C-algebras associated to product systems of Hilbert bimodules

C-algebras associated to product systems of Hilbert bimodules
C-algebras associated to product systems of Hilbert bimodules

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C ?-ALGEBRAS ASSOCIATE

D TO PRODUCT SYSTEMS OF HILBERT BIMODULES AIDAN SIMS AND TRENT YEEND Abstract.Let (G,P )be a quasi-lattice ordered group and let X be a compactly aligned product system over P of Hilbert bimodules.Under mild hypotheses we as-sociate to X a C ?-algebra which we call the Cuntz-Nica-Pimsner algebra of X .Our construction generalises a number of others:a sub-class of Fowler’s Cuntz-Pimsner algebras for product systems of Hilbert bimodules;Katsura’s formulation of Cuntz-Pimsner algebras of Hilbert bimodules;the C ?-algebras of ?nitely aligned higher-rank graphs;and Crisp and Laca’s boundary quotients of Toeplitz algebras.We show that for a large class of product systems X ,the universal representation of X in its Cuntz-Nica-Pimsner algebra is isometric.1.Introduction In this article we introduce and begin to analyse a class of C ?-algebras,which we call Cuntz-Nica-Pimsner algebras,associated to product systems of Hilbert bimodules.This work draws on and generalises a substantial body of previous work in a number of related areas:results of [14,16,18,24,26]on C ?-algebras associated to Hilbert bimodules;the study of C ?-algebras associated to product systems in [9,10,12,15];the theory of C ?-algebras associated to higher-rank graphs [19,28,30];and the theory of Toeplitz algebras (and quotients thereof)associated to quasi-lattice ordered groups [6,7,22,25].Consequently,putting our results in context and indicating their signi?cance requires some discussion.1.1.C ?-algebras associated to Hilbert bimodules.Let A be a C ?-algebra.A Hilbert A –A bimodule (or C ?-correspondence over A )is an A –A bimodule X equipped with an A -valued inner product ·,· A such that:X is complete in the norm x 2=

x,x A ;and the left action of A on X is implemented by a homomorphism φof A into the C ?-algebra L (X )of adjointable operators on X .

In [26],Pimsner associated to each Hilbert A –A bimodule X two C ?-algebras O X and T X .He showed that,as the notation suggests,the C ?-algebras O X generalise the Cuntz-Krieger algebras O A associated to {0,1}-matrices A in [8],and the algebras T X generalise their Toeplitz extensions T O A .According to [26],a representation of X in a C ?-algebra B is a pair (π,ψ)where π:A →B is a C ?-homomorphism,ψ:X →B is linear,and the pair carries the Hilbert A –A bimodule structure on X to the natural Hilbert B –B bimodule structure on B .Pimsner proved that T X is universal for representations of

2AIDAN SIMS AND TRENT YEEND

X,and that O X is universal for representations satisfying a covariance condition now

known as Cuntz-Pimsner covariance,and in particular is a quotient of T X.

In the spirit of Coburn’s Theorem for the classical Toeplitz algebra[5],Fowler and

Raeburn proved a uniqueness theorem for T X[16,Theorem3.1].A key example in their work,and an informative application of Pimsner’s ideas,is related to graph C?-

algebras.A directed graph consists of a set E0of vertices,a set E1of edges,and maps

r,s:E1→E0,called the range and source maps,which give the edges their direction:

the edge e is directed from s(e)to r(e).When we discuss graphs and their C?-algebras here,we follow the conventions of[27].

Fowler and Raeburn used their analysis of T X to investigate graph algebras by asso-

ciating a Hilbert c0(E0)–c0(E0)bimodule X(E)to each directed graph E.Previously

[20,21]C?-algebras had been associated to directed graphs E which are row-?nite(each

vertex receives at most?nitely many vertices)and have no sources(each vertex receives at least one edge).In[16],Fowler and Raeburn showed that for such graphs,T X(E)can

naturally be thought of as a Toeplitz extension of the graph C?-algebra C?(E)of[20].

They observed that E is row-?nite precisely when the homomorphismφ:A→L(X(E))

which implements the left action takes values in the algebra K(X(E))of generalised com-pact operators on X(E)and that E has no sources precisely whenφis injective.As

Pimsner’s theory does not require that the left action be by compact operators,Fowler

and Raeburn’s results suggested what is now the accepted de?nition of the graph C?-

algebra of a non-row-?nite graph[13].Results of Exel and Laca[11]were used to prove

a version of the Cuntz-Krieger uniqueness theorem for arbitrary graph C?-algebras in

[13],and direct methods were used to extend a number of other graph C?-algebraic

results to the non-row-?nite setting in[2].

Via the connection between graph C?-algebras and Cuntz-Pimsner algebras discussed

above,the uniqueness theorems of[2,13]suggested an alternate approach to Cuntz-Pimsner algebras whenφis not injective.Speci?cally,when interpreted in terms of

representations of the bimodule X(E),the gauge-invariant uniqueness theorem of[2]

suggested that one could weaken Pimsner’s covariance condition to obtain a covariance

condition for which the universal C?-algebra satis?es the following two criteria:that the universal representation is injective on A(in Pimsner’s theory,this requires thatφ

is injective);and that any representation of the universal C?-algebra which respects the

gauge action of T and is injective on A is faithful.

In[18],Katsura identi?ed such a covariance condition:it is the de?ning relation for a relative Cuntz-Pimsner algebra(see[14,24])with respect to a certain ideal of

A.Katsura’s universal algebra O X satis?es the two criteria set forth in the preceding

paragraph,and under Katsura’s de?nition,O X(E)~=C?(E)for arbitrary graphs E. 1.2.C?-algebras associated to product systems.Let P be a semigroup with iden-

tity https://www.360docs.net/doc/1c9265303.html,rmally,a product system over P of Hilbert A–A bimodules is a semigroup

X= p∈P X p such that each X p is a right-Hilbert A–A bimodule,and x?A y→xy determines an isomorphism of X p?A X q onto X pq for all p,q∈P\{e}.These objects

were introduced in this generality by Fowler in[12]as generalisations of the continu-ous product systems of Hilbert spaces introduced by Arveson in[1]and their discrete analogues introduced by Dinh in[9,10].

C?-ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS3 In[12],Fowler considered a class of compactly aligned product systems X over semi-groups P arising in quasi-lattice ordered groups(G,P).Inspired by work of Nica[25] and of Laca and Raeburn[22]on Toeplitz algebras associated to quasi-lattice ordered groups,Fowler introduced and studied what he called Nica covariant representations of X and the associated universal C?-algebra T cov(X).When P=N Nica covariance is

.Fowler’s main theorem[12,The-automatic and T cov(X)coincides with Pimsner’s T X

1

orem7.2]gave a spatial criterion for faithfulness of a representation of T cov(X).This theorem generalised both Laca and Raeburn’s uniqueness theorem[22,Theorem3.7] for C?(G,P)and Fowler and Raeburn’s uniqueness theorem[16,Theorem3.1]for the Toeplitz algebra of a single Hilbert bimodule.

