Morita equivalence for relative Cuntz–Pimsner algebras

It is defined on a 2-arroww: (F₀, V₀)⇒(F₁, V₁) byL(w) = (w⊗1OJ1,E1)^].

Let (F, V): (A,E,J)→(A₁,E₁,J₁) and (F₁, V₁): (A₁,E₁,J₁)→(A₂,E₂,J₂) be proper covariant correspondences. The isomorphism

λ((F, V),(F1, V1)):L(F1, V1)◦L(F, V)∼=L((F1, V1)◦(F, V)) is built out of the left action ofO_J^e

1,E1 on (F1⊗A₂O^e_J

2,E2)^] constructed in Proposition 5.4.2. That is, it is given by the canonical isomorphism

(F ⊗A₁O_J^e_1,E1)^]⊗_O^e

J1,E1 (F1⊗A₂O^e_J2,E2)^]∼= (F ⊗A₁F1⊗A₂O_J^e_2,E2)^].

The compatibility isomorphism for units is obtained from the nondegenerate^∗-homomorphismj_J:A→ O_J^e_,E.

5.4.2 Morita equivalence for relative Cuntz–Pimsner algebras

LetC^G denote the bicategory whose objects are C^∗-algebras carrying a coaction of G. Arrows are correspondences with a coaction ofGcompatible with those on the underlying C^∗-algebras. We refer to [18, Definition 2.10] for a precise definition. See also [18, Theorem 2.15] forC^G. Let C^G_pr be the sub-bicategory ofC^G whose arrows are proper correspondences.

Corollary 5.4.4. The construction of relative Cuntz–Pimsner algebras is functorial. There is a homomorphism of bicategoriesC^P_pr →C^G_pr which is defined on objects by

(A,E,J)7→ O_J,E.

Proof. It follows from Proposition 5.4.1 that a proper covariant correspondence between two simplifiable product systems of Hilbert bimodules gives rise to a nondegenerate proper correspondence between their relative Cuntz–Pimsner algebras with a gauge-compatible coaction ofG. This yields a homomorphism of bicategoriesC^P_pr,∗→C^G_pr which sends (B,G,IG) to OIG,G and a proper covariant correspondence (F, V): (B,G,IG) → (B₁,G1,IG1) to OF,V. Composing such a homomorphism with the reflector

obtained in Corollary 5.4.3, we obtain a homomorphismC^P_pr→C^G.

By Proposition 5.2.1,OIG,Gis naturally isomorphic to the cross sectional C^∗-algebra of the associated Fell bundle extended fromP. This establishes a canonical isomorphism

O_OJ,E,I

OJ,E

∼=O_J,E

becauseOJ,E is isomorphic to the cross sectional C^∗-algebra of the Fell bundle associated to the gauge coaction ofG. HereOOJ,E,I_OJ,E is the relative Cuntz–Pimsner algebra for Katsura’s ideals of (O^p_J,E)p∈P. From this we obtain a homomorphism of bicategoriesC^P_pr→C^G which maps (A,E,J) toOJ,E and a proper covariant correspondence (F, V): (A,E,J)→(A₁,E₁,J₁) toO_F,V. By construction, such a homomorphism is naturally equivalent to the one described in the previous paragraph.

Corollary 5.4.5. Let(A,E,J) and(B,G,JB)be objects of C^P_pr. ThenOJ,E and OJB,G are Morita equivalent if there is a covariant correspondence (F, V): (A,E,J)→(B,G,J_B)so that J_p^AF =FJ_p^B

for allp∈P and F:A;B establishes a Morita equivalence. For objects in C^P_pr,∗, this equivalence preserves amenability of Fell bundles.

Proof. First, notice that F is automatically a proper correspondence. By Corollary 5.4.4 and Lemma 3.1.9, it suffices to show thatF is an invertible arrow inC^P_pr. That is, there is a proper covariant correspondence (F^∗,Ve): (B,G,J_B)→(A,E,J) with (invertible) 2-arrows w: (F ⊗_BF^∗, V •Ve)⇒ satisfies the coherence axiom (4.3.2). To do so, letp, q∈P. SinceV_q⁻¹ intertwines the left actions ofA, the following diagram commutes:

Applying the coherence axiom (4.3.2) to (F, V), we deduce that the top-right composite of (5.4.6) is precisely the isomorphism

This corresponds to the top composite of the diagram

Gp⊗BGq⊗BF^∗

after tensoring with 1F^∗ on the left and on the right. The left-bottom composite of this diagram is obtained from that of (5.4.6) in the same way. Hence the commutativity of (5.4.6) implies that (F^∗,Ve) is a proper covariant correspondence from (B,G,J_B) to (A,E,J), as desired. The canonical

isomorphismsw:F ⊗_BF^∗∼=Aandwe:F^∗⊗_AF ∼=B are the required 2-arrows.

