TTOEPLITZ ALGEBRAS OF SEMIGROUPS
MARCELO LACA AND CAMILA F. SEHNEM
Abstract.
To each monoid P that embeds in a group we associate a universal Toeplitz C ∗ -algebra T u ( P ) defined via generators and relations; T u ( P ) is a quotient of Li’s semigroup C ∗ -algebra C ∗ s ( P )and they are isomorphic if and only if P satisfies independence. We give a partial crossed productrealization of T u ( P ) and show that several key results known for C ∗ s ( P ) when P satisfies independenceare also valid for T u ( P ) when independence fails. At the level of the reduced semigroup C ∗ -algebra T λ ( P ), we show that nontrivial ideals have nontrivial intersection with the reduced crossed productof the diagonal subalgebra by the action of the group of units of P , generalizing a result of Li formonoids with trivial unit group. We also characterize when the action of the group of units istopologically free and we show that in this case a representation of T λ ( P ) is faithful iff it is jointlyproper. This yields a uniqueness theorem for C ∗ -algebras generated by semigroups of isometriesthat unifies several classical results. We provide a concrete presentation for the covariance algebraof the product system over P with one-dimensional fibers in terms of a notion of foundation sets ofconstructible ideals that generalizes that of Sims and Yeend for quasi-lattice orders. We then showthat the covariance algebra is a full, or universal, analogue of the boundary quotient ∂ T λ ( P ). We givepurely algebraic sufficient conditions on P for the boundary quotient ∂ T λ ( P ) to be purely infinitesimple, which reduce to Starling’s conditions in the case of right LCM monoids. We discuss concreteapplications of our results to examples that include a numerical semigroup and the ax + b -monoid ofan integral domain. This application is particularly interesting in the case of nonmaximal orders innumber fields, for which we show independence always fails. In addition, we simplify and generalizeresults of Brownlowe, Larsen and Stammeier for right LCM monoids. Introduction
Let P be a submonoid of a group and consider the left regular representation L : P → B ( ‘ ( P ))determined by L p δ x = δ px on the usual orthonormal basis { δ x | x ∈ P } of ‘ ( P ). The operators L p associated to p ∈ P are isometries and generate the Toeplitz C ∗ -algebra T λ ( P ). The spatial natureof T λ ( P ) provides many useful tools for its study, such as the existence of a faithful conditionalexpectation onto a diagonal subalgebra. But along with this comes the notoriously difficult problemof estimating norms of operators, in this case, of polynomials on the generating isometries and theiradjoints. As a result, it is often quite hard to decide whether a given representation of P by isometriesgenerates a homomorphic image of T λ ( P ). One strategy that has been successfully used to get aroundthis problem is to characterize T λ ( P ) by way of generators and relations that replicate distinguishedproperties of T λ ( P ). When such a universal characterization is possible and conditions are givenfor faithfulness of the resulting representations, one has what is known as a uniqueness theorem.Examples include celebrated theorems of Coburn for the natural numbers [7], of Douglas for discretesubmonoids of the additive reals [16], and of Cuntz for the free semigroup F + n [11], as well as severalgeneralizations, see e.g. [3, 10, 25, 34, 37].It has long been clear that a universal C ∗ -algebra generated by all isometric representations ofa monoid is often too large to be of much use. Indeed, as Murphy observed in [32], already inthe case of two commuting isometries, that is, for representations of N , one obtains a nonnuclearuniversal C ∗ -algebra. In many of the aforementioned situations, however, it is possible to identify extraconditions that distinguish a specific class of isometric representations and give tractable C ∗ -algebras.Substantial progress along this path took place in the early 90’s when Nica introduced a C ∗ -algebraC ∗ ( G, P ) that is universal for a class of ‘covariant’ representations of a quasi-lattice ordered group
Date : 17 January 2021.2010
Mathematics Subject Classification.
Primary 46L55; Secondary 46L55, 20M30, 11R04, 47B35.
Key words and phrases.