In[12]Fowler also proposed a Cuntz-Pimsner covariance condition for representa-tions of compactly aligned product systems and an associated universal C?-algebra O X. Fowler de?ned a representationψof a product system X to be Cuntz-Pimsner covariant if its restriction to each X p is Cuntz-Pimsner covariant in Pimsner’s sense.Fowler showed that his Cuntz-Pimsner covariance condition implies Nica covariance under the hypothe-ses that:each pair of elements of P has a common upper bound;each X p=

4AIDAN SIMS AND TRENT YEEND

forces j X(a)=0.Moreover examples of[30,Appendix A],when interpreted in terms of product systems,show that if A does not act compactly on the left of every X p, Fowler’s O X may not satisfy criterion(B).Indeed,the examples of[30,Appendix A] show that the same problems persist even if,in Fowler’s de?nition of O X,we replace Pimsner’s covariance condition with Katsura’s.The situation is less clear when pairs in P need not have a common upper bound because in this case we have fewer guiding examples.We take the approach,di?erent from Fowler’s,that N O X should always be a quotient of T cov(X).The situation is then clearer:the C?-algebra which is universal for representations satisfying both Nica covariance and Fowler’s Cuntz-Pimsner covariance will not satisfy(A)if there exist p,q∈P with p∨q=∞.

The approach that N O X should be a quotient of T cov(X)is justi?ed by examples in isometric representation theory for quasi-lattice ordered groups.Fowler[12]showed that when each X p is a1-dimensional Hilbert space and multiplication in X is implemented by multiplication of complex numbers,his T cov(X)agrees with the C?-algebra C?(G,P) universal for Nica covariant representations of P[6,22,25].Crisp and Laca have recently studied what they call boundary quotients of Toeplitz algebras associated to quasi-lattice ordered groups[7].Their results show that the relationship between their boundary quotient and C?(G,P)is often analogous to the relationship between the Cuntz algebras and their Toeplitz extensions.In particular if G=F n is the free group on n generators,and P=F+n is its positive cone,then C?(G,P)is isomorphic to the Toeplitz extension T O n of O n and Crisp and Laca’s boundary quotient is isomorphic to O n itself.

More generally,[7,Theorem6.7]shows that when G is a right-angled Artin group with trivial centre(and P is its positive cone),the associated boundary quotient is simple and purely in?nite.Regarded as a statement about a C?-algebra associated to the product system over P with1-dimensional?bres,simplicity is equivalent to(B). Hence boundary quotients provide important motivation and a good test-case for our theory when G=Z k.In particular,the results of Section5.4suggest that we are on the right track for fairly general quasi-lattice ordered groups.

1.3.Outline of the paper.In this paper we combine ideas of[12,14,18]with intuition drawn from the theory of k-graph C?-algebras[30]to associate what we call a Cuntz-Nica-Pimsner algebra N O X to a broad class of compactly aligned product systems X of Hilbert bimodules.This N O X is universal for a class of representations which we refer to as Cuntz-Nica-Pimsner covariant(or CNP-covariant for short).By de?nition these are the representations which are Nica covariant in the sense of Fowler,and also satisfy a Cuntz-Pimsner covariance relation which looks substantially di?erent from Fowler’s (but agrees with Fowler’s under a number of additional hypotheses on X).Our ultimate aim is to verify that N O X satis?es both(A)and(B)above.

Our main result,Theorem4.1,shows that our C?-algebra N O X satis?es(A).In Sec-tions5.2,5.3and5.4,we present evidence that it also satis?es(B).In Section5.2,we prove that our construction agrees with Katsura’s when P=N,and in Section5.3,we show that given a k-graphΛand the associated product system X(Λ)as constructed in[28],our N O X coincides with the Cuntz-Krieger algebra C?(Λ)of[30].The gauge-invariant uniqueness theorems[30,Theorem4.2]and[18,Theorem6.2]then imply that our de?nition satis?es(B).In Section5.4we prove that our Cuntz-Pimsner covariance

C?-ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS5 relation is compatible with the de?ning relations for Crisp and Laca’s boundary quo-tients[7].In particular,Crisp and Laca’s uniqueness theorem[7,Theorem6.7]shows that N O X satis?es(B)when G is a right-angled Artin group with trivial centre and X is the product system over P with1-dimensional Hilbert spaces for?bres and multipli-cation implemented by multiplication of complex numbers.

We show in Section5.1that N O X coincides with Fowler’s O X when[12,Proposi-tion5.4]suggests that it might—namely when each pair in P has a common upper bound,and eachφp is injective withφp(A)?K(X p).

Acknowledgements.It came to our attention in the late stages of preparation of this manuscript that Toke Carlsen,Nadia Larsen and Sean Vittadello were also considering the question of a satisfactory notion of Cuntz-Pimsner covariance for representations of product systems.We thank them for many helpful and insightful comments on this manuscript which have signi?cantly clari?ed both our ideas and our exposition of them. We also thank Iain Raeburn for a number of helpful discussions.

2.Definitions

In this section,we recall the de?nitions and notation described in[12,Section1]. 2.1.Hilbert bimodules.We attempt to summarise only those aspects of Hilbert bi-modules of direct relevance to this paper.We refer the reader to[3,23,31]for more detail.

Let A be a C?-algebra,and let X be a complex vector space carrying a right action of A.Suppose that ·,· A:X×X→A is linear in the second variable and conjugate linear in the?rst variable and,for x,y∈X and a∈A,satis?es:

(1) x,y A= y,x ?A;

(2) x,y·a A= x,y A a;

(3) x,x A is a positive element of A;and

(4) x,x A=0??x=0.

The formula x = x,x A 1

span{x?y?:x,y∈X}is an essential ideal of L(X) whose elements we refer to as generalised compact operators on X.

A right-Hilbert A–A bimodule is a right-Hilbert A module together with a homomor-phismφ:A→L(X).We think ofφas implementing a left action of A on X,so we typically write a·x forφ(a)x.Becauseφ(a)∈L(X)for all a∈A,we automatically have a·(x·b)=(a·x)·b for all a,b∈A and x∈X.

An important special case is the bimodule A A A with inner product given by a,b A= a?b and right-and left-actions given by multiplication in A.The C?-algebra L(A A A)is

6AIDAN SIMS AND TRENT YEEND

isomorphic to the multiplier algebra M(A),and the homomorphism that takes a∈A to left-multiplication by a on X is an isomorphism of A onto K(A A A).

