C(ΩE) is equal to sup{∑_{k} g_{k}(t)φ(x_{k}): 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.3.3
A bounded bilinear map T: E × F → Z is a bilinear map for which there is a constant C such that ‖T(x, y)‖ ≤ C‖x‖‖y‖, for all x ∈ E, y ∈ F. 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 E ⊗ F 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 × F → E ⊗̂ F is a contractive bilinear map, and for any bounded bilinear T: E × F → Z, the associated linear map E ⊗ F → Z is continuous with respect to the just mentioned norm, and extends to a bounded linear map T̃ E ⊗̂ F → Z with ‖T̃‖ = ‖T‖. From this it is easy to see that
A.3.4
A bounded operator u: E → F is said to be 2summing if I _{l2} ⊗ u extends to a bounded operator from l ^{2}⊗̆E into l ^{2}(F). We set
It is not hard to check that π_{2}(·) is a (complete) norm on the space Π_{2}(E, F) of 2summing operators from E into F. We also note that π_{2}(u) is the supremum of the norms of the mappings ${I}_{{\ell}_{n}^{2}}\otimes u:{\ell}_{n}^{2}\stackrel{\u0306}{\otimes}E\to {\ell}_{n}^{2}\left(F\right)$, for n ∈ ℕ.
A.3.5
We review three tensor products related to Hilbert space factorization. Let (e_{k})k denote the canonical basis of l ^{2}. For z ∈ E ⊗ F define
(p.363) Consider $\sum}_{k=1}^{n}{e}_{k}\otimes {x}_{k$ for x _{1},…,x_{n} ∈ E as above, and let u: E* → ℓ ^{2} be the associated linear map. According to (A.1), ${\Vert {\displaystyle {\sum}_{k=1}^{n}{e}_{k}\otimes {x}_{k}}\Vert}_{{\ell}^{2}\stackrel{\u030c}{\otimes}E}$ is equal to the usual operator norm of u. Analoguously, we define ${\Vert {\displaystyle {\sum}_{k=1}^{n}{e}_{k}\otimes {x}_{k}}\Vert}_{{\displaystyle {\prod}_{2}\left({E}^{*},{\ell}^{2}\right)}}$ to be π_{2}(u). Then for z ∈ E ⊗ F we define
Next we let
It is not hard to see that ${\Vert \cdot \Vert}_{\vee}\le {\text{\gamma}}_{\text{2}}(\cdot )\le {g}_{2}(\cdot )\le {\text{\gamma}}_{\text{2}}^{\text{*}}(\cdot )\le {\Vert \cdot \Vert}_{\wedge}$ on E ⊗ F.
A.3.6
Let H, K be hilbert spaces. Then the Space K ⊗ H̅ 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.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 CBanach algebra is a Banach space A which is also an algebra such that ‖ab‖ ≤ C‖a‖‖b‖ for all a, b ∈ A. 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 (e_{t})_{t} with e_{t}a → a and ae_{t} → a. This is a contractive approximate identity (cai) if moreover ‖e_{t}‖ ≤ 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 h ∈ A 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: A → B is a unital contractive linear map between unital Banach algebras, then u(Her(A)) ⊂ Her(B).
(p.364) It is wellknown 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 A_{sa} of selfadjoint elements.
A nonzero homomorphism on a unital Banach algebra is a state, and is called a character. The maximal ideal space M_{A} 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: A → B(A), and identifying A + C1 with the span of λ(A) + C I_{A}, which is easy to see is a unital Banach subalgebra of B(A). Thus if a ∈ A and α ∈ C then
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 (e_{t})t is a cai for A then
As a consequence of (A. 10), if π: A → B is a contractive homomorphism between Banach algebras, such that (π(e_{t}))_{t} is a cai for B for some cai (e_{t})_{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 a ∈ A 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 I_{H}. 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 *: A → A such that (a*)* = a and (ab)* = b*a*, for all a, b ∈ A, which also satisfies the C*identity: ‖a*a‖ = ‖a‖^{2}, for a ∈ A. 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 b ∈ A. We write A _{+} for the set of such elements, and write a ≤ b if b − a ∈ A _{+}, 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 *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: A → B(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 onetoone, and surjective) functions between compact spaces, and unital (resp. and surjective, and onetoone ) *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 π_{u}: A → B(H_{u}) 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 π_{u} has the property that for any state ϕ of A, there exists a unit vector ξ ∈ H_{u} such that ϕ = <π_{u}(·)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 π_{u}.
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 π_{u}: A → B(H_{u}) be the universal representation of A (see A.5.5). Let $\tilde{{\text{\pi}}_{u}}:{A}^{**}\to B\left({H}_{u}\right)$ be the unique (contractive) w*continuous extension of π_{u} as in A.2.2 (thus $\tilde{{\text{\pi}}_{u}}={i}^{*}\circ {\text{\pi}}_{u}^{**}$, where i: S ^{1}(H_{u}) = B(H_{u})_{*} → B(H_{u})^{*} is the canonical injection). Let ψ be a linear functional on A of norm 1. It is wellknown 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 a ∈ A we have that
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: A → B 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 onetoone.
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 a ∈ A_{+} 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: A → B 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 onetoone 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(1_{A}) = 1_{B}. Since π(Her(A)) π Her(B) (see A.4.2),
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: E → F*. 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 ${\overline{E}}^{{w}^{*}}=\text{Ran}\left(\tilde{u}\right)$. 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 (e_{t})t in Ball(E), converging in the w*topology to η. Thus e_{t} = u(e_{t}) → 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 ${\overline{E}}^{{w}^{*}}$.
A.6 MODULES AND COHEN's FACTORIZATION THEOREM
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
Now suppose that X is a (left, say) Amodule over a Banach algebra A. We say that X is a normed (resp. Banach) Amodule if X is a normed (resp. Banach) space, and the module action A × X → X is a contractive bilinear map. This is the same as saying that the associated homomorphism π: A → B(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 ABbimodule is a left Amodule which is also a right Bmodule, such that the two actions commute. That is, a(xb) = (ax)b for a ∈ A, b ∈ B, x ∈ X. A Banach ABbimodule is a Banach space and a bimodule, which is both a left and a right Banach module.
We say that a (left) Banach Amodule X is a nondegenerate Amodule, or that A acts nondegenerately on X, if X equals the norm closure of the linear span of the products ax for a ∈ A and x ∈ X. If A has a bounded approximate identity (e_{t})_{t} then this is equivalent to saying that e_{t}x → x for all x ∈ X, 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 Amodule. Then X is a nondegenerate Amodule if and only if any x ∈ X may be written in the form x = ay for some a ∈ A,y ∈ X. 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 Amodule map f: A → X is bounded.
Proof Take a sequence a_{n} → 0 in A. Let Y = C _{0}(A), the set of sequences converging to 0 with terms in A. This is a right Banach Amodule, which is not hard to see is nondegenerate. Applying Cohen's theorem we may write a_{n} = b_{n}b, for b, b_{n} ∈ A with b_{n} → 0. Thus f(a_{n}) = b_{n}f(b) → 0.
A.6.4
If X is a (not necessarily nondegenerate) left Banach Amodule 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 a ∈ A and x ∈ X. If A has a bounded approximate identity (e_{t})_{t} then this set is clearly exactly the set of x ∈ X such that e_{t}x → x. Clearly in this case the essential part is a nondegenerate Amodule. 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: a ∈ A,x ∈ X}, and also equals $\left\{{\displaystyle {\sum}_{k=1}^{n}{a}_{k}{x}_{k}:n\in \mathbb{N},{a}_{k}\in A,{x}_{k}\in X}\right\}$. 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.