6.3 Relationship to other constructions
6.3.4 Crossed products by interaction groups
i=1
Si
Si∈ J )
.
Let C∗(∪)s (P) be the universal C∗-algebra generated by isometries{vp|p∈P}and projections{eS|S ∈ J∪} satisfying the conditions (i)–(iii) of Definition 6.3.10 with the additional relation
(iv) eS1∪S2 =eS1+eS2−eS1∩S2 for allS1, S2∈ J∪.
The C∗-algebra C∗(∪)s (P) coincides with C∗s(P) wheneverJ is independent (see [39, Proposition 2.24]). The next result generalises Proposition 6.3.11.
Corollary 6.3.13. The semigroup C∗-algebraC∗(∪)s (P) is naturally isomorphic toA×EP.
Proof. It follows from [39, Lemma 3.3] and [39, Corollary 2.22] that the C∗-subalgebra of C∗(∪)s (P) generated by theeS’s is naturally isomorphic toA. Again condition (iii) of Definition 6.3.10 implies that such a C∗-subalgebra coincides with the fixed-point algebra for the canonical coaction of G on C∗(∪)s (P). Now we may employ the same argument used in the proof of Proposition 6.3.11 to obtain an isomorphism C∗(∪)s (P)∼=A×EP.
6.3.4 Crossed products by interaction groups
In this subsection, we will show how Exel’s crossed products by interaction groups fit into our approach.
This notion of crossed products was introduced in [22] in order to study semigroups of unital and injective endomorphisms. We first recall some concepts from his work, although many details will be omitted. Aninteraction groupis a triple (A, G, V), whereAis a unital C∗-algebra,Gis a group andV is apartial representation ofGin the Banach algebra of bounded operators onA. This consists of a family{Vg}g∈G of continuous operators onAwithV1= idA and
VgVhVh−1=VghVh−1, Vg−1VgVh=Vg−1Vgh
for allg, h ∈G. It follows thatEg := VgVg−1 is an idempotent for eachg ∈ GandEgEh = EhEg, g, h∈G. The partial representation is also assumed to satisfy the following conditions:
(i) Vg is a positive map, (ii) Vg(1) = 1,
(iii) Vg(ab) =Vg(a)Vg(b) ifaorb belong to the range ofVg−1.
For allg∈G, the idempotentEg is a conditional expectation onto the range ofVg. An interaction group is said to benondegenerateifEg is faithful for all ginG. That is,Eg(a∗a) = 0 impliesa= 0 (see [22, Definition 3.3]).
Frow now on let us fix a nondegenerate interaction group (A, G, V). Given a unital C∗-algebraB, recall thatv: G→B is a∗-partial representation if it is a partial representation satisfyingvg∗=vg−1
for allg∈G. Acovariant representation of (A, G, V) inB is a pair (π, v), whereπ:A→B is a unital
∗-homomorphism andv is a∗-partial representation ofGinB such that vgπ(a)vg−1 =π(Vg(a))vgvg−1.
The Toeplitz algebra of (A, G, V), denoted byT(A, G, V), is the universal C∗-algebra for covariant representations of (A, G, V). It is generated by a copy ofAand elements{bsg}g∈Gso thatbs:g7→bsg is a∗-partial representation and the pair (jV,bs) is a covariant representation of (A, G, V) inT(A, G, V), wherejV: A→ T(A, G, V) denotes the canonical embedding.
In order to recall the notion of redundancy introduced by Exel in [22], let us first define certain subspaces ofT(A, G, V). Given a wordα= (g1, g2, . . . , gn) in G, set
bsα=bsg1bsg2· · ·bsgn.