We form the balanced tensor product X?A Y of two right-Hilbert A–A bimodules as follows.Let X⊙Y be the algebraic tensor product of X and Y as complex vector spaces.Let X⊙A Y be the quotient of X⊙Y by the subspace spanned by vectors of the form x·a⊙y?x⊙a·y where x∈X,y∈Y and a∈A.The formula

x1⊙y1,x2⊙y2 A= y1, x1,x2 A·y2 A

determines a bounded sesquilinear form on X⊙A Y.Let N=span{n∈X⊙A Y: n,n A=0A}.Then z+N =inf{ z+n,z+n A 1/2:n∈N}de?nes a norm on (X⊙A Y)/N,and X?A Y is the completion of(X⊙A Y)/N in this norm.

If X and Y are right-Hilbert A–A bimodules and S∈L(X),then there is an ad-jointable operator S?1Y(with adjoint S??1Y)on X?A Y determined by(S?1Y)(x?A y)=Sx?A y for all x∈X and y∈Y.The formula a→φ(a)?1Y there-fore determines a homomorphism of A into L(X?A Y).The notation X?A Y always refers to the right-Hilbert A–A bimodule in which the left action is implemented by this homomorphism.

2.2.Semigroups and product systems of Hilbert bimodules.Let P be a discrete multiplicative semigroup with identity e,and let A be a C?-algebra.A product system over P of right-Hilbert A–A bimodules is a semigroup X= p∈P X p such that:

(1)for each p∈P,X p?X is a right-Hilbert A–A bimodule;

(2)the identity?bre X e is equal to the bimodule A A A;

(3)for p,q∈P\{e}there is an isomorphism M p,q:X p?A X q→X pq satisfying

M p,q(x?A y)=xy for all x∈X p and y∈X q;and

(4)multiplication in X by elements of X e=A implements the actions of A on each

X p;that is ax=a·x and xa=x·a for all p∈P,x∈X p and a∈X e.

For p∈P,we denote the homomorphism of A to L(X p)which implements the left

action byφp,and we denote the A-valued inner product on X p by ·,· p

A .

By(2)and(4),for p∈P,multiplication in X induces maps M p,e:X p?A X e→X p and M e,p:X e?A X p→X p as in(3).Each M p,e is automatically an isomorphism by[31, Corollary2.7].We do not insist that M e,p is an isomorphism as this is too restrictive if we want to capture Pimsner’s theory(which does not require that

C?-ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS7 and P is a subsemigroup of G;P∩P?1={e};and with respect to the partial order p≤q??p?1q∈P,any two elements p,q∈G which have a common upper bound in P have a least upper bound p∨q∈P.We write p∨q=∞to indicate that p,q∈G have no common upper bound in P,and we write p∨q<∞otherwise.

Let(G,P)be a quasi-lattice ordered group,and let X be a product system over P of right-Hilbert A–A bimodules.We say that X is compactly aligned if,for all p,q∈P

such that p∨q<∞,and for all S∈K(X p)and T∈K(X q),we haveιp∨q

p (S)ιp∨q

q

(T)∈

K(X p∨q).Note that this condition does not imply compactness of eitherιp∨q

p (S)or

ιp∨q

q

(T).

2.3.Representations of product systems and Nica covariance.Let(G,P)be a quasi-lattice ordered group,and let X be a compactly aligned product system over P of right-Hilbert A–A bimodules.Let B be a C?-algebra,and letψbe a function from X to B.For p∈P,letψp=ψ|X

p

.We callψa representation of X if

(T1)eachψp:X p→B is linear,andψe is a C?-homomorphism;

(T2)ψp(x)ψq(y)=ψpq(xy)for all p,q∈P,x∈X p,and y∈X q;and

(T3)ψe( x,y p

A

)=ψp(x)?ψp(y)for all p∈P,and x,y∈X p.

Remark2.1.Our de?nition agrees with Fowler’s[12,De?nition2.5]:condition(4)of the de?nition of a product system together with(T1)–(T3)ensures that each(ψe,ψp) is representation of X p in the sense of Pimsner.It then follows from Pimsner’s results (see[26,p.202])that for each p∈P there is a homomorphismψ(p):K(X p)→B which satis?esψ(p)(x?y?)=ψp(x)ψp(y)?for all x,y∈X p.

We say that a representationψof X is Nica covariant if

(N)For all p,q∈P and all S∈K(X p)and T∈K(X q),

ψ(p)(S)ψ(q)(T)= ψ(p∨q) ιp∨q p(S)ιp∨q q(T) if p∨q<∞

0otherwise.

Results of[12]show that there exist a C?-algebra T cov(X)and a Nica covariant rep-resentation i of X in T cov(X)which are universal in the sense that:

(1)T cov(X)is generated by{i(x):x∈X};and

(2)ifψis any Nica covariant representation of X on a C?-algebra B then there is a

unique homomorphismψ?:T cov(X)→B such thatψ??i=ψ.

3.Cuntz-Nica-Pimsner covariance and the Cuntz-Nica-Pimsner algebra In this section we present our de?nition of Cuntz-Pimsner covariance for a compactly aligned product system X.We then introduce the Cuntz-Nica-Pimsner algebra N O X. To present our de?nition we begin by introducing a collection of bimodules X p,p∈P which we associate to a product system X over P.The X p do not form a product system;rather,they play a r?o le similar to that played by the setsΛ≤m for a locally convex k-graph in[29].We?rst recall some standard notation for direct sums of Hilbert modules.

Let J be a set,let X j be a right-Hilbert A–A bimodule for each j∈J,and let X= j∈J X j.LetΓc(J,X)be the space of all?nitely-supported sections x:J→X;that is

8AIDAN SIMS AND TRENT YEEND

x(j)∈X j for all j∈J.The formula x,y A= j∈J x(j),y(j) A de?nes an A-valued inner-product onΓc(J,X).The completion ofΓc(J,X)in the norm arising from this inner product is called the direct sum of the X j and denoted j∈J X j.Endowed with pointwise left-and right-actions of A, j∈J X j is itself a right-Hilbert A–A bimodule. Let X be a right-Hilbert A module,and let I be an ideal of A.Let X·I denote {x·a:x∈X,a∈I}.It is well-known that

(3.1)X·I={x∈X: x,x A∈I}=

C?-ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS9 Since p=e,each M p,r is an isomorphism,and the result follows. Example3.3.LetΛbe a?nitely aligned k-graph and let X=X(Λ)be the corresponding product system over N k as in[28].Then A=c0(Λ0),and for n∈N k,I n=

span{δμ:μ∈Λm,s(μ)Λe i=?whenever m i

In the language of[29,30],this spanning set is familiar:δμ∈X m·I n?m if and only if μ∈Λm∩Λ≤n.That is,as a vector space, X n=c0(Λ≤n).