If (F, V): (A,E,I_E)→(B,G,I_G) is an equivalence inC^P_pr,∗ and ( ˆGg)g∈G is amenable, then ( ˆEg)g∈G

is also amenable. Indeed, by functoriality,O_F,V: O_IE,E ;O_IG,G is an imprimitivity bimodule. In particular,O_IE,E ∼= C^∗ ( ˆEg)g∈Gacts faithfully on O_F,V. Since C^∗ ( ˆGg)g∈G∼= C^∗r ( ˆGg)g∈Gthrough the regular representation,F^∗⊗AC^∗r ( ˆEg)g∈G is a faithful proper correspondence C^∗ ( ˆGg)g∈G

→ C^∗r ( ˆEg)g∈G

because the conditional expectation from C^∗ ( ˆGg)g∈G

onto B is faithful and the con-tinuous projection fromF^∗⊗AC^∗r ( ˆEg)g∈G

ontoF^∗⊗AA∼=F^∗ provides the image of C^∗ ( ˆGg)g∈G inB F^∗⊗_AC^∗r(( ˆE_g)g∈G)with a topologicalG-grading. From this we obtain a faithful and proper correspondence

O_F,V ⊗_C∗(( ˆGg)g∈G)F^∗⊗AC^∗r(( ˆEg)g∈G): C^∗(( ˆEg)g∈G);C^∗r(( ˆEg)g∈G). Hence the isomorphism

OF,V ⊗_C∗(( ˆGg)g∈G)F^∗⊗AC^∗r(( ˆEg)g∈G)∼=F ⊗BF^∗⊗AC^∗r(( ˆEg)g∈G)∼= C^∗r Eˆg)g∈G yields an injective^∗-homomorphism C^∗ ( ˆEg)g∈G

→C^∗r ( ˆEg)g∈G

when composed with the^∗ -homomor-phism

C^∗ ( ˆEg)g∈G

→B OF,V ⊗_C∗(( ˆGg)g∈G)F^∗⊗AC^∗r(( ˆEg)g∈G) .

But such a^∗-homomorphism coincides with the regular representation, since (F^∗,Ve)◦(F, V)∼= (A, ι_E).

Therefore, ( ˆEg)g∈G is amenable.

Remark 5.4.7. The fact that an equivalence between objects in C^P_pr,∗preserves amenability could also be derived from [2] and Theorem 5.2.11.

Example 5.4.8. LetAandB be C^∗-algebras and letF: A→B be an imprimitivityA, B-bimodule. A compactly aligned product systemE = (Ep)p∈P overAinduces a compactly aligned product system G= (Gp)p∈P overB as follows. We setGp:=F^∗⊗AEp⊗AF. The multiplication mapµep,q:Gp⊗AGq ∼= Gpq is defined using the isomorphismF ⊗AF^∗∼=A. More explicitly, it is given by

Gp⊗BGq =F^∗⊗AEp⊗AF ⊗BF^∗⊗AEq⊗AF

∼=F^∗⊗AEp⊗AEq⊗AF (1F^∗⊗µp,q⊗1F)

∼=F^∗⊗AEpq⊗AF =Gpq.

The multiplication maps {µe_p,q}p,q∈P satisfy the coherence axiom required for product systems be-cause{µp,q}p,q∈P do so.

We claim thatG is compactly aligned. Indeed, letp, q∈P withp∨q <∞. Notice thatK(Gp) is canonically isomorphic toF^∗⊗AK(Ep)⊗AF through the identification

F^∗⊗_AE_p⊗_AF ⊗_B(F^∗⊗_AE_p⊗_AF)^∗∼=F^∗⊗_AE_p⊗_AF ⊗_BF^∗⊗_AE_p^∗⊗_AF

∼=F^∗⊗AEp⊗AE_p^∗⊗AF

∼=F^∗⊗AK(Ep)⊗AF.

So takeT ∈K(Ep) andS ∈K(Eq). Letζ1, ζ2, η1, η2∈ F and letη^∗⊗ξ⊗ζ be an elementary tensor of F^∗⊗AEp∨q⊗AF. We have that

ι^p∨q_q (η₂^∗⊗S⊗ζ2)(η^∗⊗ξ⊗ζ) =η₂^∗⊗ι^p∨q_q (S) ϕp∨q(hhζ2|ηii)(ξ)

⊗ζ.

