This paper extends Morita equivalence concepts from $C^{*}$-correspondences to $W^{*}$-correspondences, demonstrating that weak Morita equivalence of the correspondences implies weak Morita equivalence of their Hardy algebras, with special focus on $W^{*}$-graph correspondences.
Contribution
It introduces the weak$^{*}$ version of Morita equivalence for $W^{*}$-correspondences and proves that this equivalence implies the same for their Hardy algebras, expanding the theory to dual operator algebras.
Findings
01
Weak Morita equivalence of $W^{*}$-correspondences implies weak Morita equivalence of their Hardy algebras.
02
Results specialized to $W^{*}$-graph correspondences.
03
Established connections between $W^{*}$-correspondences and their Hardy algebras.
Abstract
Muhly and Solel developed a notion of Morita equivalence for C∗- correspondences, which they used to show that if two C∗-correspondences E and F are Morita equivalent then their tensor algebras T+(E) and T+(F) are (strongly) Morita equivalent operator algebras. We give the weak∗ version of this result by considering (weak) Morita equivalence of W∗-correspondences and employing Blecher and Kashyap's notion of Morita equivalence for dual operator algebras. More precisely, we show that weak Morita equivalence of W∗-correspondences E and F implies weak Morita equivalence of their Hardy algebras H∞(E) and H∞(F). We give special attention to W∗-graph correspondences and show a number of results related to their Morita equivalence.
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Morita Equivalence of W∗-Correspondences and Their Hardy Algebras
Rene Ardila
Department of Mathematics, Grand Valley State University, Allendale, Michigan 49401
Muhly and Solel developed a notion of Morita equivalence for C∗- correspondences, which they used to show that if two C∗-correspondences E and F are Morita equivalent then their tensor algebras T+(E) and T+(F) are (strongly) Morita equivalent operator algebras. We give the weak*∗* version of this result by considering (weak) Morita equivalence of W∗-correspondences and employing Blecher and Kashyap’s notion of Morita equivalence for dual operator algebras. More precisely, we show that weak Morita equivalence of W∗-correspondences E and F implies weak Morita equivalence of their Hardy algebras H∞(E) and H∞(F). We give special attention to W∗-graph correspondences and show a number of results related to their Morita equivalence.
Given a von Neumann algebra A and a W∗-correspondence E over A, Muhly and Solel constructed an algebra H∞(E) which they called the Hardy algebra of E [MS04]. This algebra is a noncommutative generalization of the classic Hardy algebra H∞(T) of bounded analytic functions on the open unit disc. More precisely, when E=A=C, H∞(E) is the classical Hardy space H∞(T). When A=C and E=Cn, H∞(E) is the free semigroup algebra Ln studied by Popescu [Pop91], Davidson, Pitts [DP98] and others. This Hardy algebra is a dual operator subalgebra of L(F(E)), the adjointable operators of the Fock space of E, generated by diagonal and creation operators. When E is a correspondence derived from a directed graph G, H∞(E) is a dual operator algebra version of what algebraists call the path algebra of G.
Kiiti Morita’s 1958 groundbraking paper [Mor58] contains the main ideas of what later became known as Morita equivalence, an extremely important concept in the study of the algebraic structure of rings. Following the dissemination of Morita’s ideas, mainly by H. Bass and P. Gabriel in the early 1960s, many other notions of Morita equivalence have been developed, including notions of Morita equivalence for
selfadjoint algebras, operator algebras, groupoids, group *-algebras, finite groups, Poisson manifolds, non commutative smooth tori, tensor categories, semigroups and star products.
In [MS00], Muhly and Solel introduced a notion of (strong) Morita equivalence for C∗-correspondences, which they used to show that if two C∗-correspondences E and F are (strongly) Morita equivalent then their tensor algebras T+(E) and T+(F) are (strongly) Morita equivalent operator algebras. At that time however, there was no clear notion of Morita equivalence for dual operator algebras. Such notions were developed ten years later in the work of Blecher, Kashyap, Eleftherakis and Paulsen ([BK08], [EP08], [Ele08]). Motivated by Muhly and Solel’s work, we consider
(weak) Morita equivalence of W∗-correspondences, and use Blecher and Kashiap’s notion of Morita equivalence for dual operator algebras to show that if two W∗-correspondences E and F are (weakly) Morita equivalent then their Hardy algebras H∞(E) and H∞(F) are (weakly) Morita equivalent dual operator algebras.
In the last section,
we concentrate on Morita equivalence of W∗-graph correspondences, their Hardy algebras and their representations. We show that if (E,A) is a W∗-graph correspondence then any two faithful normal representations σ and τ of A give rise to Morita equivalent dual corresoondences (Eσ, σ(A)′) and (Eτ, τ(A)′). Then we consider the induced representations σF(E) and τF(E) of the Hardy algebra H∞(E) and show that the commutants of σF(E)(H∞(E)) and τF(E)(H∞(E)) are (weakly) Morita equivalent dual operator algebras. We also study equivalence bimodules and the relation between graphs and the Morita equivalence of their W∗-correspondences.
2. Preliminaries
A right C∗-module E over a C∗-algebra A is said to be selfdual if every continuous A-module map f:E→A is of the form f(⋅)=⟨y,⋅⟩, for some y∈E. We say that E is a right W∗-module if E is a selfdual right C∗-module over a W∗-algebra.
We write LA(E) (or simply L(E)) for the space of adjointableA-module maps on E. An A−BW∗-correspondence is a right W∗-module E over B for which there exists a unital normal ∗-homomorphism φ: A→LB(E). We then say that E is a W∗-correspondence from A to B, and we denote it by AEB. If A=B then we say that E is a W∗-correspondence over A. In this case, we might also denote the correspondence by (E,A). The center of a W∗-correspondence (E,A) is the set Z(E)={x∈E:a⋅x=x⋅a for all a∈A}. We will sometimes abbreviate ”weak*∗*” to ”w∗”.
The W∗-module tensor product⊗A (sometimes written as the composition tensor productX⊗σY) is defined to be the selfdual completion (the weak*∗*-completion) of the C∗-module interior tensor product X⊗σY. When there is no risk of confusion, we will simply write X⊗Y. The W∗-module tensor product is functorial and associative. If E is a W∗-correspondence from A to B and F is a W∗-correspondence from B to C then E⊗AF is a W∗-correspondence from A to C with inner product given by ⟨x1⊗y1,x2⊗y2⟩E⊗F=⟨y1,φ(⟨x1,x2⟩E)y2⟩F and left/right actions given by a⋅(x⊗y)⋅c=(a⋅x)⊗(y⋅c)=(φE(a)x)⊗(y⋅c). In particular, given a W∗-correspondence E over A and a Hilbert space H equipped with a normal representation σ of A, we can form the Hilbert space E⊗σH, where we have ⟨x1⊗h1,x2⊗h2⟩=⟨h1,σ(⟨x1,x2⟩E)h2⟩.
A W∗-correspondence isomorphism between two W∗-correspondences (E1,A1) and (E2,A2) is a pair (σ,ψ) where σ:A1→A2 is an isomorphism of W∗-algebras and ψ:E1→E2 is a vector space isomorphism, where for e,f∈E1 and a,b∈A1, we have ψ(a⋅e⋅b)=σ(a)⋅ψ(e)⋅σ(b) and ⟨ψ(e),ψ(f)⟩=σ(⟨e,f⟩). Such ψ must be a weak*∗*-homeomorphism because the predual of a W∗-module is unique.
If A and B are W∗-algebras, then an A-BW∗-equivalence bimodule is an A-BW∗-bimodule X which is a w∗-full right W∗-module over B and a w∗-full left W∗-module over A, such that the two (left and right) inner products of X are compatible in the sense that A⟨x,y⟩⋅z=x⋅⟨y,z⟩B for all x,y,z∈X. If AXB and CYD are W∗-equivalence bimodules then a W∗-equivalence bimodule isomorphism (as defined in [EKQR06, Definition 1.16 and Remark 1.19]) is a triple (σ,ϕ,π), where σ:A→C and π:B→D are W∗-algebra isomorphisms and ϕ:X→Y is a vector space isomorphism such that ϕ(a⋅e⋅b)=σ(a)⋅ϕ(e)⋅π(b), ⟨ϕ(e),ϕ(f)⟩D=π(⟨e,f⟩B) and C⟨ϕ(e),ϕ(f)⟩=σ(A⟨e,f⟩).