We thank Sean Vittadello for pointing out a simpli?cation of the proof of the following Lemma.

Lemma3.4.Let(G,P)be a quasi-lattice ordered group,let X be a compactly aligned product system over P of right-Hilbert A–A bimodules,and?x p,q,r∈P such that pr≤q and p=e.Then X pr·I(pr)?1q is invariant underιpr p(S)for all S∈L(X p). Proof.If x∈X p,y∈X r and i∈I(pr)?1q,then

ιpr p(S)(xy·i)=(Sx)(y·i)=((Sx)y)·i∈X pr·I(pr)?1q

by de?nition ofιpr p,axiom(3)for product systems,and associativity of multiplication in X.Since vectors of the form xy·i where x∈X p,y∈X r and i∈I(pr)?1q span a dense subspace of X pr·I(pr)?1q,and sinceιpr p(S)∈L(X pr)is bounded and linear,the result follows. Remark3.5.Sinceιr e=φr,each X r·I r?1q is also invariant underιr e(a)for all a∈K(X e). Notation3.6.Let(G,P)be a quasi-lattice ordered group,and let X be a compactly aligned product system over P of right-Hilbert A–A bimodules.Recall thatιq p is the zero homomorphism from L(X p)to L(X q)when p≤q.Lemma3.4implies that for all p,q∈P with p=e there is a homomorphism from L(X p)to L( X q)determined by

S→ r≤qιr p(S)for all S∈L(X p).

We denote this homomorphism by ιq p;it is characterised by( ιq p(S)x)(r)=ιr p(S)x(r). As with theιq p,when p=e,we write ιp e for the homomorphism from K(X e)to L( X q) obtained from φp and the isomorphism K(X e)~=A.

Remark3.7.Note that ιq p(S)is the zero operator on those summands X r·I r?1q of X q such that p≤r.Thus ιq p can alternatively be characterised by

ιq p(S)= r≤q,p≤r0L(X r·I r?1q) ⊕ p≤r≤qιr p(S)|X r·I r?1q .

.

In particular,given S∈L(X p)and q∈P such that p≤q,we have ιq p(S)=0

L(e X q)

To formulate our Cuntz-Pimsner covariance condition,we require another de?nition. De?nition3.8.Let(G,P)be a quasi-lattice ordered group.We say that a predicate statement P(s)(where s∈P)is true for large s if:for every q∈P there exists r∈P such that q≤r and P(s)holds for all s≥r.

10AIDAN SIMS AND TRENT YEEND

We now present our de?nition of Cuntz-Pimsner covariance.We give a de?nition only in the situation that the homomorphisms φq:A→L( X q)are all injective.We will see in Lemma3.15that this is automatically true for extensive classes of product systems. We will also see in Example3.16that this is a necessary assumption for our de?nition to satisfy criterion(A)of Section1.2.

De?nition3.9.Let(G,P)be a quasi-lattice ordered group,and let X be a compactly aligned product system over P of right-Hilbert A–A bimodules in which the homo-morphisms φq of De?nition3.1are all injective.Letψbe a representation of X in a C?-algebra B.We say thatψis Cuntz-Pimsner covariant if (CP) p∈Fψ(p)(T p)=0B for every?nite F?P,and every choice of generalised compact operators{T p∈K(X p):p∈F}such that p∈F ιs p(T p)=0for large s. Remark3.10.The idea is that as the indices q∈P become arbitrarily large,the associ-ated X q approximate the“boundary”of the product system.That is(CP)is intended to encode relations that we would expect to hold if we could let all the K(X p)act on the boundary of X.

Our primary object of study in this paper will be the C?-algebra which is universal for representations which are both Nica covariant and Cuntz-Pimsner covariant.

De?nition3.11.We shall call a representation which satis?es both(N)and(CP)a Cuntz-Nica-Pimsner covariant(or CNP-covariant)representation.

Proposition3.12.Let(G,P)be a quasi-lattice ordered group,and let X be a compactly aligned product system over P of right-Hilbert A–A bimodules such that the homomor-phisms φq of De?nition3.1are all injective.Then there exist a C?-algebra N O X and a CNP-covariant representation j X of X in N O X such that:

(1)N O X=

C?-ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS11 representation then it is,in particular,a Nica covariant representation of X and it follows from[12,Theorem6.3]that there is a homomorphismψ?:T cov(X)→B such thatψ=ψ??i X.Sinceψis Cuntz-Pimsner covariant,we have I?ker(ψ?),and it follows thatψ?descends to a homomorphismΠψ:N O X→B,establishing(2).

All that remains to be proved is the uniqueness claim,for which we give the following standard argument.If(C,ρ)is another such pair then(2)for N O X implies that there is a homomorphismΠρ:N O X→C such thatΠρ(j X(x))=ρ(x)for all x.Statement(2)for C implies that there is a homomorphismΠj X:C→N O X such thatΠj X(ρ(x))=j X(x) for all x.Applications of(1)then show thatΠρandΠj X are surjective and are mutually inverse. Remark3.13.No obvious notation for the homomorphismΠψof Proposition3.12(2) occurred to us.We were loathe to re-de?ne Fowler’s notationψ?:ourΠψis induced from Fowler’sψ?by regarding N O X as a quotient of T cov(X),so employing the same notation would lead to confusion in any situation where representations of both T cov(X) and N O X are discussed.We settled onΠψon the basis that it might bring to mind the integrated formπ×ψof a representation(π,ψ)of a single bimodule.

Remark3.14.It should be emphasised that N O X may di?er from Fowler’s O X even should our notion of Cuntz-Pimsner covariance and Fowler’s coincide.The algebras N O X and O X will only coincide for product systems such that:the two versions of Cuntz-Pimsner covariance are equivalent;and Cuntz-Pimsner covariance implies Nica covariance.

We conclude this section by showing that the hypothesis that the φq are all injective is automatic for broad classes of product systems;but we also show by example that there exist product systems for which this hypothesis fails.

Speci?cally,we shall show that the φq are all injective whenever theφq are all injective, and also whenever the quasi-lattice ordered pair(G,P)has the property that every nonempty bounded subset of P contains a maximal element in the following sense.

(3.5)If S?P is nonempty and there exists q∈P such that p≤q for all p∈S,then there exists p∈S such that p≤p′for all p′∈S\{p}.

Lemma3.15.Let(G,P)be a quasi-lattice ordered group,and let X be a compactly aligned product system over P of right-Hilbert A–A bimodules.If eachφp is injective, or if P satis?es(3.5),then φq:A→L( X q)is injective for each q∈P. Proof.Suppose?rst that eachφp is injective.Then X p~=X p for all p and this isomor-phism intertwines φp andφp,so the φp are injective.

