Relative Cuntz–Pimsner algebras

et^(p)(T)et^(q)(S)−et^(p∨q)(ι^p∨q_p (T)ι^p∨q_q (S))|p, q∈P, p∨q <∞, T ∈K(Ep) andS∈K(Eq) with

et^(p)(T)et^(q)(S)|p, q∈P, p∨q=∞, T ∈K(Ep) andS∈K(Eq) .

Put N TE := TE/IN and let ¯π = {π¯p}p∈P be the representation of E in N TE obtained from the composition ofetwith the quotient map. So ¯πis Nica covariant. It is also injective because the Fock representation is Nica covariant. Hence the pair (N TE,π¯) satisfies all the required properties.

We callN T_E theNica–Toeplitz algebraofE.

4.2.2 Relative Cuntz–Pimsner algebras

LetE = (Ep)p∈P be a product system. For eachp∈P, letJp/ Abe an ideal that acts by compact operators onEpand setJ ={Jp}p∈P. We say that a representation ψ={ψp}p∈P isCuntz–Pimsner covariant onJ if, for all p∈P and allainJ_p,

ψ^(p)(ϕ_p(a)) =ψ_e(a).

Repeating the argument employed in the proof of Proposition 4.2.5, we obtain the following:

Proposition 4.2.6. Let (G, P) be a quasi-lattice ordered group and let E be a compactly aligned product system overP. LetJ ={Jp}p∈P be a family of ideals inAwithϕp(Jp)⊆K(Ep)for allp∈P.

Then there is aC^∗-algebraOJ,E and a Nica covariant representation j={jp}p∈P ofE inOJ,E that is also Cuntz–Pimsner covariant on J and such that

(i) O_J,E is generated by j(E)as aC^∗-algebra;

(ii) given a Nica covariant representationψ={ψp}p∈P ofEin aC^∗-algebraB that is Cuntz–Pimsner covariant onJ, there is a unique^∗-homomorphismψ¯_J: OJ,E →B such thatψ¯_J ◦j_p=ψ_p for all p∈P.

Moreover, the pair(O_J,E, j)is unique up to canonical isomorphism.

Definition 4.2.7. GivenE andJ as above, we callO_J,E therelative Cuntz–Pimsner algebra deter-mined byJ.

We emphasize two particular cases. IfJp={0}for allp∈P, thenO_J,E =N T_E. If (G, P) = (Z,N), Eis a product system of Hilbert bimodules andJp=hhEp| Epiifor allpinP, thenOJ,E is the C^∗-algebra studied by Katsura in [29]. He proved that the canonical^∗-homomorphism from A toOJ,E is an isomorphism onto the fixed-point algebra ofOJ,E. In this case,E extends to a semi-saturated Fell bundle overZ(see [1]). We will generalise this to a certain class of compactly aligned product systems of Hilbert bimodules over semigroups arising from quasi-lattice orders.

Remark 4.2.8. Fowler defined the Cuntz–Pimsner algebra of a product systemE to be the universal C^∗-algebra for representations of E that are Cuntz–Pimsner covariant on J = {Jp}_p∈P, where Jp =ϕ⁻¹_p (K(Ep)) for allp∈P (see [26]). Here we consider the class of compactly aligned product systems and define the relative Cuntz–Pimsner algebra with respect to a family of ideals as a quotient of the Nica–Toeplitz algebra ofE. This provides the construction of relative Cuntz–Pimsner algebras with a special feature and will allow us to generalise most of the results obtained in Chapter 3 to quasi-lattice ordered groups. Our approach applies to Fowler’s Cuntz–Pimsner algebras of proper product systemsE= (Ep)p∈P if (G, P) is a quasi-lattice ordered group andP is directed. This is so because, in this case, a Cuntz–Pimsner covariant representation ofE in the sense of Fowler is also Nica covariant [26, Proposition 5.4].

A product system of Hilbert bimodulesE = (E_p)p∈P gives rise to a product systemE^∗ overP^op by settingE^∗:= (E_p^∗)p∈P,whereE_p^∗is the Hilbert bimodule adjoint toEp. We will identifyAwith its adjoint Hilbert bimoduleA^∗through the isomorphisma7→ae^∗implemented by the involution operation onA, whereae^∗ is the image ofa^∗ inA^∗ under the canonical conjugate-linear map. The multiplication mapE_p^∗⊗AE_q^∗∼=E_qp^∗ is given by the isomorphismE_p^∗⊗AE_q^∗∼= (Eq⊗AEp)^∗,ξ^∗⊗η^∗7→(η⊗ξ)^∗, followed by the multiplication mapµq,p. In addition,E^∗∗ =E. Before providing more concrete examples of relative Cuntz–Pimsner algebras, we need the following lemma.