Given a representation σ of A, an operator T∈L(E) and an operator S∈σ(A)′, the map x⊗h→Tx⊗Sh defines a bounded operator on E⊗σH denoted by T⊗S. In particular, the representation of L(E) resulting from letting S=I, is Rieffel’s induced representation of L(E) induced by σ. This representation is denoted by σE. That is, σE(T)=T⊗I. Likewise, we say that the composition σE∘φ is the representation of A on E⊗σH induced by E.
Let N0=N∪{0}. If E is a W∗-correspondence over a W∗-algebra A then we can form the tensor powers E⊗n, n≥0, where E⊗0=A. For each n, E⊗n is a W∗-correspondence over A with the inner product defined inductively.
The ultraweak direct sum F(E):=⨁n∈N0wcE⊗n is a W∗-correspondence over A called the Fock space over E. The left action of A on F(E) is given by the map φ∞ defined by φ∞(a)= diag(a,φ(a),φ(2)(a),φ(3)(a),⋯) where φ(n)(a)(x1⊗x2⊗⋯⊗xn)=(φ(a)x1)⊗x2⊗⋯⊗xn∈E⊗n. Given x∈E, the creation operatorTx∈L(F(E)) is defined by Tx(η)=x⊗η, η∈F(E)). That is,
φ∞(a)=φ(0)(a)000φ(1)(a)0⋯⋯⋱ and
Tx=0Tx(1)0000Tx(2)0⋯⋯⋯⋱0000
where Tx(n)(x1⊗x2⊗⋯⊗xn−1)=x⊗x1⊗x2⊗⋯⊗xn−1.
The tensor algebra over E, denoted T+(E) is defined to be the norm closed subalgebra of L(F(E)) generated by φ∞(A) and {Tx:x∈E}. The ultraweak closure of T+(E) in L(F(E)) is called the Hardy Algebra of E, and is denoted by H∞(E).
As shown in [MS04], the completely contractive representations of H∞(E) are determined by pairs (T,σ) where σ:A→B(H) is a normal ∗-representation of A and T:E→B(H) is a linear, completely contractive w∗-continuous representation of E satisfying T(axb)=σ(a)T(x)σ(b) for all x∈E and a,b∈A. The linear map T defined on the algebraic tensor product E⊗H by T(x⊗h)=T(x)h extends to an operator of norm at most 1 on the completion E⊗σH. The pairs (T,σ) are called the completely contractive covariant representations of E. The bimodule property of T is equivalent to the equation T(σE∘φ(a))=T(φ(a)⊗I)=σ(a)T for all a∈A, which means that T intertwines the representations σ and σE∘φ of A on H and E⊗H respectively. The space composed of all these intertwiners is called the intertwining space, and it is usually denoted as I(σE∘φ,σ) or (Eσ)∗. Furthermore, for each completely contractive covariant representation (T,σ) of a correspondence E over a W∗ algebra A, there is a unique completely contractive representation ρ of the algebra T+(E) satisfying ρ(Tx)=T(x) and ρ(φ∞(a))=σ(a) for each x∈E,a∈A. The map (T,σ)↦ρ is bijective and onto the set of all completely contractive representations of T+(E) whose restrictions to φ(A) are ultraweakly continuous. If ∣∣T∣∣<1 then ρ extends to an ultraweakly continuous representation σ×T of H∞(E).
That is, the ultraweakly continuous completely contractive representations of H∞(E) are parametrized by the elements in the unit ball
[TABLE]
The space Eσ is itself a W∗-correspondence over σ(A)′ with the actions given by a⋅η=(IE⊗a)η and η⋅a=ηa for η∈E and a∈A.
The σ(A)′-valued inner product is given by ⟨η,ξ⟩=η∗ξ.
A dual operator algebra is an operator algebra A which is also a dual operator space. Any weak*∗-closed subalgebra of B(H) is a dual operator algebra and conversely, for any dual operator algebra A, tA dual operator algebra is an operator algebra A which is also a dual operator space. Any weak∗-closed subalgebra of B(H) is a dual operator algebra and conversely, for any dual operator algebra A, there is a Hilbert space H and a w∗-continuous completely isometric homomorphism φ:A→B(H). By the Krein-Smulian theorem, φ(A) is a weak∗*-closed subalgebra of B(H), so we can identify A with φ(A) as dual operator algebras. A normal representation of a dual operator algebra is a completely contractive, w∗-continuous homomorphism φ:A→B(H). The category of normal representations of A is denoted by AM. The objects of AM are pairs (H,φ) where H is a Hilbert space and φ:A→B(H) is a unital completely contractive, w∗-continuous homomorphism. If (Hi,φi),i=1,2, are objects in AM, the morphisms are given by HomA(H1,H2)={T∈B(H1,H2):Tφ1(a)=φ2(a)T,a∈A}.
If A and B are dual operator algebras, a dual operator A−B-bimodule is a nondegenerate operator A−B-bimodule X, which is also a dual operator space, such that the module actions are separately weak*∗*-continuous. If X and Y are right operator modules over
B, then we write CBB(X,Y) for the set of completely bounded right B-module maps from X to Y. If X and Y are left operator modules over
A, then we write ACB(X,Y) for the set of completely bounded left A-module maps from X to Y. Similarly, we write w∗CBB(X,Y) for the set of w∗-continuous completely bounded right B-module maps from X to Y.
3. Morita Equivalence of W∗-correspondences and
Hardy Algebras
In 2000, Muhly and Solel introduced a notion of (strong) Morita equivalence for C∗-correspondences [MS00, Definition 2.1]. This notion can be extended to W∗-correspondences in the following way: W∗-correspondences AEA and BFB are called (weakly) Morita equivalent if the W∗-algebras A and B are weakly Morita equivalent via a W∗-equivalence bimodule X for which there is an A-BW∗-correspondence isomorphism W from X⊗BF onto E⊗AX. In this case, we will write E∼WMEXF. Recall that X⊗BF and E⊗AX are the self dual completions of the balanced C∗-module interior tensor products. Remember also that the C∗-module interior tensor product coincides with the module Haagerup tensor product while the W∗-module tensor product ⊗A coincides with the module weak*∗* Haagerup tensor product ⊗w∗h, which is the same as the extended Haagerup tensor product ⊗eh [BL04, 8.5.40, 1.7.1.5].
Throughout this section, AEA and BFB are W∗-correspondences over the W∗-algebras A and B.
In [BMP00], Blecher, Muhly and Paulsen generalized Rieffel’s strong Morita equivalence of C∗-algebras ([Rie74b]) to general operator algebras. Their generalization is a natural variation of the theory of Morita equivalence that one finds in pure algebra, where the description of Morita equivalence is given in terms of Morita contexts (these contexts are also found in the pure algebra literarture under the name: sets of pre-equivalence data). Their definition is the following: Let A and B be unital or approximately unital operator algebras. Let X be an A-B operator bimodule, and let Y a B-A operator bimodule. Let (⋅,⋅) be a completely bounded bilinear map from X×Y to A, balanced over B. Let [⋅,⋅] be a completely bounded bilinear map from Y×X to B, balanced over A. The 6-tuple (A,B,X,Y,(⋅,⋅),[⋅,⋅]) is called a (strong) Morita context for A and B if the module actions are completely contractive and:
•
(x1,y)⋅x2=x1⋅[y,x2]x1,x2∈X,y∈Y.
[y1,x]⋅y2=y1⋅(x,y2)y1,y2∈Y,x∈X.
•
The linear map from X⊗hY to A determined by (⋅,⋅) is a complete quotient map onto A.
•
The linear map from Y⊗hX to B determined by [⋅,⋅] is a complete quotient map onto B.
As shown in [BMP00], a (strong) Morita context determines a pair of equivalence functors between the categories of operator modules of both operator algebras in the context. It also determines an equivalence between the categories of Hilbert modules of both operator algebras. Furthermore, the Morita context gives rise to an isomorphism between the lattices of ideals of both operator algebras in the context.
One important shortcoming of this notion of strong Morita equivalence is that if the two operator algebras A and B that we are comparing, are dual operator algebras then the strong Morita context does not capture this duality. More precisely, the two functors that are derived from the context, do not give an equivalence between the categories of normal representations of A and B.
In [Ele08], Eleftherakis formulated a version of Morita theory for dual operator algebras using ternary rings of operators (TROs) and a relation called Δ-equivalence. In [EP08], Eleftherakis and Paulsen showed that this notion of Δ-equivalence is equivalent to the notion of weak*∗* stable isomorphism of dual operator algebras. In [BK08], Blecher and Kashyap introduced a new notion of weak Morita equivalence of dual operator algebras which includes most of the examples of Morita-like equivalence (in the dual setting) found in the literature. Also, this approach contains the notion of stable isomorphism given by Eleftherakis and Paulsen. That is, if two unital dual operator algebras are weak*∗* stably isomorphic then they are weakly Morita equivalent in the sense of [BK08].