Now suppose that P satis?es(3.5).Fix a∈A\{0}and q∈P.We must show that φq(a)=0;that is,φp(a)|X p·I p?1q=0for some p≤q.We haveφe(a)=0because φe(a)(a?)=aa?=0.Let S={p∈P:p≤q,φp(a)=0}.Then S is nonempty,and is bounded above by q.Since P satis?es(3.5),it follows that S contains a maximal element p.Since p∈S,we haveφp(a)=0,so we may?x x∈X p such thatφp(a)x=0. Since p is maximal in S,if e

12AIDAN SIMS AND TRENT YEEND

Let {μλ:λ∈Λ}be an approximate identity for I p ?1q .By the preceding paragraph,{φp (a )x ·μλ}λ∈Λis norm-convergent to φp (a )x =0,so there exists λ∈Λsuch that φp (a )x ·μλ=0.Setting y =x ·μλ,we have y ∈X p ·I p ?1q and φp (a )y =0.That is φp (a )|X p ·I p ?1q =0,so φq (a )=0by de?nition.

The hypotheses of Lemma 3.15are not just an artifact of our proof.To see why,consider the following example.Example 3.16.Let G =Z ×Z ,and let P =((N \{0})×Z )∪({0}×N ).Then P is a subsemigroup of G satisfying P ∩?P ={(0,0)}.The partial order ≤on G de?ned by m ≤n ??n ?m ∈P is the lexicographic order on Z ×Z ,and in particular (G,P )is a quasi-lattice ordered group.Note that (G,P )does not satisfy (3.5):let S denote the subset S ={0}×N .Then S is bounded above by (1,0),but has no maximal element.Consider the right-Hilbert C 2module C 2C 2with the usual right action and inner-

product.Then L (C 2C 2)=K (C 2C 2)~=C 2:the element (z 1,z 2)∈C 2acts by point-wise left multiplication on C 2C 2.De?ne homomorphisms φS ,φP \S :C 2→L (C 2C 2)by φS =id C 2

and φP \S (z 1,z 2)=(z 1,z 1).We write X S (respectively X P \S )for C 2C 2regarded as a

right-Hilbert C 2–C 2bimodule with left action implemented by φS (respectively φP \S ).There are isomorphisms

X S ?C 2X S ~=X S

determined by (z 1,z 2)?C 2(w 1,w 2)→(z 1w 1,z 2w 2),X S ?C 2X P \S ~=X P \S

determined by (z 1,z 2)?C 2(w 1,w 2)→(z 1w 1,z 1w 2),X P \S ?C 2X S ~=X P \S

determined by (z 1,z 2)?C 2(w 1,w 2)→(z 1w 1,z 2w 2),and X P \S ?C 2X P \S ~=X P \S determined by (z 1,z 2)?C 2(w 1,w 2)→(z 1w 1,z 1w 2);

to see this,one checks that each of these formulae preserves inner-products of elementary tensors.

For p ∈S ,let X p =X S ,and for p ∈P \S ,let X p =X P \S .With multiplication maps de?ned as above,X is a product system over P .Note that for p ∈P ,φp =φS if p ∈S ,and φp =φP \S if p ∈S .Since L (X p )=K (X p )for all p ,the left action of C 2on each ?bre is by compact operators,and in particular X is compactly aligned.

We have (0,1)≤p for all p ∈P \{(0,0)}.Since ker(φ(0,1))={0},it follows that

I p ={0}for p =(0,0).Hence X

q =X q and φq =φq .In particular, φ(1,0)=φ(1,0)=φP \S is not injective.

Observe that ιs (0,0)((0,1))=0L (e X s )for large s .Hence every representation ψof X satisfying (CP)satis?es ψe ((0,1))=0.In particular the algebra universal for such representations does not satisfy criterion (A)of Section 1.2.

Remark 3.17.Subject to failure of the hypothesis of Lemma 3.15,Example 3.16is as well-behaved as possible:(G,P )is countable and totally ordered,and the natural order topology is discrete;A and the X p are all ?nite-dimensional,so that in particular the action on each ?bre is by compact operators;and each X p is essential as a left A -module in the sense that φp (A )X p =X p .

4.Injectivity of the universal CNP-covariant representation

In this section we prove our main theorem.

C?-ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS13 Theorem4.1.Let(G,P)be a quasi-lattice ordered group,and let X be a compactly aligned product system over P of right-Hilbert A–A bimodules.Suppose that the ho-momorphisms φq of De?nition3.1are all injective.Then the universal CNP-covariant representation j X:X→N O X is isometric: j X(x) = x for all x∈X.In particular, the conclusion holde if eachφp is injective,or if P satis?es(3.5).

We now introduce a modi?cation,based on the X p,of Fowler’s Fock representation. Notation4.2.Let(G,P)be a quasi-lattice ordered group,and let X be a product

system over P of right-Hilbert A–A bimodules.As on[12,page340],we let F(X)= p∈P X p,and call it the Fock space of X.We also de?ne F(X)= p∈P X p,and call it the augmented Fock space of X.

Fowler shows[12,page340]that for x∈X p there is an adjointable operator l(x)on F(X)determined by

(l(x)y)(q)= x(y(p?1q))if p≤q

0otherwise.

He shows further that l:X→L(F(X))is a Nica covariant representation of X.The next lemma shows that we obtain a parallel result if we replace F(X)with F(X). Lemma 4.3.Let(G,P)be a quasi-lattice ordered group,and let X be a compactly aligned product system over P of right-Hilbert A–A bimodules.

(1)Let p,q,r∈P,with q≤r,and let x∈X p and z∈X q·I q?1r.Then xz∈

X pq·I(pq)?1pr.

(2)For p∈P and x∈X p,there is an adjointable operator l(x)∈L( F(X))which

satis?es

( l(x)y)(q)= x(y(p?1q))if p≤q

0otherwise

for all y∈ F(X).Moreover if y∈ F(X)satis?es y(s)=0for all s≥p,then l(x)?y=0.

(3)The map x→ l(x)of(2)is a Nica covariant representation of X in L( F(X)). Proof.To prove(1),write z=z′·i where z′∈X q and i∈I q?1r?A=X e.By associativity of multiplication in X and condition(4)of the de?nition of a product system we have xz=x(z′·i)=(xz′)·i∈X pq·I(pq)?1(pr).

For(2),observe that each X q is a sub-module of F(X),and that the restriction of l(x) to this submodule agrees with l(x).Since l(x)is an adjointable operator on F(X),the ?rst part of(2)follows.For the second part,observe that by linearity and continuity of l(x)?,we may assume that y has just one nonzero coordinate.That is,we may assume that there exists r∈P such that r≥p and y(s)=0for s=r.For z∈ F(X),we then have

l(x)?y,z A= y, l(x)z A= s∈P y(s),( l(x)z)(s) e X s A= y(r),( l(x)z)(r) e X r A=0 by the?rst part of(2).