Applyingι^p∨q_p (η₁^∗⊗T⊗ζ₁) to both sides of the above equality, we deduce that ι^p∨q_p (η₁^∗⊗T⊗ζ₁)ι^p∨q_q (η₂^∗⊗S⊗ζ₂)(η^∗⊗ξ⊗ζ)

=η₁^∗⊗ι^p∨q_p (T) ϕp∨q(hhζ1|η2ii)ι^p∨q_q (S) ϕp∨q(hhζ2|ηii)(ξ)

⊗ζ.

DefineT⁰∈K(Ep∨q) byT⁰=ι^p∨q_p (T)ϕ_p∨q(hhζ1|η₂ii)ι^p∨q_q (S). Then η₁^∗⊗T⁰ ϕ_p∨q(hhζ₂|ηii)(ξ)

⊗ζ= (η₁^∗⊗T⁰⊗ζ₂)(η^∗⊗ξ⊗ζ). SoG is also compactly aligned, as claimed.

Givenp∈P, an elementb∈B is compact onGpif and only ifbF^∗⊆ F^∗ϕ⁻¹_p (K(Ep)), providedF^∗ is an equivalence. The bijection between the lattices of ideals ofA andB, respectively, obtained from the Rieffel correspondence, yields a one-to-one correspondence between ideals inA acting by compact operators onEp and ideals in B mapped to compact operators onGp. Precisely, this sends J_p^A/ ϕ⁻¹_p (K(Ep)) to J_p^B=hJ_p^AF | F i.Its inverse maps an idealJ_p^B /ϕe⁻¹_p (K(Gp)) toJ_p^A=hhFJ_p^B| F ii. The equivalenceF may be turned into a proper covariant correspondence (F, V): (A,E,JA)→ (B,G,JB), whereV ={Vp}p∈P andVp: Ep⊗AF ∼=F ⊗BGp arises from the canonical isomorphism

Ep⊗AF ∼=F ⊗BF^∗⊗AEp⊗AF =F ⊗BGp. HereJA andJB are related by the bijection described above.

It follows from Corollary 5.4.5 that (F, V) is invertible inC^P_pr and produces a Morita equivalence between OJA,E and OJB,G. Therefore, up to equivariant Morita equivalence, the relative Cuntz–

Pimsner algebras associated toE correspond bijectively to those associated toG. In particular, ifE is a simplifiable product system of Hilbert bimodules, the cross sectional C^∗-algebra of the Fell bundle associated toE is Morita equivalent to that ofG. This is so because the family of Katsura’s idealsI_E corresponds toI_G under the Rieffel correspondence.

The next proposition characterises equivalences between product systems built out of semigroups of injective endomorphisms with hereditary range as in Example 4.2.10. This generalises [42, Proposition 2.4]. The idea of the proof is also taken from there.

Proposition 5.4.9. Let α:P → End(A) and β:P → End(B) be actions by extendible injective endomorphisms with hereditary range. Let _αA and _βB be the associated product systems of Hilbert bimodules over P^op. There is an equivalence (F, V): (A, A_α ,I_αA) → (B, B_β ,I_βB) if and only if there are an imprimitivityA, B-bimoduleF and a semigroup homomorphismp7→Up fromP to the semigroup ofC-linear isometries on F such that, for all p∈P andξ, η∈ F, becauseβ_p⁻¹ is an injective^∗-homomorphism between C^∗-algebras. Its composition withU_p⁰ yields a linear mapF → F, which we denote by U_p. Givenξ, η ∈ F, we have that hξ|ηi=hU_p⁰(ξ)|U_p⁰(η)i, that is,U_p⁰ preserves inner products. From this we deduce

hU_p(ξ)|U_p(η)i=β_p hU_p⁰(ξ)|U_p⁰(η)i=β_p(hξ|ηi). Similarly,hhU_p⁰(ξ)|U_p⁰(η)ii=hhξ|ηii and we see thathhUp(ξ)|Up(η)ii=αp(hhξ|ηii).