In the following definitions, A and B are dual operator algebras, X and Y are dual operator bimodules, X is an A-B bimodule and Y is an B-A-bimodule. The following two definitions of Morita equivalence for unital dual operator algebras are found in [BK08, section 3]. A and B are called weak∗ Morita equivalent, with equivalence bimodules X and Y, if A≅X⊗BY as dual operator A-bimodules, and similarly B≅Y⊗BX as dual operator B-bimodules.
(A,B,X,Y) is called a weak∗ Morita Context.
For the next definition, assume that (⋅,⋅):X×Y→A and [⋅,⋅]:Y×X→B are separately weak*∗*-continuous completely contractive bilinear maps. We will use the 6-tuple, or context,(A,B,X,Y,(⋅,⋅),[⋅,⋅]). We say that A is weakly Morita equivalent to B if there exists a 6-tuple as above, there exist w∗-dense operator algebras A′ and B′ in A and B respectively, there exists a w∗-dense operator A′-B′-submodule X′ in X, and a w∗-dense operator B′-A′-submodule Y′ in Y, such that the subcontext (A′,B′,X′,Y′), together with restrictions of the maps (⋅,⋅) and [⋅,⋅], is a strong Morita context in the sense of [BMP00, Definition 3.1]. (A,B,X,Y,(⋅,⋅),[⋅,⋅]) is called a weak Morita Context.
Our goal in this section is to show that if two W∗-correspondences (E,A) and (F,B) are (weakly) Morita equivalent then their Hardy algebras H∞(E) and H∞(F) are weakly Morita equivalent and weak*∗* Morita equivalent.
Lemma 3.1**.**
Let X be an A-BW∗-equivalence bimodule. Then the pairs (IA,mA) and (IB,mB), where mA:X⊗BX→A and mB:X⊗AX→B are defined by mA(x⊗y)=A⟨x,y⟩ and mB(x⊗y)=⟨x,y⟩B, are W∗-correspondence isomorphisms.
Proof.
The two identities are obviously W∗-algebra isomorphisms.
[TABLE]
[TABLE]
[TABLE]
That is, mA preserves both left and right inner products. So it is isometric, hence injective with closed range. Since mA is defined in terms of the left inner product of X, and by definition, X is a w∗-full left W∗-module over A, mA has w∗-dense range in A. So mA is surjective, thus (IA,mA) is a W∗-correspondence isomorphism. Similarly, (IB,mB) is also a W∗-correspondence isomorphism.
∎
Theorem 3.2**.**
If two W∗-correspondences (E,A) and (F,B) are (weakly) Morita equivalent then their Hardy algebras H∞(E) and H∞(F) are weakly Morita equivalent and weak∗ Morita equivalent (as dual operator algebras) in the sense of [BK08].
Proof.
We model our proof on the proof given in [MS00] for the C∗ case. A and B are weakly Morita equivalent W∗-algebras via a W∗-equivalence bimodule AXB and there is a W∗-correspondence isomorphism W:X⊗BF→E⊗AX. Let I (and Iw) denote the norm closure in A (and the w∗-closure in A) of the span of the range of the A-valued inner product on X. Form the linking W∗-algebra L for X. That is, let
[TABLE]
Let Y1 be the first column, (BX). This is a W∗-module over B since it is the column sum of B and X. Likewise, the second column, Y2=(XA), is a W∗-module over A. CBB(X)=BB(X)=LB(X)=M(KB(X))≅M(I)=Iw=A and CBB(B)=B. Since X is a selfdual space, CBB(B,X)≅X and CBB(X,B)≅X. So we have that L≅CBB(Y1). Similarly, CBA(X)≅B, CBA(A,X)≅X, CBA(X,A)≅X and CBA(A)≅A. So L≅CBA(Y2). That is, CBB(Y1) and CBA(Y2) are W∗-algebras. For more information on the previous identifications, see [BL04, 8.5.5, 8.1.15, 8.5.3, 8.5.1, 8.5.13, 8.5.5 (1), 2.6.1, 3.5.4 (2)] for example.
Now we have that CBB(Y1,F) is a module over the W∗-algebra CBB(Y1)=L, where right multiplication is given by composition and the inner product is given by ⟨T,S⟩=T∗S∈CBB(Y1)=L. By [Ble97, equation (††)], CBB(Y1,F)=(Y1⊗^BF∗)∗, where
⊗^B is the module operator space projective tensor product. So CBB(Y1,F) is a W∗-module over CBB(Y1)=L (by [BL04, corollary 8.5.7]). Similarly, CBA(Y2,E) is a W∗-module over CBA(Y2)=L. So their sum
[TABLE]
is a W∗-module over L. Since CBB(Y1,F) can be written as (CBB(B,F),CBB(X,F)) and CBA(Y2,E) is (CBA(X,E),CBA(A,E)), we have
[TABLE]
The right action of L on Z is realized as the usual matrix multiplication, and the inner product is given by
[TABLE]
Since CBB(B,F)≅F, CBB(X,F)≅F⊗X, CBA(X,E)≅E⊗X and ACB(A,E)≅E, we have
[TABLE]
The right action then becomes
[TABLE]
and the inner product is
[TABLE]
Let φZ:L→L(Z) be defined by
[TABLE]
where W:X⊗AE→F⊗BX is the isomorphism defined in [MS00, Section 2, pg 116]. By [MS00, Proposition 2.6], φZ is a ∗-homomorphism, and since W is w∗-continuous (being a W∗-correspondence isomorphism), φZ is normal. Thus Z is a W∗-correspondence over L (where the left action of L on Z is given by φZ).
Replacing the C∗-module interior tensor product (which is the same as the module Haagerup tensor product) with the W∗-module tensor product, and replacing the direct sum with the ultraweak direct sum in [MS00, Lemmas 2.7, 2.8 and 2.10], we have that the Fock space F(Z) can be written in the form
[TABLE]
Form the operator algebras T(Z),T+(Z) and H∞(Z) associated with Z. Consider the submodule
[TABLE]
By the definition of the map φZ, F′(Z) is invariant for the diagonal operators in L(F(Z)). By [MS00, Lemma 2.9], F′(Z) is also invariant for the creation operators in L(F(Z)). Thus, F′(Z) is invariant for H∞(Z). Furthermore, by [MS00, Lemma 3.1], the representation of T(Z) obtained by restricting the action of T(Z) to F′(Z) is faithful. That is, we can study the action of T(Z) on F(Z) by just studying
the action of T(Z) on F′(Z).
Write p for the projection in L(F′(Z)) onto (F(F)0) and q for the projection onto (0F(E)⊗AX).
Next, we show that pH∞(Z)p≅H∞(F) and qH∞(Z)q≅H∞(E). Let
[TABLE]
and let f∈F⊗l. We can view f as the element
(f0)∈(F⊗l0)⊂F′(Z). By [MS00, Lemma 2.9],
[TABLE]
where Wk=(IE⊗Wk−1)(W1⊗IF⊗(k−1)). Hence
[TABLE]
which we write as pTξp=Th1. For
[TABLE]
we have
[TABLE]
Hence
[TABLE]
So pφ∞(λ)p=φ∞(b). That is, the generators of the algebra pT+(Z)p are identified with the generators of T+(F). Thus pT+(Z)p≅T+(F) and pH∞(Z)p≅H∞(F).
Similarly, we can view the element e⊗u∈E⊗l⊗X as the element (0e⊗u00)∈F′(Z). So by [MS00, Lemma 2.9],
[TABLE]
where c:F⊗m⊗X×X⊗F⊗l→F⊗m⊗F⊗l is a bilinear map which is not relevant for our purposes.
So
[TABLE]
which we write as qTξq=Tk2⊗Ix. For
[TABLE]
we have
[TABLE]
Hence
[TABLE]
So qφ∞(λ)q=φ∞(a)⊗IX. Since the map from L(F(E)⊗AX) to L(F(E)) taking T⊗IX to T is an isomorphism ([MS00, Lemma 2.12]), we have that qT+(Z)q≅T+(E) and qH∞(Z)q≅H∞(E).