14AIDAN SIMS AND TRENT YEEND

For (3),recall that l satis?es (T1)–(T3)and (N).It follows that l does as well.

Let B be the C ?-algebra generated by { l (x ):x ∈X }?L ( F (X )).Let I e F (X )be the

ideal in B generated by p ∈F l (p )(T p ):F ?P is ?nite ,

T p ∈K (X p )for each p ∈F ,and

p ∈F ιs p (T p )=0for large s .

Let q e F (X ):B →B/I e F (X )denote the quotient map.Then ψ:=q F (X )? l is a CNP-

covariant representation of X in B/I F (X ).Proposition 4.4.Let (G,P )be a quasi-lattice ordered group,and let X be a compactly aligned product system over P of right-Hilbert A –A bimodules.Suppose that each φq is injective.Then q F (X )? l :X →B/I F (X )is faithful on A .

Proof.We must show that I e F (X )∩ l (A )={0}.

For a ∈A ,the restriction l (a )|e X q of l (a )to the q -summand of F (X )is given by l (a )x = φ

q (a )x .Since φq is injective and hence isometric,it follows that the operator norm l (a )|e X q is equal to the C ?-norm a of a ∈A .It therefore su?ces to show that (4.1)for each ψ∈I e F (X )and each ε>0there exists s ∈P such that ψ|e X

s <ε.We do this in stages.Let K ?I e F (X )be the subset K =

p ∈F

l (p )(T p ):F ?P is ?nite ,T p ∈K (X p )for each p ∈F ,and

p ∈F ιs p (T p )=0for large s .Then K is a subspace of I e F (X ),and a continuity argument shows that for each ψ∈

span { l (x ) l (y )?:x,y ∈X }.Consequently,

I e F (X )=

K,x,y,x ′,y ′,∈X }.Fix k ∈

x y x ′ y ′ for all s ′≥r ′.Let r =

p (y ′)p (x ′)?1r ′≥q .Then for each s ≥r ,we have p (x ′)p (y ′)?1

s ≥r ′,and hence

l (x ) l (y )?k l (x ′) l (y ′)?|e X s <ε.Now suppose that p (y ′)∨q =∞.Then s ≥q implies s ≥p (y ′),so Lemma

4.3(2)implies that l (y ′)?|e X s =0.Consequently r =q satis?es l (x ) l (y )?k l (x ′) l (y ′)?|e X s =0for every s ≥r .

The assertion (4.1)now follows by linearity and continuity.

C ?-ALGEBRAS ASSOCIATE

D TO PRODUCT SYSTEMS 15

Remark 4.5.Since

each summand of F (X )is a sub-module of F (X ),it was not necessary to introduce F

(X )at all.Essentially the argument of Proposition 4.4is valid if we work instead with Fowler’s l :X →F (X )followed by the appropriate quotient map.However using F (X )makes the argument clearer,and helps give some intuition for the signi?cance of the X

p .Proof of Theorem 4.1

.Proposition 4.4together with he universal property of N O X im-plies that j X is injective on X e =A .As j X |X e is a C ?-homomorphism,it follows that j X is isometric on A .For any x ∈X ,we therefore have

x 2= x,x A = j X ( x,x A ) = j X (x )?j X (x ) = j X (x ) 2,

completing the proof. Remark 4.6.Note that N k ,and indeed every right-angled Artin semigroup,satis?es (3.5),so Theorem 4.1applies when P is any of these semigroups.

Remark 4.7.Theorem 4.1shows that if each φ

q is injective,and in particular if P

satis?es (3.5),then N O X satis?es criterion (A)of Section 1.2.In the next section,we

will show that it also satis?es (B)in a number of motivating examples.We will achieve this indirectly by combining Theorem 4.1with the uniqueness theorems of [7,18,30].We have not veri?ed (B).This is done in [4]for certain pairs (G,P )using a careful analysis of the ?xed-point algebra N O δX =

16AIDAN SIMS AND TRENT YEEND

(2)Ifφp(A)?K(X p)for each p∈P,andψis Cuntz-Pimsner covariant in the

sense of[12,De?nition2.5],thenψsatis?es(CP).

Proof.We begin with some observations which we will use to prove both(1)and(2).Let a∈A.Recall that X e is equal to A,and thatφe(a):X e→X e is the left-multiplication operator b→ab,which we denote by L a∈K(X e).Every element T of K(X e)can be written as T=L a for some a∈A(see[31,Lemma2.26]).By de?nition ofιp e,we have (5.1)ιp e(L a)=φp(a)for all p∈P,a∈A.

We now prove(1)and(2)separately.

For(1),let p∈P and a∈A,and suppose thatφp(a)∈K(X p).We must show that

ψe(a)?ψ(p)(φp(a))=0.Equation(5.1)implies thatιs e(L a)?ιs p(φp(a))=0L(X

s)for all

s≥p.Since each pair in P has a least upper bound,it follows that for each q∈P,we haveιs e(L a)?ιs p(φp(a))=0L(X

s)

for all s≥p∨q.

Since eachφp is injective,we have I p={0}for p=e.Hence X p=X p and ιq p=ιq p for all p≤q∈P.Thus ιs e(L a)? ιs p(φp(a))=0for large s.Sinceψsatis?es(CP),we deduce thatψ(e)(L a)?ψ(p)(φp(a))=0.A straightforward computation using an approximate identity for A and thatψe is a homomorphism shows thatψ(e)(L a)=ψe(a).Hence ψe(a)=ψ(p)(φp(a)),andψis Cuntz-Pimsner covariant in the sense of[12,De?nition2.5]. Now for(2),?x a?nite subset F?P and compact operators T p∈K(X p),p∈F such that p∈F ιs p(T p)=0for large s.An inductive argument using that every pair in P has a least upper bound shows that there is a least upper bound q= F for F in P.Since eachφp is injective,we haveιs q injective for s≥q,so we must have p∈Fιq p(T p)=0.

We may assume without loss of generality that e∈F,and rearrange to obtain p∈F\{e}ιq p(T p)=?ιq e(T e).As observed above,T e=L a for some a∈A,so(5.1) implies thatιq e(T e)=φq(a).Since the left action of A on X q is by compact operators, we haveφq(a)∈K(X q),and hence Fowler’s Cuntz-Pimsner covariance condition forces (5.2)ψ(q) p∈F\{e}ιq p(T p) =?ψe(a).

Since A acts compactly on the left of each X p,[26,Corollary3.7]shows that each ιq p(T p)∈K(X q),and the argument of[26,Lemma3.10]shows thatψ(q)(ιq p(S))=ψ(p)(S). Hence(5.2)implies that

p∈F\{e}ψ(p)(T p)=?ψe(a).