LetMcα=jV(A)bsαjV(A) and eα:=bsαbsα−1, where α−1= (gn−1,· · ·, g−12 , g1−1). ThensbαjV(a)bsα−1 = jV(Vα(a))eαand, by [22, Proposition 2.7],eα is also an idempotent. The subspaceZbαassociated to the wordαwill be the closed linear span of elements of the form
jV(a0)bsg1jV(a1)bsg2· · ·bsgnjV(an)
witha0, a1, a2, . . . , an∈A. We setZbα=jV(A) in caseαis the empty word. Observe that we always haveMcα⊆Zbα. We also associate a finite subset ofGto the wordαby letting
µ(α) ={e, g1, g1g2, . . . , g1g2· · ·gn},
so thatµ(α) ={e}if αis the empty word. We further letα. =g1g2· · ·gn. Ifα. =e, it follows that µ(α) =µ(α−1). We denote byWα the set of all wordsβ inGwithµ(β)⊆µ(α) andβ. =eand let
Zbµ(α):= spann cβ
cβ∈Zbβ, β∈ Wα
o .
This is a C∗-subalgebra ofT(A, G, V) sinceβ∈ Wαif and only ifβ−1∈ WαandWαis also closed under concatenation of words (see [22, Proposition 4.7] for further details). In addition,Zbµ(α)Mcα⊆Mcα. Definition 6.3.14. Letαbe a word in G. We say thatc∈Zbµ(α)is anα-redundancy ifcMcα={0}.
The crossed product of A by G under V, denoted by AoGV, is the universal C∗-algebra for covariant representations that vanish on all redundancies. ThusAoGV is isomorphic to the quotient of T(A, G, V) by the ideal generated by all redundancies. A covariant representation of (A, G, V) that vanishes on such an ideal was calledstrongly covariant by Exel. He was able to prove thatA is embedded intoAoGV. The crossed product carries a canonicalG-grading, and a representation of AoGV is faithful on its fixed-point algebra if and only if it is faithful onA.
IfP is a subsemigroup ofG, sometimes an action ofP on a C∗-algebraA may be enriched to an interaction group (A, G, V) so thatVp=αpfor allp∈P. Under certain assumptions,V is unique if it exists andAoGV is generated byAand isometries{vp}p∈P [22, Theorem 12.3]. We will see that ifP is reversible, in the sense thatpP∩qP 6=∅andP p∩P q6=∅for allp, q∈P, andG=P−1P =P P−1, then AoGV can be obtained from a covariance algebra of a certain product system if {Vp}p∈P
generates the image ofGunderV. So we will assume thatV is an interaction group which extends an action ofP by endomorphisms of AandVp−1◦αp = idA. This holds if and only if the ∗-partial representation ofGinAoGV restricts to an isometric representation ofP.
Lemma 6.3.15. Let(i, s)denote the representation of (A, G, V)inAoGV. Then sp is an isometry if and only ifVp−1◦αp= idA.
Proof. Suppose thatVp−1◦αp= idA. Let us prove thatsb∗pbsp−1 vanishes onMc(p−1)=jV(A)sbp−1jV(A).
Sincesbis a ∗-partial representation ofG, one has thatbsp−1 =sb∗p. Putβ1= (p−1, p) andβ2= (e). So bothβ1 andβ2 belong toW(p−1)and hencebs∗pbsp−1∈Zb{e,p−1}. Thus all we must do is prove that
(bs∗pbsp−1)jV(A)bs∗pjV(A) ={0}.
To do so, leta∈A. Then
bs∗pbspjV(a)bs∗p=bs∗pjV(Vp(a))bspbs∗p=jV(Vp−1(αp(a)))bs∗pbspbs∗p=jV(a)bs∗p. This proves thatbs∗pbsp−1 is a redundancy. Hencesp is an isometry inAoGV.
Now assume thatsp is an isometry. For eachainA,
i(a) =s∗pspi(a)s∗psp=i(Vp−1(αp(a))). This shows thatVp−1◦αp= idA becauseAis embedded intoAoGV.
Thus in order to build a product system overP so that it encodes the interaction group, we suppose thatVp−1◦αp= idA for allp∈P. It follows from [22, Lemma 2.3] that, for all p, q∈P, we have
Vq−1Vp−1 =Vq−1p−1, Vp−1q=Vp−1Vq.