Next, we show that H∞(E) and H∞(F) are weakly Morita equivalent (as dual operator algebras) in the sense of [BK08]. First note that pH∞(Z)p and qH∞(Z)q are unital dual operator algebras with identities pφ∞(1L)p=φ∞(1B) and qφ∞(1L)q=φ∞(1A) respectively.
Let (⋅,⋅):pH∞(Z)q×qH∞(Z)p⟶pH∞(Z)p and [⋅,⋅]:qH∞(Z)p×pH∞(Z)q⟶qH∞(Z)q be given by:
[TABLE]
respectively, and let (⋅,⋅)t:pT+(Z)q×qT+(Z)p⟶pT+(Z)p and [⋅,⋅]t:qT+(Z)p×pT+(Z)q⟶qT+(Z)q be the respective restrictions of (⋅,⋅) and [⋅,⋅].
is a strong Morita context in the sense of [BMP00, Definition 3.1]. In particular, the multiplication maps (⋅,⋅) and [⋅,⋅] are completely contractive bilinear maps.
Since for any Hilbert space H, the product in B(H) is separately weak*∗-continuous, we have that (⋅,⋅) and [⋅,⋅] are separately weak∗*-continuous (here we are using the identification of H∞(Z), an abstract dual operator algebra, with a subalgebra of B(H) via a complete isometric homomorphism which is a w∗-homeomorphism).
Then, since pT+(Z)pw∗=pH∞(Z)p , qT+(Z)qw∗=qH∞(Z)q, pT+(Z)qw∗=pH∞(Z)q and qT+(Z)pw∗=qH∞(Z)p, we have that
[TABLE]
is a weak Morita context. Thus H∞(E)≅qH∞(Z)q and H∞(F)≅pH∞(Z)p are weakly Morita equivalent in the sense of [BK08, Definition 3.2]. By [BK08, Corollary 3.4], H∞(E) and H∞(F) are also weak*∗* Morita equivalent.
∎
In the same way that a strong Morita context gives rise to a linking algebra L (see [BMP00, Chapter 3]), a weak*∗* Morita context (and therefore a weak Morita context also) gives rise to a weak linking algebra Lω, which is a dual operator algebra. The construction of Lω is given in [BK08, Section 4]. In our case, the linking operator algebra of the strong Morita context (T+(F),T+(E),pT+(Z)q,qT+(Z)p) is
[TABLE]
and the weak linking algebra of the weak Morita context (H∞(F),H∞(E),pH∞(Z)q,qH∞(Z)p) is
[TABLE]
L can be identified completely isometrically with a weak*∗-dense subalgebra of Lω.
Adapting [BK08, 4] to our algebras, we have that (H∞(F),Lω,H∞(F)⊕rpH∞(Z)q,H∞(F)⊕cqH∞(Z)p) and (H∞(E),Lω,H∞(E)⊕rqH∞(Z)p,H∞(E)⊕cpH∞(Z)q) are weak∗* Morita contexts. The next corollary follows.
Corollary 3.3*.*
If two W∗-correspondences (E,A) and (F,B) are weakly Morita equivalent, then H∞(F) and H∞(E) are weakly Morita equivalent to Lω.
Furthermore, applying the map LN in [BK08, Theorem 3.6] to the weak linking algebras of the weak*∗* Morita contexts (H∞(F),Lω,H∞(F)⊕rpH∞(Z)q,H∞(F)⊕cqH∞(Z)p) and (H∞(E),Lω,H∞(E)⊕rqH∞(Z)p,H∞(E)⊕cpH∞(Z)q), we have:
Corollary 3.4*.*
If two W∗-correspondences (E,A) and (F,B) are weakly Morita equivalent,
then w∗CBH∞(F)(H∞(F)⊕cqH∞(Z)p)≅Lω≅w∗CBH∞(E)(H∞(E)⊕cpH∞(Z)q) completely isometrically and w∗-isomorphically.
In [BMN99], it was shown that a strong Morita equivalence of operator algebras gives a subcontext of a strong Morita equivalence (in the sense of Rieffel) of containing C∗-algebras.
In [BK08] and [Kas08], Blecher and Kashyap, presented an extension of all this theory to the setting of dual operator algebras and weak*∗* Morita equivalence.
Basically, if (A,B,X,Y) is a weak*∗* Morita context of dual operator algebras A and B, then any W∗-algebra C generated by A induces a W∗-algebra D generated by B, such that C and D are Morita equivalent, via the W∗-equivalence bimodule C⊗AσhX.
Like in the norm-closed case, we also have that the correspondence C↦F(C)=Y⊗AσhC⊗AσhX taking W∗-algebras generated by A to W∗-algebras generated by B is bijective and order preserving ([Kas08, Theorems 5.3.5 and 5.3.6]).
Let Wmax∗(A) denote the maximal W∗-algebra of a dual operator algebra A, as defined in [BS04, section 4]. Using Z=(FE⊗AXF⊗BXE), as in the proof of theorem 3.2, we have the following:
Corollary 3.5*.*
If two W∗-correspondences (E,A) and (F,B) are weakly Morita equivalent, and C is a W∗-algebra generated by H∞(E), then
(1)
pH∞(Z)q⊗H∞(E)C⊗H∞(E)qH∞(Z)p is a W∗-algebra generated by H∞(F), which is Morita equivalent to C via the equivalence bimodule pH∞(Z)q⊗H∞(E)C. In particular, pH∞(Z)q⊗H∞(E)Wmax∗(H∞(E))⊗H∞(E)qH∞(Z)p=Wmax∗(H∞(F)) is Morita equivalent to Wmax∗(H∞(E)) via the equivalence bimodule Wmax∗(H∞(E))⊗H∞(E)qH∞(Z)p.
(2)
Wmax∗(H∞(E)) and Wmax∗(H∞(F)) are Morita equivalent W∗-algebras.
(3)
pH∞(Z)q⊗H∞(E)C≅F(C)⊗H∞(E)pH∞(Z)q. In particular, pH∞(Z)q⊗H∞(E)Wmax∗(H∞(E))≅Wmax∗(H∞(F))⊗H∞(E)pH∞(Z)q.
Proof.
(1) folows directly from Theorem 3.2 and [Kas08, Theorem 5.3.5]. Parts (2) and (3) follow from theorem 3.2 and [BK08, Theorem 5.2].
∎
4. W∗-Graph Correspondences
A directed graphG=(G0,G1,r,s) consists of two countable sets G0,G1 and functions r,s:G0→G1 identifying the range and source of each edge.
The W∗-correspondence (E,A) associated to G=(G0,G1,r,s) is given by:
A=ℓ∣G0∣∞E={x:G1→C∣v∈G0sup{s(e)=v∑∣x(e)∣2}<∞}
The left and right actions are given by (a⋅x⋅b)(e)=a(r(e))x(e)b(s(e)), where a,b∈A and x∈E. The inner product is given by ⟨x,y⟩A(v)=s(e)=v∑x(e)y(e). This definition of a W∗-graph correspondence is equivalent to the one given by Solel in [Sol04, pg 3], where the W∗-correspondence is defined in terms of matrices. Let I,J be indexing sets with ∣I∣=∣G0∣ and ∣J∣=∣G1∣. We will write the elements of A as a=(ai) or as a=i∈I∑aiδvi, and the elements of E as (zj) or as x=j∈J∑ziδej, (δvi and δej denote the point masses of a vertex and an edge respectively). Note that A is the w∗-closure of c0(G0) (by ℓ∞=c0∗∗ and Goldstine’s theorem). The norm of A is given by ∣∣a∣∣=i∈Isup∣ai∣. The norm of E is given by ∣∣x∣∣=∣∣⟨x,x⟩A∣∣21=(v∈G0sup{s(e)=v∑∣x(e)∣2})21. Note that E is a subspace of ℓ∣G1∣∞, which may also be viewed as a disjoint union v∈G0⨆ℓ∣s−1(v)∣2.
For each vertex vi∈G0, we have δvi2=δvi=δvi∗. That is, for each i∈I, δvi is a projection. If σ is a faithful normal representation of A on a Hilbert space H, then σ(δvi) is also a projection. So for any a=(ai)∈A, we have σ(a)=i∈I⨁aiIHi, a direct sum of uniformly bounded operators on H=i∈I⨁Hi, the Hilbert space direct sum. Then σ(A)=⨁i∞CiIHi, where ⨁i∞ denotes the ∞-direct sum (for more information on this sum, see for example [BL04, 1.2.17]). Since σ is faithful, it is isometric (∣∣σ(a)∣∣=isupaiIHi=isupai=∣∣a∣∣), and so are its amplifications σn. So σ is completely isometric. The dimension of each Hilbert space Hi is the multiplicity mi of the one-dimensional representation σ(i=1∑∣G0∣aiδvi)=ai. So σ is completely determined up to unitarily equivalence by the sequence (m1,m2,⋯) of these multiplicities. Since σ is faithful, 0<mi≤∞. Thus H can be written as H=i∈I⨁Cmi, where C∞ is interpreted as ℓ2.