Lemma 4.2.9. Let E= (Ep)p∈P be a product system of Hilbert bimodules. For eachp∈P, let I_Ep:=

hhEp| Epii and set I_E = {I_Ep}_p∈P. A representation ψ = {ψp}_p∈P of E in a C^∗-algebra B that is Cuntz–Pimsner covariant on I_E naturally induces a representation of E^∗ = (E_p^∗)p∈P that is Cuntz–

Pimsner covariant onI_E^∗, where I_E^∗={I_E^∗_p}_p∈P and I_Ep^∗=hhE_p^∗| E_p^∗ii=hEp| Epi. As a consequence, representations ofE that are Cuntz–Pimsner covariant on I_E are in one-to-one correspondence with representations ofE^∗ that are Cuntz–Pimsner covariant on I_E^∗.

Proof. For p= e, put ψ^∗_e := ψ_e. Given p∈ P \ {e}, define ψ_p^∗: E_p^∗ → B by ψ^∗_p(ξ^∗) :=ψ_p(ξ)^∗ and setψ^∗={ψ_p^∗}_p∈P. Then, for allξ∈ E_p andη∈ E_q,

ψ^∗_p(ξ^∗)ψ^∗_q(η^∗) =ψp(ξ)^∗ψq(η)^∗=ψqp(µq,p(η⊗ξ))^∗=ψ_qp^∗ (µq,p(η⊗ξ)^∗). Sinceψis Cuntz–Pimsner covariant onIE, it follows that

ψ^∗_p(ξ^∗)^∗ψ_p^∗(η^∗) =ψ_p(ξ)ψ_p(η)^∗=ψ^(p)(|ξihη|) =ψ_e(hhξ|ηii) =ψ^∗_e(hhξ|ηii)

for all ξ, η ∈ E_p. That ψ^∗ is Cuntz–Pimsner covariant on I_E^∗ follows from the fact that ψ is a representation ofE. So the last statement is obtained from the identityE =E^∗∗.

Example 4.2.10. Let α: P → End(A) be an action by injective extendible endomorphisms with hereditary range. LetAα= (Aα_p)p∈P be the product system of Hilbert bimodules built out ofαas in Example 4.1.3. Although it is not clear whenAαis compactly aligned,_αAalways is so. The idealIp/ A given by the left inner product ofAα_pis preciselyAαp(1)A. Given a nondegenerate representationψ= {ψp}_p∈P ofAαin a C^∗-algebraB, we obtain a strictly continuous unital^∗-homomorphism ¯ψe:M(A)→ M(B) by nondegeneracy ofψe. In addition, we define a semigroup homomorphism from P to the semigroup of isometries inM(B) by setting

vp:= lim

λ ψp(uλαp(1)).

Here the limit is taken in the strict topology ofM(B). It indeed exists becausekψp(uλαp(1))k ≤1 for eachλand, fora∈Aandb∈B,

limλ ψp(uλαp(1))(ψe(a)b) = lim

λ ψp(uλαp(a))b=ψp(αp(a))b

and lim

λ (bψe(a))ψp(uλαp(1)) = lim

λ bψp(auλαp(1)) =bψp(aαp(1)). To see thatv^∗_pvp = 1, observe that

v_p^∗vp(ψe(a)b) = lim

λ ψp(uλαp(1))^∗ψp(αp(a))b= lim

λ ψe α⁻¹_p (αp(1)uλαp(a))

b=ψe(a)b.

The semigroup of isometries{vp|p∈P} and the^∗-homomorphism ¯ψe:M(A)→M(B) satisfy the relation

vp·ψ¯e(c) = ¯ψe(αp(c))vp

for allc∈M(A) andp∈P. Hence

ψ¯_e(α_p(c))v_pv^∗_p=v_pψ¯_e(c)v^∗_p. (4.2.11) In addition, ψ_p(aα_p(1)) = ψ_e(a)v_p for all a∈ A and p∈ P. Ifψ is Cuntz–Pimsner covariant on IA_α={Ip}_p∈P,it follows that for allc∈M(A) andp∈P,

ψ¯e(αp(c)) = ¯ψe(αp(c))vpv_p^∗. Indeed, forc inAandaαp(1) inAα_p, we compute

αp(c^∗c)aαp(1) =αp(c^∗)αp(α_p⁻¹(αp(c)aαp(1)))

=αp(c^∗)αp(hαp(c^∗)αp(1)|aαp(1)i)

=αp(c^∗)· hαp(c^∗)αp(1)|aαp(1)i

=|αp(c^∗)ihαp(c^∗)|(aαp(1)). Hence Cuntz–Pimsner covariance gives us

ψe(αp(c^∗c)) =ψp(αp(c^∗))ψp(αp(c^∗))^∗=ψe(αp(c^∗))vpv_p^∗ψe(αp(c))

=ψe(αp(c^∗))vpψe(c)v_p^∗=ψe(αp(c^∗c))vpv^∗_p.