Let us now describe the product system associated toV. This is defined as in Example 4.2.10. Unlike in [38], here we do not requireP to be abelian since we assumeVp−1◦αp= idA. We setEp:=A, endowed with the right action ofAthrougha·b:=aαp(b) and theA-valued inner productha|bi=Vp−1(a∗b). This provides Ep with a structure of right Hilbert A-module because Vp−1(a∗a) = 0⇔ a= 0 and Vp−1(ab) =Vp−1(a)Vp−1(b) wheneverb lies in the range ofαp. The∗-homomorphism ϕp:A→B(Ep) is given by the multiplication onA, so thatϕp(a)·b=abfor alla∈A,b∈ Ep. The correspondence isomorphism µp,q: Ep ⊗AEq ∼= Epq sends an elementary tensor a⊗b to aαp(b). Using that αp is an endomorphism ofA, we deduce thatµp,q preserves the bimodule structure. It is also surjective becauseαp is unital for allp∈P.
Lemma 6.3.16. E= (Ep)p∈P is a product system.
Proof. We will prove thatµp,qpreserves the inner product and that the multiplication inEis associative.
Leta0, a1, b0, b1∈A. Then
ha0⊗b0|a1⊗b1i=Vq−1(b∗0Vp−1(a∗0a1)b1)
=Vq−1(Vp−1(αp(b0)∗)Vp−1(a∗0a1)Vp−1(αp(b1)))
=Vq−1(Vp−1(αp(b0)∗a∗0a1αp(b1)))
=V(pq)−1(αp(b0)∗a∗0a1αp(b1))
=hµp,q(a0⊗b0)|µp,q(a1⊗b1)i.
This completes the proof thatµp,qis an isomorphism of correspondences for allp, q∈P. Now lets∈P, a∈ Ep,b∈ Eq andc∈ Es. Then
(µpq,s(µp,q⊗1)) a⊗b⊗c
=aαp(b)αpq(c) =aαp(bαq(c))
= (µp,qs(1⊗µq,s)) a⊗b⊗c .
Lemma 6.3.17. There is a covariant representation of (A, G, V) in A×E P. It sends g=p−1q to vg :=jp(1p)∗jq(1q) and a toje(a). Moreover, given a wordβ = (g1, g2, . . . , gn)in G, the map a7→
je(a)vβ is injective, wherevβ=vg1vg2· · ·vgn.
Proof. We begin by proving thatjp(1p)∗jq(1q) =jp0(1p0)∗jq0(1q0) for allp, q, p0, q0∈P such thatp−1q= p0−1q0. To do so, we use thatPis also left reversible. We can finds∈Pwiths∈(pP∩qP)∩(p0P∩q0P). Since (A, G, V) is nondegenerate,E is faithful and henceIr−1(r∨s)={0}for allr∈P such thatr6∈sP. So
E{s}= M
r∈sP
Er. Now givenr∈sP, we writebr for an element inEr. We compute
t{s} et(1p)∗et(1q)
(br) =t{s} et(1p)∗)(αq(br)⊗1
=Vp−1(αq(br)) =Vp−1q(br) =Vp0 −1q0(br)
=t{s} et(1p0)∗et(1q0) (br).
Therefore,jp(1p)∗jq(1q) =jp0(1p0)∗jq0(1q0) and the mapg=p−1q7→jp(1p)∗jq(1q) is well defined. This gives a partial representation ofGinA×EP becauseV is a partial representation. Giveng=p−1q∈G, vg−1 =jq(1q)∗jp(1p) =vg∗. Sog7→vg indeed defines a∗-partial representation ofG.
Let us prove that (je, v) is covariant. Takeg=p−1q∈Ganda∈A. Again we use the assumption thatP is left reversible and chooses∈pP ∩qP. Thus it suffices to show that
t{s} et(1p)∗et(1q)et(a)et(1q)∗et(1p)
=t{s} et(Vg(a))et(1p)∗et(1q)et(1q)∗et(1p) onEr forr∈sP. Indeed, givenbr∈ Er, one has
t{s} et(1p)∗et(1q)et(a)et(1q)∗et(1p)
(br) =Vg(aVg−1(br)) =Vg(a)Vg(Vg−1(br))
=t{s} et(Vg(a))et(1p)∗et(1q)et(1q)∗et(1p)(br), so that (je, v) is a covariant representation of (A, G, V).