As stated in section 2,
attached to each faithful normal representation σ of A, there is a dual correspondence Eσ, which is a W∗- correspondence over σ(A)′. Furthermore,
the ultraweakly continuous representations of H∞(E) are parametrized by the elements of D((Eσ)∗). In the following theorem, we identify the elements of (Eσ)∗ when (E,A) is a W∗-graph correspondence.
Theorem 4.1**.**
If (E,A) is a W∗-graph correspondence and σ:A→B(H) is a faithful normal representation of A, then the elements of (Eσ)∗ are block matrices (Tij) where Tij∈B(Hs(ej),Hvi) and Tij=0 if r(ej)=vi.
Proof.
Let x∈E, a∈A and h∈H. σ(a)=σ(i=1∑∣G0∣aiδvi)=i=1∑∣G0∣aiσ(δvi). So σ(A)=⨁i∞aiIHi and H=i=1⨁∣G0∣Hi. Sometimes, for clarity, we might also write σ(A)=⨁i∞aviIHvi and H=i=1⨁∣G0∣Hvi. Let x⊗h∈E⊗σH.
So E⊗σH=j=1⨁∣G1∣Cδej⊗Hs(ej)≅j=1⨁∣G1∣C⊗Hs(ej)≅j=1⨁∣G1∣Hs(ej). Then (Eσ)∗=I(σE∘φ,σ)⊂B(E⊗σH,H)=B(j=1⨁∣G1∣Hs(ej),i=1⨁∣G0∣Hvi).
Let η∗∈(Eσ)∗. So η∗ is a block matrix (Tij), where Tij∈B(Hs(ej),Hvi).
Sometimes, for clarity, we will write η∗=(Tij) as η∗=(Tviej), where Tviej∈B(Hs(ej),Hvi).
[TABLE]
That is, σE∘φ(a)(j=1∑∣G1∣zjδej⊗hs(ej))=j=1∑∣G1∣zjar(ej)δej⊗hs(ej) ,or isomorphically, σE∘φ(a)(j=1∑∣G1∣zjhs(ej))=j=1∑∣G1∣zjar(ej)hs(ej). So σE∘φ(a)=j=1⨁∣G1∣ar(ej)Is(ej). Since an intertwiner η∗∈(Eσ)∗ satisfies η∗(σE∘φ(a))=σ(a)η∗, we have (Tij)(j=1⨁∣G1∣ar(ej)Is(ej))=(i=1⨁∣G0∣aviIHvi)(Tij). So (ar(ej)Tij)=(aviTij).
Then, since each edge ej has a unique range r(ej), we must have Tij=0 if r(ej)=vi. In other words, if we write the blocks Tij as Tviej, we have that the only (possible) nonzero blocks of an intertwiner (Tviej)∈(Eσ)∗ are the blocks Tr(ej)ej∈B(Hs(ej),Hr(ej)). That is, the only nonzero block of each column ej of η∗∈(Eσ)∗ is the block on row r(ej).
∎
Furthermore, since ∣∣η∗∣∣=∣∣η∗η∣∣21, we have ∣∣η∗∣∣=∣∣(Tij)∣∣=∣∣(Tij)(Tij)∗∣∣21=∣∣i=1⨁∣G0∣r(ej)=vi∑TijTij∗∣∣. So D((Eσ)∗)={(Tij)∣Tij∈B(Hs(ej),Hvi),Tij=0ifr(ej)=vi,and∣∣i=1⨁∣G0∣r(ej)=vi∑TijTij∗∣∣<1}.
Corollary 4.2*.*
If (E,A) is a W∗-graph correspondence and σ:A→B(H) is a faithful normal representation of A, then the elements of Z((Eσ)∗) are block matrices (Tij) such that T_{ij}=\left\{\begin{array}[]{lll}z_{j}I_{H_{s(e_{j})}}&&\mbox{if }e_{j}\mbox{ is a loop}\\
0&&\mbox{otherwise}.\end{array}\right.
Proof.
It follows from [MS08, Lemma 4.12 (2)] and the definition of left and right actions in a W∗-graph correspondence.
∎
4.1. Commutants of Induced Representations of The Hardy Algebra
Our next goal is to show that if (E,A) is a W∗-correspondence derived from a directed graph G and σ,τ are two faithful normal representations of A, then (σF(E)(H∞(E)))′ is weakly Morita equivalent to (τF(E)(H∞(E)))′.
Lemma 4.3**.**
The W∗-algebras σ(A)′ and τ(A)′ are weakly Morita equivalent.
Proof.
If σ and τ are faithful normal representations of A on H and K respectively, then
H=q∈I⨁HqandK=q∈I⨁Kq ,
σ(A)=⨁q∞CqIHqandτ(A)=⨁q∞CqIKq ,
σ(A)′=⨁q∞B(Hq)andτ(A)′=⨁q∞B(Kq).
Let X=⨁q∞B(Kq,Hq). Let q∈I⨁Tq,q∈I⨁Sq∈X. We show that X with inner products
[TABLE]
is a σ(A)′ -τ(A)′ equivalence bimodule, where the left and right actions are given by regular matrix multiplication. First, we check that the two previous equations do define inner products on X. Let x=q∈I⨁Tq,y=q∈I⨁Sq,z=q∈I⨁Uq∈X, a=q∈I⨁Rq∈τ(A)′ and λ,μ∈C. Then
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Since each diagonal entry (Tq∗Tq)jj of Tq∗Tq is the sum i∑((Tq)ij)2 of the squares of the entries on column j of Tq, we have that Tq∗Tq=0 implies Tq=0. So ⟨x,x⟩τ(A)′=⟨q∈I⨁Tq,q∈I⨁Tq⟩τ(A)′=q∈I⨁Tq∗Tq=0 implies Tq=0 for all q∈I. So x=q∈I⨁Tq=0. Thus ⟨⋅,⋅⟩τ(A)′ is a right inner product on X. Similarly, σ(A)′⟨⋅,⋅⟩ is a left inner product on X.
Now we show that X is a w∗-full left Hilbert σ(A)′-module and a w∗-full right Hilbert τ(A)′-module.
Let M∈τ(A)′. So M=q∈I⨁Mq (where Mq∈B(Kq)). Assume for the moment, each Hilbert space Kq has finite dimension nq. For each q∈I, Mq=i,j=1∑nqmij(q)Eij(q) , where mij(q)∈C and {Eij(q)}i,j is the usual (matrix unit) basis for B(Kq). That is, Eij(q) is the nq×nq matrix with 1 in the i,j entry and zeros everywhere else. Let {Tij(q)}i,j be a matrix basis for B(Kq,Hq). Then Eij(q)=T1i(q)∗T1j(q)=⟨T1i(q),T1j(q)⟩τ(A)′. So Mq=i,j=1∑nqmij(q)Eij(q)=i,j=1∑nqmij(q)⟨T1i(q),T1j(q)⟩τ(A)′.
Likewise, since Eij(q)=Ti1(q)T1j(q)=Ti1(q)Tj1(q)∗=σ(A)′⟨Ti1(q),Tj1(q)⟩, we have M=q∈I⨁Mq=q∈I⨁i,j=1∑nqmij(q)Eij(q)=q∈I⨁i,j=1∑nqmij(q)σ(A)′⟨Ti1(q),Tj1(q)⟩. Then, since for each q, K(Kq) is w∗ dense in B(Kq) (by Goldstine’s theorem),
we have that X=q∈I⨁B(Kq,Hq) is a w∗-full left Hilbert σ(A)′-module and a w∗-full right Hilbert τ(A)′-module.
Next, we show that σ(A)′ acts as adjointable operators on Xτ(A)′ and τ(A)′ acts as adjointable operators on σ(A)′X. Let x,y∈X, a∈σ(A)′ and b∈τ(A)′. Then
[TABLE]
and
[TABLE]
Next, we show that the two inner products are compatible.