Sinceψe(a)=ψ(e)(L a)=ψ(e)(T e),we have p∈Fψ(p)(T p)=0.Soψsatis?es(CP). Corollary5.2.Let(G,P)be a quasi-lattice ordered group,and let X be a compactly aligned product system over P of right-Hilbert A–A bimodules.Suppose that each pair in P has a least upper bound.Suppose that eachφp:A→L(X p)is injective with φp(A)?K(X p).Letψ:X→B be a representation of X.Thenψis CNP-covariant if and only ifψ(p)?φp=ψe for all p∈P.

Proof.Since eachφp(A)?K(X p),the conditionψ(p)?φp=ψe for all p∈P is precisely the Cuntz-Pimsner covariance condition of[12,De?nition2.5].Proposition5.4of[12]

C?-ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS17 implies that ifψis Cuntz-Pimsner covariant in this sense,then it is also Nica covariant. Hence the result follows from Proposition5.1.

5.2.Katsura’s C?-algebras associated to Hilbert bimodules.Let X be a right-Hilbert A–A bimodule.There is a product system X?over N of right-Hilbert A–A bimodules such that X?0=A,and X?n=X?n for1≤n∈N.For nonzero m,n,the isomorphism M m,n:X?m?X?n→X?m+n implementing the multiplication in the system is the natural isomorphism X?m?X?n~=X?m+n.As N is totally ordered,X?is compactly aligned.

If(ψ,π)is a representation of X,then there is a representationψ?of X?given by ψ?e=πandψ?n=ψ?n for n≥1.This representation is automatically Nica covariant, and every representation of X?is of this form.Proposition2.11of[12]says that if the homomorphismφ:A→X implementing the left action is injective or takes values in K(X)thenψis covariant in Pimsner’s sense if and only ifψ?is Cuntz-Pimsner covariant in Fowler’s sense.Hence Pimsner’s O X[26]and Fowler’s O X?[12]coincide.

A key goal of our construction was to achieve the same outcome with respect to Kat-sura’s reformulation of Cuntz-Pimsner covariance for a single Hilbert bimodule.That is,given an arbitrary Hilbert bimodule X,we desire that N O X?should coincide with Katsura’s O X[17,18].

Recall that if I?A is an ideal in a C?-algebra,then I⊥denotes the ideal{a∈A:ab= 0for all b∈I}.Recall also that Katsura’s O X is the universal C?-algebra generated by

a representation(i A,i X)of X such that i(1)

X (φ(a))=i A(a)for all a∈ker(φ)⊥such that

φ(a)∈K(X).

Proposition5.3.Let X be a right-Hilbert A–A bimodule.Let(i A,i X)be the universal

representation of X on O X,and j X?be the universal representation of X?on N O X?.

(1)There is an isomorphismθ:O X→N O X?satisfyingθ(i A(a))=j X?(a)and

θ(i X(x))=j X?(x)for all a∈A and x∈X.

(2)Let(π,ψ)be a representation of X and letψ?be the corresponding Nica covariant

representation of X?.Then(π,ψ)is covariant in the sense of Katsura if and

only ifψ?satis?es(CP).

Proof.Statement(2)follows from(1)and the universal properties of O X and N O X?,

so it su?ces to prove(1).

We have ker(φ)?ker(φ?1n?1)for n≥1,so I n is equal to A if n=0and is

equal to ker(φ)if n=0.Let(j0,j1)denote the representation of X determined by the universal representation of X?on N O X?.If a∈ker(φ)⊥∩φ?1(K(X)),then φ1(a)= 0ker(φ)⊕φ(a).Let S=φ(a)∈K(X).Then ι10(φ0(a))? ι11(S)=0,and it follows

that ιn0(φ0(a))? ιn1(S)=0for all n≥1.Since j X?satis?es(CP),we therefore have j0(a)?j(1)1(S)=0;that is j0(a)=j(1)1(φ(a)).Thus(j0,j1)is covariant in the sense of Katsura,and the universal property of O X implies that there is a homomorphism θ:O X→N O X determined byθ?i A=j0andθ?i X=j1.Moreover,θis surjective because

N O X=

18AIDAN SIMS AND TRENT YEEND

5.3.Cuntz-Krieger algebras of?nitely aligned higher-rank graphs.In this sec-tion,we use the notation and conventions of[30]for higher-rank graphs.In[28],a product system X(Λ)over N k of right-Hilbert c0(Λ0)–c0(Λ0)bimodules is associated to each k-graphΛ.WhenΛis row-?nite and has no sources,the homomorphismφp

implementing the left action of c0(E0)on X(Λ)p is an injective homomorphism into the compact operators on X(Λ)p.Corollary4.4of[28]shows that C?(Λ)coincides with O X(Λ)as de?ned in[12]and hence with N O X(Λ)by Proposition5.1.A key goal of our construction is to extend this to arbitrary?nitely aligned k-graphs and the correspond-

ing compactly aligned product systems of bimodules.We show in this section that we have achieved this aim.

We brie?y recall some salient point about k-graphs from[30].A k-graph is a countable categoryΛtogether with a functor d:Λ→N k satisfying the factorisation property:for all m,n∈N k andλ∈d?1(m+n)there exist uniqueμ∈d?1(m)andν∈d?1(n)such

thatλ=μν.Each d?1(n)is denotedΛn.The elements ofΛ0are called vertices,and are in bijection with the objects ofΛ,so the codomain and domain maps in the category Λdetermine maps r,s:Λ→Λ0.Forμ,ν∈Λ,we write MCE(μ,ν)for the collection {λ∈Λ:d(λ)=d(μ)∨d(ν),λ=μμ′=νν′for someμ′,ν′∈Λ}.For n∈N k,we write

Λ≤n for the set{λ∈ m≤nΛm:d(λ)

a Toeplitz-Cuntz-KriegerΛ-family if

(CK1){s v:v∈Λ0}is a set of mutually orthogonal projections;

(CK2)sμsν=sμνwhenever s(μ)=r(ν);and

(CK3)s?μsν= μμ′=νν′∈MCE(μ,ν)sμ′s?ν′for allμ,ν∈Λ.

It is called a Cuntz-KriegerΛ-family if it additionally satis?es

(CK4) λ∈F(s v?sλs?λ)=0for all v∈Λ0and all nonempty?nite exhaustive sets F?r?1(v).

The Cuntz-Krieger algebra C?(Λ)is the universal C?-algebra generated by a Cuntz-

KriegerΛ-family.