It remains to verify thatp7→Upis a semigroup homomorphism fromP to the semigroup ofC-linear isometries onF. First, let αq(1)a ∈ _αqA and notice that, given an elementary tensor ξ⊗αp(1)b

of (µ^α_q,p⊗1) αq(1)a⊗AV_p⁻¹(ξ⊗βp(1)b)

From the above observation and from the fact thatV_p⁻¹ andV_q⁻¹intertwine the right actions ofB, we conclude that

U_p⁰U_q(ξu_λu_λb) = (µ^α_q,p⊗1)(1⊗V_p⁻¹⊗1B_βp) V_q⁻¹(ξ⊗β_q(u_λ))⊗β_pq(u_λ)⊗β_pq(b) .

Combining this with the coherence condition (4.3.2) we may replace the right-hand side of the above equality by

Conversely, suppose that we are given an imprimitivityA, B-bimodule F and a semigroup homo-morphismp7→Up fromP to the semigroup of C-linear isometries on F satisfying (5.4.10). For each p∈P,ξ∈ F andb∈B, we have thatUp(ξb) =Up(ξ)βp(b) because product. In addition, it intertwines the left and right actions ofAandB.

Now we letVep:F ⊗B_βB where the isomorphism on the left-hand side comes from the identification

α_pA⊗_AαA

p

∗∼=hh_αA

p |_αA

p ii=A.

ThenVep is an isometry between correspondencesA;B. To see that it is indeed unitary, we need to prove that it is also surjective.

First, observe that

αp(hhξ|ηii)ζ=hhUp(ξ)|Up(η)iiζ=Up(ξ)hUp(η)|ζi.

This implies αp(A)F = Up(F)hUp(F)| F i, provided hhF | F ii = A. Again we let (uλ)λ∈Λ be an approximate identity forA and fixλ∈Λ. Letc∈Abe such that uλ=c^∗c. Takea∈Aandξ∈ F. Then

αp(uλ)a⊗Aξ=αp(c^∗)⊗Aαp(c)(aξ)∈αp(c^∗)⊗AUp(F)hUp(F)| F i.

Using thatUp(F) =αp(A)Up(F), we deduce thatαp(uλ)a⊗Aξ belongs to the image ofVep. This has closed range and henceα_p(1)a⊗ξalso lies in Ve_p(F ⊗_{B β}B

p ). Applying again the fact thatVe_p has closed range, we conclude that it is indeed unitary.

We let Vp = Ve_p^∗ and V = {Vp}_p∈P. We shall now prove that (F, V) is a proper covariant correspondence. In this case, it suffices to show that it satisfies the coherence axiom (4.3.2) and thatVe

is the canonical isomorphism obtained from the left and right actions ofA andB, respectively. This latter fact follows from the identities

hhUe(ξ)|ηii=hhξ|Ue(η)ii=hhξ|ηii=hhUe(ξ)|Ue(η)ii, so thatUe= idF. The above equalities may be derived from the computation

hhUe(ξ)|ηii=α_e(hhUe(ξ)|ηii) =hhUe(U_e(ξ))|U_e(η)ii=hhUe(ξ)|U_e(η)ii=hhξ|ηii.

Finally, givena, c∈A,b, d∈B andξ∈ F, we have

cU_q(aU_p(ξ)b)d=cα_q(a)U_q(U_p(ξ))β_q(b)d=cα_q(a)U_qp(ξ)β_q(b)d.

This leads to a commmutative diagram for Vep, Veq andVeqp as in (4.3.2). By reversing arrows, we conclude that (F, V) also makes such a diagram commute. This completes the proof.

C ^∗ -algebras for product systems over subsemigroups of groups

In this chapter, we treat product systems over semigroups that can be embedded in groups. We combine ideas of Exel and Sims and Yeend (see [22, 55]) to construct a C^∗-algebra A×E P out of a product systemE so that a representation ofA×E P is faithful on its fixed-point algebra for the canonical coaction of a group containingP if and only if it is faithful on the coefficient algebra. We begin by considering a family of representations of the Toeplitz algebra ofE. We use its topological grading from Lemma 4.1.6 to define an ideal inTE, so that the coaction descends to the corresponding quotient. We then prove thatAembeds into this quotient. This is done in Section 6.1.

In Section 6.2, we show that a representation of such a quotient ofT_E is faithful on its fixed-point algebra if and only if it is faithful on A. We apply this to prove that this construction does not depend on the choice of the group containing P. Then in Theorem 6.2.5 we introduce what we call the covariance algebra of E. We finish this chapter with examples of C^∗-algebras that can be described as covariance algebras of product systems. We also discuss the relationship of these algebras to Cuntz–Nica–Pimsner algebras.

Im Dokument On C*-algebras associated to product systems (Seite 67-73)