Letβ = (g1, . . . , gn) be a word inG. In order to prove that the map a7→je(a)vβ is injective, take s∈Kµ(β)−1. That is,
s∈P∩g−1n P∩(g−1n gn−1−1 )P· · · ∩(g−1n gn−1−1 · · ·g1−1)P.
It exists becauseG=P P−1. Using that Vg is unital for allg∈G, we deduce that je(a)vβvs=je(a)vβjs(1) =je(a)j.
βs(Vβ(1)) =je(a)j.
βs(1) =j.
βs(a).
Since the representation of E in A×E P is injective, the right-hand side above is nonzero. This guarantees thata7→je(a)vβ is an injective map.
The following is the main result of this subsection.
Proposition 6.3.18. LetP be a subsemigroup of a groupGwithG=P−1P =P P−1. Let(A, G, V) be a nondegenerate interaction group extending an action α:P → End(A) by unital and injective endomorphisms. Suppose, in addition, that Vp−1◦αp= idA for all p∈P. Then AoGV is isomorphic toA×EP, whereE is the product system constructed out of V.
Proof. We begin by proving that (je, v) factors throughAoGV. The pair (je, v) induces a∗ -homomor-phismφb:T(A, G, V)→A×EP. Lemma 6.3.17 says that the mapa7→je(a)vβ is injective for each wordβ inG. Hence [22, Proposition 10.5] implies thatφbis injective onMcα. In particular, ifc∈Zbµ(α) is anα-redundancy,φb(c)jr(Er) ={0}for allr∈Kµ(α) because
jr(a) =je(a)jr(1) =je(a)vαjα.−1
r(1)∈φb Mcα jα.−1
r(1)
for allainA. Soφb(c) must be zero inA×EP. This induces a∗-homomorphismφ: AoGV →A×EP that is faithful onAand preserves theG-grading ofAoGV. Proposition 4.6 of [22] says thatφis also faithful on the fixed-point algebra ofAoGV. Now by Lemma 6.3.15, spis an isometry in AoGV for allp∈P. Moreover, [22, Lemma 2.3] says thatspsq =spqfor allp, q∈P. Hence one can show that the mapsEp31p7→sp anda7→i(a) give rise to a representation ofE. By applying the injectivity ofφon the fibres and the usual argument that the induced∗-homomorphismTE →AoGV preserves theG-grading, we conclude that such a representation must factor throughA×E P. The resulting
∗-homomorphism is the inverse ofφ.
Remark 6.3.19. LetP be a reversible cancellative semigroup and letGbe its enveloping group. LetA be a unital C∗-algebra and let α:P → End(A) be an action by injective endomorphisms. Given a not-necessarily nondegenerate interaction group (A, G, V) extendingαwithVp−1◦αp = idA, the equalityVq−1Vp−1=Vq−1p−1 still holds by [22, Lemma 2.3]. Hence one may build a product system as above by lettingEp :=Aαp(1) and µp,q(aαp(1)⊗Abαq(1)) :=aαp(b)αpq(1) (see [38]). Thus the covariance algebra of such a product system may be viewed as the crossed product of A underV, generalising Exel’s construction to interaction groups satisfyingVq−1Vp−1 = Vq−1p−1 that are not necessarily nondegenerate. For instance, the product system built in the previous subsection fits into this setting, whereVg(χS) :=χgS∩P for allS∈ J andg∈G.
General theory of Hilbert modules
In this appendix, we recall some basic aspects of the theory of Hilbert modules. We state some results that were needed in the main text of this work. This appendix is based on [36] and [52].