Theorem4.2and Proposition6.4of[28]show that Nica covariant representations of X(Λ)are in bijective correspondence with Toeplitz-Cuntz-KriegerΛ-families.To describe this bijection,we must brie?y recall the de?nition of X(Λ)(see[28]for details). For n∈N k,we endow c c(Λn)=span{δλ:λ∈Λn}with the structure of a pre-Hilbert c0(Λ0)-bimodule via the following formulae:(a·x·b)(λ):=a(r(λ))x(λ)b(s(λ));and (v)= s(λ)=v

x,y c

0(Λ0)

C?-ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS19

(1)There is an isomorphismθ:C?(Λ)→N O X satisfyingθ(sλ)=j X(δλ)for all

λ∈Λ.

(2)Letψ:X→B be a Nica-covariant representation of X,and let{tλ:λ∈Λ}be

the corresponding Toeplitz-Cuntz-KriegerΛ-family tλ=ψ(δλ).Thenψsatis?es De?nition3.9if and only if{tλ:λ∈Λ}satis?es the Cuntz-Krieger relation[30, De?nition2.5(iv)].

Proof.Statement(2)follows from(1)and the universal properties of C?(Λ)and N O X. So it su?ces to prove(1).

For eachλ∈Λ,let jλ=j X(δλ),so{jλ:λ∈Λ}is a Toeplitz-Cuntz-KriegerΛfamily which generates N O X.We claim that{jλ:λ∈Λ}is a Cuntz-KriegerΛ-family.

Fix v∈Λ0and a?nite exhaustive set F?vΛ.We must show that

μ∈F(j v?jμj?μ)=0.

For a subset G?F,we will denote by∨d(G)the element μ∈G d(μ)of N k.If λ,μ∈Λsatisfyλ=μμ′for someμ′∈Λ,we say thatλextendsμ.Recall from [28]that for a nonempty subset G of F,MCE(G)denotes the set{λ∈Λ:d(λ)=∨d(G),λextendsμfor allμ∈G}.Recall also that∨F:= G?F MCE(G)is?nite and is closed under minimal common extensions.We have

μ∈F(j v?jμj?μ)=j v+ ? =G?F

(?1)|G|jλj?λ

λ∈MCE(G)

(δv?δ?v)+ ? =G?Fλ∈MCE(G)(?1)|G|j(∨d(G))X(δλ?δ?λ).

=j(e)

X

Since j X(Λ)satis?es(CP)it su?ces to show that for each q∈N k there exists r≥q such that for all s≥r we have

ιs e(δv?δ?v)+ ? =G?Fλ∈MCE(G)(?1)|G| ιs∨d(G)(δλ?δ?λ)=0.

For this,?x q∈N k,let r=q∨(∨d(F))and?x s≥r.Since X s=⊕t≤s

20AIDAN SIMS AND TRENT YEEND

The factorisation property implies thatτextends eachμin G if and only if there exists λin MCE(G)such thatτextendsλ.The factorisation property also implies that if there does exist such aλ∈MCE(G)then it is necessarily unique.We therefore have μ∈G ιs d(μ)(δμ?δ?μ) (δτ)= λ∈MCE(G) ιs∨d(G)(δλ?δ?λ) (δτ)

Since the?xed nonempty subset G of F in the preceding paragraph was arbitrary,we may now calculate:

μ∈F ιs e(δv?δ?v)? ιs d(μ)(δμ?δ?μ) (δτ)

= ιs e(δv?δ?v)+ ? =G?F (?1)|G| μ∈G ιs d(μ)(δμ?δ?μ) (δτ)

= ιs e(δv?δ?v)+ ? =G?Fλ∈MCE(G)(?1)|G| ιs∨d(G)(δλ?δ?λ) (δτ).

Since F is exhaustive,there existsν∈F such that MCE(τ,ν)=?;sayττ′∈MCE(τ,ν). Since d(ν),d(τ)≤s,we have d(ττ′)≤s.Sinceτ∈Λ≤s,this forces d(τ′)=0,soτextendsν.Thus(5.4)implies that

μ∈F ιs e(δv?δ?v)? ιs d(μ)(δμ?δ?μ) (δτ)

= μ∈F\{ν}( ιs e(δv?δ?v)? ιs d(μ)(δμ?δ?μ)) ιs e(δv?δ?v)? ιs d(ν)(δν?δ?ν) (δτ)=0,

establishing(5.3).Hence{jλ:λ∈Λ}is a Cuntz-KriegerΛ-family as claimed.

Since the jλgenerate N O X,the universal property of C?(Λ)implies that there is a surjective homomorphismθ:C?(Λ)→N O X satisfyingθ(sλ)=jλ=j X(δλ)for all λ∈Λ.By Theorem4.1,we have j v=0for all v∈Λ0.Sinceθintertwines the gauge actions of T k on N O X and C?(Λ),the gauge-invariant uniqueness theorem for C?(Λ) [30,Theorem4.2]therefore implies thatθis an isomorphism.

5.4.Boundary quotients of Toeplitz algebras.In this section,we consider product systems whose?bres are isomorphic to C C C.Nica covariant representations of such product systems amount to Nica covariant representations of(G,P)in the sense of [6,22,25].This prompts us to explore the connection between our N O X and the boundary quotients of T(G,P)studied by Crisp and Laca in[7].

The?rst part of the following proposition follows from results of Fowler and Raeburn [15],but we include it for completeness.We?rst make the following de?nition:if(G,P) is a quasi-lattice ordered group,we say that a?nite subset F of P is a foundation set for P if,for every q∈P there exists p∈F such that p∨q<∞.

Notation5.5.Given a semigroup P,we denote by C P the product system of right-Hilbert C–C bimodules determined by X p=C C C for all p with multiplication in X given by multiplication of complex numbers.As a notational convenience,when we are regarding a complex number z as an element of C P p,we shall denote it z p.

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苏泊尔电磁炉维修手册大全苏泊尔电磁炉维修手册大全电磁加热原理电磁灶是一种利用电磁感应原理将电能转换为热能的厨房电器。在电磁灶内部,由整流 电路将50/60hz的交流电压变成直流电压,再经过控制电路将直流电压转换成频率为 20-40khz的高频电压,高速变化的电流流过线圈会产生高速变化的磁场,当磁场内的磁力线 通过金属器皿(导磁又导电材料)底部金属体内产生无数的小涡流,使器皿本身自行高速发热, 然后再加热器皿内的东西。 1.2 458系列筒介 458系列是由建安电子技术开发制造厂设计开发的新一代电磁炉,介面有led发光二极管显示模式、led数码显示模式、lcd液晶显示模式、vfd 莹光显示模式机种。操作功能有加热 火力调节、自动恒温设定、定时关机、预约开/关机、预置操作模式、自动泡茶、自动煮饭、 自动煲粥、自动煲汤及煎、炸、烤、火锅等料理功能机种。额定加热功率有700~3000w的不 同机种,功率调节范围为额定功率的85%,并且在全电压范围内功率自动恒定。200~240v机种 电压使用范围为160~260v, 100~120v机种电压使用范围为90~135v。全系列机种均适用于50、

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