For all x=q∈I⨁Tq,y=q∈I⨁Sq,z=q∈I⨁Uq∈X, we have
[TABLE]
Finally, each B(Kq,Hq) is the dual of the space S1((Hq,Kq) of trace class operators. So X=⨁q∞B(Kq,Hq)=(⨁q∈I1S1(Kq,Hq))∗ is a dual space. Here, ⨁q∈I1 denotes the 1-direct sum. For more information on this direct sum and its ℓ1-norm, see [BL04, 1.4.13] for example.
Thus X is a
σ(A)′-τ(A)′W∗-equivalence bimodule. That is, σ(A)′ and τ(A)′ are weakly Morita equivalent.
∎
Theorem 4.4**.**
If (E,A) is a W∗-graph correspondence and σ:A→B(H), τ:A→B(K) are faithful normal representations of A, then (Eσ, σ(A)′)∼WMEX(Eτ, τ(A)′).
Proof.
Let X=j=1⨁∣G0∣B(Ki,Hi) be the σ(A)′-τ(A)′ equivalence bimodule given in lemma 4.3. We show that the map
[TABLE]
gives a W∗-correspondence isomorphism, where x,y∈X and (IE⊗x∗):E⊗σ(A)H→E⊗τ(A)K is defined by (IE⊗x∗)(ξ⊗h)=ξ⊗x∗h.
Let x,y∈σ(A)′Xτ(A)′, η∈Eσ, a,b∈τ(A)′. Note that φ is well defined, since for any c,d∈σ(A)′, we have φ(x⋅c⊗η⋅d⊗y)=φ(c∗x⊗ηd⊗y)=(IE⊗x∗c)ηdy=(IE⊗x∗)(IE⊗c)ηdy=(IE⊗x∗)(c⋅η)(d⋅y)=φ(x⊗c⋅η⊗d⋅y).
[TABLE]
Let x1⊗η1⊗y1, x2⊗η2⊗y2∈X⊗Eσ⊗X. Then
[TABLE]
That is, φ preserves the inner product. So it is isometric, hence injective with closed range. Now we show φ is surjective. Each element S∈Eτ has ∣L∣ nonzero blocks Sq, where L is a set with ∣G0∣≤∣L∣≤∣G1∣. Each Sq∈B(Ks(eq),Kr(eq)). For Q∈B(Ks(eq),Hs(eq)),P∈B(Hs(eq),Hr(eq)),R∈B(Hg,Kg), RPQ∈B(Ks(eq),Kr(eq)). Let MQ∈X be an element with all zero blocks except for Q, MP∈Eσ be an element with all zero blocks except for P, and MR∈(IE⊗X) be an element with all zero blocks except for R. Assume for the moment that all Hilbert spaces Hj and Kj are finite dimensional. That is, the multiplicity of the representation of δv is finite for all v∈G0. Sq=i,j⨁sijEij, where sij∈C and {Eij}i,j is a matrix basis for B(Ks(eq),Kr(eq)). Let {Tij}i,j be a matrix basis for B(Ks(eq),Hs(eq)), {Yij}i,j be a matrix basis for B(Hs(eq),Hr(eq)), and {Zij}i,j be a matrix basis for B(Hr(eq),Kr(eq)). Then Eij=ZijYj1T1j. So for x,y∈X,η∈Eσand H,K finite dimensional, the products (IE⊗x)ηy span Eτ. If the representation of δs(eq) or δr(eq) on K is not finite dimensional then since
B(Ks(eq),Kr(eq))=K(Ks(eq),Kr(eq))∗∗, the span of the finite dimensional products Eij=ZijYj1T1j is w∗-dense in B(Ks(eq),Kr(eq)) (by Goldstine’s theorem).
Summing over all q∈L, we have that the span of these finite dimensional products is w∗-dense in Eτ. So φ is surjective, thus a W∗-correspondence isomorphism.
Since by lemma 3.1, X⊗τ(A)′X≅σ(A)′ and Eσ≅σ(A)′⊗σ(A)′Eσ as W∗-correspondences, we have:
[TABLE]
as W∗-correspondences. So (Eσ, σ(A)′)∼WMEX(Eτ, τ(A)′) .
∎
In [MS99], Muhly and Solel defined the induced representationsρ of H∞(E), which play a central role in the study of Hardy algebras. Indeed, these induced representations (in the sense of Rieffel [Rie74a]) appear in most of the work related to Hardy algebras. In [MS09], Muhly and Solel showed how the commutant of ρ(H∞(E)) can be expressed in terms of induced representations of H∞(Eσ). More precisely, let σ:A→B(H) be a normal representation of A on a Hilbert space H and form the Hilbert space F(E)⊗σH. The induced covariant representation of E determined by σ is the representation (T,φ∞⊗IH) where T:E→B(F(E)⊗σH) is defined by T(ξ)(η⊗h)=(ξ⊗η)⊗h for ξ∈E and η⊗h∈F(E)⊗σH. The representation of H∞(E), induced by σ, denoted by σF(E), is the integrated form of (T,φ∞⊗IH). For X∈H∞(E), σF(E)(X) is also written as X⊗IH. Define a map U:F(Eσ)⊗ιH→F(E)⊗ιH (where ι denotes the identity representation of σ(A)′ in B(H)) by
[TABLE]
By [MS04, Lemma 3.8], U is a Hilbert space isometric isomorphism and by [MS04, Theorem 3.9], the representation ρ of H∞(Eσ) on F(E)⊗σH, defined by the formula
[TABLE]
is an ultraweakly homeomorphic, completely isometric isomorphism from H∞(Eσ) onto (σF(E)(H∞(E)))′. Likewise, the map υ, defined by
[TABLE]
is an ultraweakly homeomorphic, completely isometric isomorphism from H∞(E) onto (ιF(Eσ)(H∞(Eσ)))′.
Theorem 4.5**.**
If (E,A) is a W∗-graph correspondence and σ:A→B(H), τ:A→B(K) are faithful normal representations of A, then (σF(E)(H∞(E)))′∼WME(τF(E)(H∞(E)))′.
Proof.
By theorem 4.4, (Eσ, σ(A)′)∼WME(Eτ, τ(A)′). Then by theorem 3.2, H∞(Eσ))∼WMEH∞(Eτ). So by the isomorphism ψ above, we have
[TABLE]
∎
Note also that if (E,A) is a graph correspondence and σ:A→B(H), τ:A→B(K) are faithful normal representations of A, then the map υ above, gives us
[TABLE]
4.2. Morita Equivalence of W∗-Graph Correspondences
Let X be a countable set, A=C(X) (with the sup norm) and let C(X)XC(X) = AXA be a W∗-equivalence bimodule. By [Pas73, Theorem 3.11] and Zorn’s lemma, X has an orthonormal basis A consisting of mutually orthogonal non zero partial isometries. That is, for each ei∈A, ⟨ei,ei⟩ is a nonzero orthogonal projection in A, and for each g∈X, g=i∑ei⟨ei,g⟩. In particular, i∑\Upthetaei,ei=IX where \Upthetaei,ei is the usual rank-one operator in K(X). The elements of A are linearly independent, otherwise there would be ej∈A such that ej=i=j∑ziei (zi∈C). But then we would have 0<⟨ej,ej⟩A<⟨ej,i=j∑ziei⟩A=i=j∑zi<⟨ej,ei⟩A=i=j∑zi(0)=0.
Since A=ℓ∞ can be identified with C(βN), where βN denotes the Stone Cech compactification of N, the maximal ideals of C(X) are {Ix}x∈X where
Ix={y∈X∑ayδy:ay∈C,sup∣ay∣<∞andy=x}. The maximal C(X)-C(X)-submodules of X are {Xj}j∈{1,⋯,n} where Xj={i∑ziei:zi∈C,sup∣zi∣<∞andi=j}. Since the Rieffel correspondence of AXA, pairs maximal ideals of A=C(X) with maximal submodules of X, we have that dim(X)=∣A∣=dim(C(X))=∣X∣.