SinceAis spanned by positive elements, the same relation holds for allc∈Aand thus for allc∈M(A) if we replaceψeby its extension ¯ψe. So combining this with (4.2.11), we deduce the relation

ψe(αp(c)) =vpψe(c)v_p^∗ for allc∈A. The same holds for ¯ψ_eandc inM(A).

Conversely, we claim that a nondegenerate^∗-homomorphismπ: A→B together with a semigroup of isometries{v_p|p∈P}satisfying the relation

π(α_p(a)) =v_pπ(a)v^∗_p (4.2.12)

yields a representation ofAα that is Cuntz–Pimsner covariant onIA_α. First, notice that the projec-tionvpv_p^∗coincides with ¯π(αp(1)), where ¯πis the strictly continuous^∗-homomorphism M(A)→M(B) extendingπ. For each p∈ P anda ∈ A, we set ψp(aαp(1)) := π(a)vp. Putψ = {ψp}_p∈P. Then

¯

π(αp(1)) =vpv_p^∗ implies thatψis Cuntz–Pimsner covariant onIp=Aαp(1)Afor allp∈P, since ψe(aαp(1)b) =π(aαp(1)b) =π(a)¯π(1)π(b) =π(a)vpv^∗_pπ(b) =π(a)vp(π(b^∗)vp)^∗

=ψ_p(aα_p(1))ψ_p(b^∗α_p(1))^∗=ψ^(p)(|aαp(1)ihb^∗α_p(1)|)

for allaandbinA. Moreover, (4.2.12) tells us thatψ_e(α_p(a))v_p =v_pψ_e(a) for alla∈Aandp∈P. This also gives

ψp(aαp(1))ψq(bαq(1)) =ψpq(aαp(b)αpq(1)) =ψpq µp,q(aαp(1)⊗bαq(1)) . Again by (4.2.12),

ψe α_p⁻¹(αp(1)a^∗bαp(1))=v_p^∗vpψe α_p⁻¹(αp(1)a^∗bαp(1)) v_p^∗vp

=v_p^∗ψe(αp(1)a^∗bαp(1))vp

=v_p^∗ψ_e(a^∗b)v_p.

This shows thatψ is a representation ofA_α that is Cuntz–Pimsner covariant onI_Aα.

As a result, the crossed productAoαP ofAby the semigroup of endomorphisms provided byαhas a description as the universal C^∗-algebra of representations ofA_α that are Cuntz–Pimsner covariant onI_Aα. By Lemma 4.2.9,AoαP may also be described as the universal C^∗-algebra for representations of_αA that are Cuntz–Pimsner covariant onI _A

α . IfP^opis the positive cone of a quasi-lattice order and is also directed, a representation of_αA that is Cuntz–Pimsner covariant onI_αA is also Nica covariant by [26, Proposition 5.4]. In this caseO_{I A}

α , A_α ∼=Ao^αP. In general,Ao^αP is the Cuntz–Pimsner algebra of_αA as defined by Fowler [26]. See, for instance, [35] and [37] for constructions of crossed products by semigroups of endomorphisms. We also refer the reader to [38] for this and further constructions of crossed products out of product systems.

Example4.2.13. We may attach a C^∗-algebra to a quasi-lattice ordered group (G, P) by considering the trivial product system overP. TheToeplitz algebra of (G, P) as introduced by Nica [47], denoted by C^∗(G, P), is the Nica–Toeplitz algebra of the trivial product system overP. This is the relative Cuntz–Pimsner algebra with respect to the trivial family of idealsJp={0}for allp∈P. In fact, there is also a description of C^∗(G, P) as a semigroup crossed product as in the previous example (see [35]

and also Subsection 6.3.3). This is the universal C^∗-algebra generated by a family of isometries{vp}p∈P

subject to the relation

vpv_p^∗vqv^∗_q =

(v_p∨qv^∗_p∨q ifp∨q <∞,

0 otherwise.

The Fock representation in this case is the canonical representation ofP by isometries on`<sub>2</sub>(P). The image of C<sup>∗</sup>(G, P) in B(`₂(P)) under the Fock representation is calledWiener–Hopf algebra [47].

Im Dokument On C*-algebras associated to product systems (Seite 43-46)