A.1 Adjointable operators on Hilbert modules
Definition A.1.1. LetE be a complex vector space andAa C∗-algebra. We say thatE is a (right) pre-HilbertA-moduleifE is a rightA-module equipped with a maph· | ·i:E × E →A, that is linear in the second variable and conjugate-linear in the first, satisfying for allξ, η, ζ ∈ E anda∈A,
(i) hξ|ηai=hξ|ηia; (ii) hξ|ηi∗=hη|ξi; (iii) hξ|ξi ≥0 inA; (iv) hξ|ξi= 0⇒ξ= 0.
The maph· | ·iis referred to asinner product.
Remark A.1.2. The axioms (i) and (iii) imply thathξa|ηi=a∗hξ|ηi. In particular, the closure of hE | Ei= span{hξ|ηi|ξ, η∈ E}
is a closed ideal inA.
Aleft pre-HilbertA-module is defined in a similar way. We require the inner product to beA-linear in the first variable and thus conjugate-linear in the second. We use the notationhh· | ·iifor the inner product of a left pre-HilbertA-module.
A pre-HilbertA-module is calledfull if the idealhE | Ei is dense inA.
Lemma A.1.3(Cauchy–Schwarz inequality). Let E be a pre-HilbertA-module andξ, η∈ E. Then hξ|ηi∗hξ|ηi ≤ khξ|ξikhη|ηi.
Corollary A.1.4. If E is a pre-Hilbert A-module, then k · k:ξ7→ kξk:=khξ|ξik12 is a norm onE for whichkξak ≤ kξkkak. Moreover,
EhE | Ei= span{ξhη|ζi|ξ, η, ζ∈ E}
is dense inE.
Definition A.1.5. A Hilbert A-module is a pre-Hilbert A-moduleE that is complete in the norm coming from theA-valued inner product.
Example A.1.6. A Hilbert space Hmay be viewed as a Hilbert C-module. It is also a left Hilbert K(H)-module with left inner product given by
hhξ|ηii:=|ξihη|,
where |ξihη| denotes the compact operator on H determined by the vectors ξ and η. That is,
|ξihη|(ζ) =ξhη|ζifor allζ∈ H.
Example A.1.7. A C∗-algebraAhas a canonical structure of right HilbertA-module with right module action implemented by the multiplication inAand inner product
(a, b)7→a∗b.
Taking (a, b)7→ab∗ as inner product,Abecomes a left HilbertA-module with left action given by left multiplication. A closed idealI / Amay be turned into right and left HilbertA-modules in a similar way.
Example A.1.8 (Direct sum). LetE andG be HilbertA-modules. ThenE ⊕ G is a Hilbert A-module with right action ofA andA-valued inner product defined coordinatewise. More generally, given a family of HilbertA-modules (Eλ)λ∈Λ,then the algebraic direct sumL
λ∈ΛEλis a pre-HilbertA-module with the structure defined coordinatewise. Its completion is a HilbertA-module.
Definition A.1.9. LetE andG be HilbertA-modules. A mapT:E → G isadjointablel if there exists a mapT∗:G → E such that for allξ∈ E andη∈ G,
hT(ξ)|ηi=hξ|T∗(η)i.
This is unique if it exists. We say thatT∗ is theadjoint ofT.
Lemma A.1.10. An adjointable mapT:E → G isA-linear and continuous.
Remark A.1.11. There are continuousA-module maps that are not adjointable.
Given HilbertA-modulesE andG, we denote byB(E,G) the set of all adjointable operators fromE toG.We writeB(E) in caseE=G.
Proposition A.1.12. IfE is a HilbertA-module, thenB(E)is aC∗-algebra with respect to the operator norm.
Corollary A.1.13. LetE be a HilbertA-module andT ∈B(E). Then, for allξ∈ E, hT(ξ)|T(ξ)i ≤ kTk2hξ|ξi.
We may attach to elementsξ∈ G andη∈ E an adjointable operator E → G defined by
|ξihη|:ζ7→ξhη|ζi.
This is thecompact operator determined byξandη. Its adjoint is |ηihξ| ∈B(G,E). The closed linear span of operators of this form is denoted byK(E,G). An element ofK(E,G) is said to be compact. If E=G, K(E) =K(E,E) is an ideal ofB(E).