If the corresponding submodule (under the Rieffel correspondence) for the maximal ideal Ix is the maximal submodule Xj, then Xj=X⋅Ix [RW98, Lemma 3.23]. So ei⋅δy=x=ej for all ei∈A. But by Cohen’s factorization theorem, ej=e⋅a for some e∈X, a∈C(X). So we must have ei⋅δx=ej for some ei∈A. Then we have ei=ej (otherwise we would have ei⋅δx=ej for i=j and 0<⟨ej,ej⟩C(X)=⟨ej,ei⋅δx⟩C(X)=⟨ej,ei⟩C(X)⋅δx=0⋅δx=0. Thus the element x∈X (and therefore δx∈C(X)) gets uniquely paired up with the element ej∈A. Likewise, each basis element δy∈C(X) gets uniquely paired up with a basis element ei∈X by the right action relation ei⋅δy=ei. So we have a bijection R between the basis elements {ei} in X and the basis elements {δy} in C(X). Applying the same analysis to the Rieffel correspondence between C(X) and X, but now with C(X) giving the left action on X, we have another bijection L:{δy}y∈X→A. Thus σ=R∘L is a permutation of {δx:x∈X}, or equivalently, σ is a permutation of X given by the Rieffel correspondence of C(X)XC(X). Note that δy⋅L(δx)⋅δw=L(δx) if y=x and w=σ(x). If we let AAσA denote the algebra A (viewed as a bimodule over itself) with a modified right action and right inner product (given by σ), we obtain the following result:
Lemma 4.6**.**
C(X)XC(X)=AXA* is a W∗-equivalence bimodule if and only if AXA is of the form AAσA, where σ=R∘L is the permutation given by the Rieffel correspondence of AXA.*
Proof.
Let AAσA denote the algebra A (viewed as a W∗-bimodule over itself) with the right and left actions given by:
[TABLE]
and the right and left inner products given by:
[TABLE]
So that if x∈X∑axδx,x∈X∑cxδx∈Aσ and y∈X∑byδy∈A, then the right and left actions and inner products are:
[TABLE]
First we check that AAσA is a W∗-equivalence bimodule:
Since {δx:x∈X} spans C(X)=A and ⟨δσ−1(x),δσ−1(x)⟩C(X)=δx and C(X)⟨δx,δx⟩=δx, we have that Aσ is a w∗-full left Hilbert A-module and a w∗-full right Hilbert A-module.
Let s,t∈Aσ, a∈A.
[TABLE]
[TABLE]
Let r,s,t∈Aσ.
[TABLE]
Since A has an operator space predual (being a W∗-algebra), it is a selfdual C∗-module over itself.
Thus AAσA is a W∗-equivalence bimodule.
Let ψ:AAσA→AXA be the linear extension of the bijection L:{δy}y∈X→A that we encountered above when we studied the Rieffel correspondence of AXA. For any element e=x∈X∑zxδx∈Aσ, we have ψ(e)=ψ(x∈X∑zxδx)=x∈X∑zxL(δx). We show now that ψ is a W∗-equivalence bimodule isomorphism. Recall that δy⋅L(δx)⋅δw=L(δx) in AXA if y=x and w=σ(x). Let a,b∈A and e∈Aσ. Then
[TABLE]
So ψ is a bimodele map. Note that if the Rieffel correspondence pairs up ej∈X and δx∈C(X) by ej⋅δx=ej, then since 1=∣∣⟨ej,ej⟩C(X)∣∣=∣∣⟨ej⋅δx,ej⋅δx⟩C(X)∣∣=∣∣δx⟨ej,ej⟩C(X)δx∣∣=∣∣δx⟨ej,ej⟩C(X)∣∣ and δxδy=0 for all x=y, we must have ⟨ej,ej⟩C(X)=δx=R(ej). Likewise, for any ei∈X and δx∈C(X) paired up by δx⋅ej=ej, we must have C(X)⟨ej,ej⟩=δx. Thus
[TABLE]
So if e,f∈Aσ, we have
[TABLE]
[TABLE]
Thus ψ preserves both inner products (so it is injective). Since A spans X, and ψ(L−1(ei))=L((L−1(ei))=ei, ψ is surjective. Thus an isomorphism.
∎
We can also view AAσA as a graph correspondence. More precisely, let Gσ=(Gσ0,Gσ1,r,s) be the directed graph given by Gσ0=X, Gσ1={ex}x∈X, r,s:Gσ1→Gσ0 given by r(ex)=x and s(ex)=σ(x). Then the graph correspondence C(Gσ0)C(Gσ1)C(Gσ0) associated to Gσ with the usual actions and inner products:
[TABLE]
is isomorphic to AAσA≅AXA via the map ω:C(Gσ1)→Aσ given by ω(δex)=δx.
Note that ∣Gσ1∣=∣Gσ0∣ and r,s are bijections. So if the graph Gσ is finite then Gσ is either a cycle or a disconnected union of cycles (given by the cycle decomposition of σ). Note also that each permutation σ of X gives an equivalence bimodule AC(Gσ1)A≅AAσA.
If [X] denotes the isomorphism class of AXA, then we have:
Lemma 4.7**.**
P={[X]:A∼WMEXA}* is a group with the operation given by [X]∗[Y]=[X⊗AY]*
Proof.
First we show that if X and Y are A-AW∗-equivalence bimodules, then so is X⊗AY. X is a w∗-full right A-module, and by Cohen’s factorization theorem, Y=A⋅Y. Thus ⟨X,X⟩A⋅Y is w∗-dense in Y. So ⟨X⊗AY,X⊗AY⟩A=⟨Y,⟨X,X⟩A⋅Y⟩A is w∗-dense in ⟨Y,Y⟩A, which is w∗-dense in A, since Y is a w∗-full right A-module. So ⟨X⊗AY,X⊗AY⟩A is w∗-dense in A. Thus AX⊗AYA is a full right Hilbert A-module. Likewise, AX⊗AYA is a full left Hilbert A-module.
Let x,y∈AX⊗AYA and a,b∈A. Then ⟨a⋅x,y⟩A=⟨a⋅(x1⊗y1),y1⊗y2⟩A=⟨a⋅x1⊗y1,y1⊗y2⟩A=⟨y1,⟨a⋅x1,x2⟩A⋅y2⟩A=⟨y1,⟨x1,a∗⋅x2⟩A⋅y2⟩A=⟨x1⊗y1,a∗⋅x2⊗y2⟩A=⟨x1⊗y1,a∗⋅(x2⊗y2⟩A)=⟨x,a∗⋅y⟩A and A⟨x⋅b,y⟩=A⟨(x1⊗y1)⋅b,x2⊗y2⟩=A⟨x1⊗y1⋅b,x2⊗y2⟩=A⟨x1,x2⋅A⟨y1⋅b,y2⟩∗⟩=A⟨x1,x2⋅A⟨y1,y2⋅b∗⟩∗⟩=A⟨x1⊗y1,x2⊗y2⋅b∗⟩=A⟨x1⊗y1,(x2⊗y2)⋅b∗⟩=A⟨x,y⋅b∗⟩.
Let x,y,z∈AX⊗AYA. Then A⟨x,y⟩⋅z=A⟨x1⊗y1,x2⊗y2⟩⋅x3⊗y3=A⟨x1⋅A⟨y1,y2⟩,x2⟩⋅x3⊗y3=x1⋅A⟨y1,y2⟩⋅⟨x2,x3⟩A⊗y3=x1⊗A⟨y1,y2⟩⟨x2,x3⟩A⋅y3=x1⊗A⟨y1,⟨x2,x3⟩A∗⋅y2⟩⋅y3=x1⊗y1⋅⟨⟨x2,x3⟩A∗⋅y2,y3⟩=x1⊗y1⋅⟨y2,⟨x2,x3⟩A⋅y3⟩A=x1⊗y1⋅⟨x2⊗y2,x3⊗y3⟩A=x⋅⟨y,z⟩A.
Thus AX⊗AYA is a W∗-equivalence bimodule. Since AA⊗AEA≅AEA≅AE⊗AAA, the identity of P is [A]. By lemma 3.1, [X]−1=[X]. Thus P is a group.
∎
Lemma 4.8**.**
If σ,τ∈SX then AAσA≅AAτA as W∗-equivalence bimodules.
Proof.
If a=x∈X∑axδx∈A, denote x∈X∑aσ(x)δx by aσ. Consider the triple (ι,ι,π):AAσA→AAτA where ι is the identity map on A and π:A→A is given by π(a)=aτ−1σ. That is, π(x∈X∑zxδx)=x∈X∑zτ−1(σ(x))δx. Then
[TABLE]
So (ι,ι,π) is a bimodule homomorphism.
[TABLE]
and
[TABLE]
So (ι,ι,π) preserves inner products. Thus (ι,ι,π) is a W∗-equivalence bimodule isomorphism.
∎
By lemma 4.6 and lemma 4.8, P={[X]:A∼WMEXA} consists of only one element:
Theorem 4.9**.**
If A=C(X) for some set X, then P={[X]:A∼WMEXA}={[A]}.
Now consider the W∗-equivalence bimodule AAσ⊗AAτA. Since this bimodule is balanced over A, we have that δx⋅δz⊗δy=δx⊗δz⋅δy if and only if σ(x)=z=y. Thus the non zero elements of AAσ⊗AAτA are of the form ∑(zxδx⊗wσ(x)δσ(x)), where zx, wσ(x)∈C. Note that if σ,τ∈SX, then lemma 4.7 and lemma 4.8 say that AAσ⊗AAτA is isomorphic to AAA. Here we give an explicit W∗-isomorphism between these two W∗-equivalence bimodules. Consider the triple
[TABLE]
where ω:A→A is given by ω(a)=aσ−1 (that is, π(x∈X∑axδx)=x∈X∑aσ−1(x)δx), ψ:Aσ⊗AAτ→A is given by ψ(x∈X∑zxδx⊗Ax∈X∑wxδx)=ψ(x∈X∑zxδx⊗wσ(x)δσ(x))=x∈X∑zxwσ(x)δσ(x) and π:A→A is given by π(a)=aτ. That is, π(x∈X∑axδx)=x∈X∑aτ(x)δx. Then
[TABLE]
So (ω,ψ,π) is a bimodule homomorphism.
[TABLE]
and
[TABLE]
So (ω,ψ,π):AAσ⊗AAτA→AAA preserves inner products. Thus, it is injective. For each x∈X∑axδx∈A, ψ(x∈X∑axδσ−1(x)⊗x∈X∑δx)=ψ(x∈X∑axδσ−1(x)⊗δx)=x∈X∑axδx. So (ω,ψ,π) is surjective. Thus a W∗-equivalence bimodule isomorphism.
Lemma 4.10**.**
Let AEA and BFB be two W∗-correspondences. If AEA≅BFB then AEA∼WMEBFB.
Proof.
If AEA≅BFB then there is a W∗-correspondence isomorphism (π,ϕ):AEA→BFB, where ϕ is a vector space isomorphism and π:A→B is a W∗-algebra isomorphism. Then B is an A-BW∗-equivalence bimodule with the left action given by a⋅b=π(a)b, right action given by multiplication in B and inner products given by ⟨b1,b2⟩B=b1∗b2 and A⟨b1,b2⟩=π−1(b1b2∗).
We show that (ι,φ):BB⊗AE⊗ABB→BFB is a W∗-correspondence isomorphism, where ι is the identity map and φ:B⊗AE⊗AB→F is defined by φ(b⊗e⊗c)=b∗⋅ϕ(e)⋅c. Let e,g∈E and a,b,c,d,α,β∈B. Then
[TABLE]
So (ι,φ,ι) is a correspondence homomorphism.
[TABLE]
So (ι,φ,ι) preserves the inner product. Thus it is injective. Since for each f∈F, there is e∈E such that ϕ(e)=f, we have φ(1⊗e⊗1)=f. So φ is surjectve, thus a W∗-correspondence isomorphism.
Since BB⊗AE⊗ABB≅BFB, we have AB⊗BF≅AB⊗BB⊗AE⊗ABB≅AA⊗AE⊗ABB≅AE⊗ABB. Thus AEA∼WMEBFB.
∎
Theorem 4.11**.**
If AEA and BDB are W∗-graph correspondences then AEA∼WMEBDB if and only if AEA≅BDB.
Proof.
One direction was already shown in lemma 4.10. Now we show the converse. If AEA∼WMEBDB then we have A∼WMEB, and since A and B are commutative, we have A≅B (recall that if two W∗-algebras are Morita equivalent then their centers are isomorphic). So there is a W∗-algebra isomorphism α:B→A, such that (α,ι):BDB→ADA is a W∗-correspondence isomorphism. Then by lemma 4.10, we have AEA∼WMEADA. By lemma 4.6, a W∗-equivalence bimodule AXA is isomorphic to AAσA, where AAσA is the same as AAA but with a modified right action and right inner product determined by some permutation σ of S. Then AEA∼WMEADA implies Aσ⊗AD≅E⊗AAσ. So using lemma 3.1, we have
[TABLE]
Thus, to show that AEA≅ADA, all we need to show is that Aσ⊗AE⊗AAσ≅AEA.
Consider the pair (π,ϕ) where ϕ:Aσ⊗AE⊗AAσ→AEA is given by ϕ(a⊗x⊗b)=a∗⋅x⋅b and π:A→A is given by π(c)=cσ. That is, π(x∈X∑ciδvi)=x∈X∑cσ(i)δvi. Clearly, π is a W∗-isomorphism. Now we show that (π,ϕ):AAσ⊗AE⊗AAσA→AEA is a W∗-correspondence isomorphism. Let a,b,c,d∈Aσ, α,β∈A and x,y∈E.
[TABLE]
and
[TABLE]
So ϕ is isometric, thus injective. Since for each e∈E, ϕ(1A⊗e⊗1A)=1⋅e⋅1=e, ϕ is surjective. Thus (π,ϕ) is a W∗-correspondence isomorphism.
∎
Two directed graphs G=(G0,G1,s1,r1) and F=(F0,F1,s2,r2) are isomorphic if there are two bijections α:G1→F1 and β:G0→F0 such that for each edge e∈G1, s2(α(e))=β(s1(e)) and r2(α(e))=β(r1(e)).
Clearly, if we draw a directed graph G=(G0,G1,s1,r1) and relabel its edges and its vertices then we produce a new graph F=(F0,F1,s2,r2) whose identical drawing implies that the two relabeling bijections γ:G1→F1 and λ:G0→F0 satisfy s2(γ(e))=λ(s1(e)) and r2(γ(e))=λ(r1(e)). So we obtain an isomorphic graph. In particular, if G1=F1 and G0=F0 then γ and λ are permutations.
Theorem 4.12**.**
Let AEA and BDB be W∗-graph correspondences associated to the directed graphs G=(G0,G1,s1,r1) and F=(F0,F1,s2,r2) respectively. AEA≅BDB if and only if G≅F.
Proof.
First note that G≅F is a particular case of having three bijections α:G1→F1 and β,γ:G0→F0 such that for each edge ei∈G1, s2(α(ei))=γ(s1(ei)) and r2(α(ei))=β(r1(ei). More precisely, G≅F is the special case when β=γ.
If G and F are isomorphic graphs, then there are two bijections α:G1→F1 and β:G0→F0 such that for each edge ei∈G1, s2(α(ei))=β(s1(ei)) and r2(α(ei))=β(r1(ei) then let φ:E→D be given by φ(δei)=δα(ei) and ω:A→B be given by ω(δvi)=δβ(vi). Then
[TABLE]
and
[TABLE]
So
[TABLE]
and
[TABLE]
Since for each δei∈D, φ(δα−1(ei))=δei, φ is surjective. Thus (ω,φ):AEA→BDB is a W∗-correspondence isomorphism.
Now we show the converse. If AEA≅BDB, then there is a W∗-correspondence isomorphism (ω,φ):AEA→BDB. Since ω and φ are bijections, we have ∣G1∣=∣F1∣ and ∣G0∣=∣F0∣. Since relabeling vertices and edges gives an isomorphic graph, we may assume that G0=F0 and G1=F1. Since each δvi is a projection, ω(δvi)=ω(δvin)=ω(δvi)n for all positive integers n. So ω(δvi) is of the form j∑δvj, and since ω is an isometry, we have ω(δvi)=δvt for some vertex vt.
Let ei∈G1, ω(δs1(ei))=δvk and φ(δei)=∑zjδej. Since ∣φ(δei)∣=1 (being an isometry) and φ(δei)=φ(δei⋅δs(ei))=φ(δei)⋅ω(δs(ei))=(∑zjδej)⋅(δvk)=zδs2−1(vk), we must have, φ(δei)=δs2−1(vk). Thus ω and φ are given by permutations.
Let β:G0→F0 be the permutation given by β(vj)=vk if ω(δvj)=δvk. Let α:G1→F1 be the permutation given by α(ei)=em if φ(δei)=δem. In BDB, we have:
δβ(r1(ei))⋅δα(ei)⋅δβ(s1(ei))=ω(δr1(ei))⋅φ(δei)⋅ω(δs1(ei))=φ(δr1(ei)⋅δei⋅δs1(ei))=φ(δei)=δα(ei). So β(s1(ei))=s2(α(ei)) and β(r1(ei))=r2(α(ei)). G anf F are isomorphic graphs.
∎
Corollary 4.13*.*
Let AEA and BDB be W∗-graph correspondences associated to the directed graphs G=(G0,G1,s1,r1) and F=(F0,F1,s2,r2) respectively. AEA∼WMEBDB if and only if G≅F.
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