Toeplitz C ∗ -algebra; semigroup C ∗ -algebra; constructible ideals; boundary quotient, orders.This research was partially supported by NSERC Discovery Grant RGPIN-2017-04052.C.F. Sehnem was supported by the Marsden Fund of the Royal Society of New Zealand, grant No.18-VUW-056. a r X i v : . [ m a t h . OA ] J a n MARCELO LACA AND CAMILA F. SEHNEM ( G, P ). These are pairs (
G, P ) consisting of a submonoid P of a group G such that P ∩ P − = { e } and for which the intersection xP ∩ yP of any two cones with vertices in G is either empty orequal to another cone; this condition is not exactly Nica’s definition of quasi-lattice order, but isequivalent to it, see [9, Definition 6 and Lemma 7]. The covariance relation used by Nica stemmedfrom the observation that the multiplication of range projections of the generating isometries in theleft regular representation replicates the intersection of cones with vertices in P . Nica showed thatC ∗ ( G, P ) is actually isomorphic to T λ ( P ) in many interesting cases, including the C ∗ -algebra generatedby k ∗ -commuting isometries, which is universal for Nica covariant representations of N k , and theToeplitz–Cuntz algebra T O n which is universal for the free monoid F + n .In general if P is a submonoid of a group G , the set of cones having vertices in P does not have tobe closed under intersection, so Nica-covariance cannot be imposed ipsissimis verbis . Thus, when theinterest in monoids that are not quasi-lattice ordered began to surge in the mid 2000’s, sparked mainlyby semigroups arising from algebraic number theory, it became clear that a new idea was needed togo beyond the ad hoc analysis of examples. The major breakthrough was achieved by Li in [28, 29],and was later summarized and extended in [15, Chapter 5]. Motivated by what happens for ax + b semigroups of algebraic integers, Li’s insight was to realize that the idea of mirroring the behaviour ofprojections in the left regular representation does carry over to the general situation. The key to thiswas to replace the cones by a collection of subsets of P that he called constructible right ideals , whichform a semilattice under intersection.We pause briefly to carry out a concrete computation that illustrates how the constructible rightideals arise naturally, and what their role is in the definition of Li’s semigroup C ∗ -algebra; thiscomputation also motivates the notation we introduce in Section 2. Assume throughout that P isa submonoid of a group and suppose p, q, r, s ∈ P satisfy p − qr − s = e . Let us work out what theproduct L ∗ p L q L ∗ r L s really is in the left regular representation by computing it on a basis vector δ x ofthe standard orthonormal basis of ‘ ( P ). Clearly L ∗ p L q L ∗ r L s δ x = L ∗ p L q L ∗ r δ sx vanishes unless sx ∈ rP ,in which case L ∗ p L q L ∗ r δ sx = L ∗ p δ qr − sx . This, in turn, vanishes unless qr − sx ∈ pP , in which case L ∗ p L q L ∗ r L s δ x = δ p − qr − sx = δ x , because p − qr − s = e . Thus, L ∗ p L q L ∗ r L s δ x = ( δ x if x ∈ P ∩ s − rP ∩ s − rq − pP, L ∗ p L q L ∗ r L s is (the operator of multiplication by) the characteristic function of theset P ∩ s − rP ∩ s − rq − pP , which is a typical constructible right ideal. To define the semigroupC ∗ -algebra of a submonoid of a group, [28, Definition 3.2], Li associates an isometry v p to each p ∈ P and a projection e S to each constructible right ideal S , and then imposes relations that say that theisometries v p multiply as the elements of P and that a product such as v ∗ p v q v ∗ r v s is the projection e S where S is the constructible right ideal P ∩ s − rP ∩ s − rq − pP mentioned above, see Definition 3.21for the precise statement. The C ∗ -algebra with this presentation has many nice features, and isisomorphic to T λ ( P ) in several interesting examples, notably those arising from ax + b semigroups ofalgebraic integers. However, in addition to the unavoidable issue of amenability, the construction isfully satisfactory only when P satisfies independence, equivalently, when the characteristic functionsof ideals are linearly independent.Here we introduce a universal Toeplitz algebra T u ( P ), for P a submonoid of a group, that worksjust as well even if P does not satisfy independence. The presentation of T u ( P ) includes extra relationsthat only apply when independence fails, and we show that T u ( P ) is the quotient of Li’s semigroupC ∗ -algebra under a canonical homomorphism whose kernel reflects the failure of independence. Wealso relate T u ( P ) to various other C ∗ -algebras associated to semigroups, giving, in particular, a partialcrossed product realization. Our main results are a characterization of faithful representations of T λ ( P ) in terms of the action of units on the diagonal and a generalized uniqueness theorem for theC ∗ -algebra generated by a collection of elements satisfying the presentation of T u ( P ). Furthermore, wegive a presentation of the covariance algebra of the one-dimensional product system over P that allowsus to realize it as a universal boundary quotient. We also give conditions on P that are equivalentto topological freeness of the partial action on the boundary, and so are sufficient for the boundaryquotient to be purely infinite simple when P is nontrivial. In the setting of submonoids of groups,our results represent significant improvements and in some cases conceptual simplifications of earlier OEPLITZ ALGEBRAS OF SEMIGROUPS 3 work on C ∗ -algebras of monoids that have trivial unit group or satisfy independence [15], and ofright LCM monoids [3]. Throughout our work, we make a point of formulating the presentations andcharacterizations in terms of the original data, with an eye towards direct applications.We describe next the contents of this paper, highlighting the main results along the way. In Section 2we review the set of constructible right ideals, which we view as the range of a map α K ( α ) fromwords to ideals, through which, e.g., the ideal P ∩ s − rP ∩ s − rq − pP corresponds to the word( p, q, r, s ). We also state explicitly and in detail the properties of this map that are needed in theremaining sections. In Section 3 we begin by showing that if v : P → B is a unital map from P to aC ∗ -algebra B and the products such as v ∗ p v q v ∗ r v s only depend on the constructible ideal associatedto the word ( p, q, r, s ), rather than on the word itself, then p v p is an isometric representationsatisfying important additional properties, Proposition 3.2. We give the presentation of the universalToeplitz algebra T u ( P ) in Definition 3.6, and we then proceed to examine it in relation to the reducedToeplitz algebra T λ ( P ). We establish that T λ ( P ) is a canonical quotient of T u ( P ), Proposition 3.12,and in Proposition 3.18 we show that a representation of T u ( P ) is faithful on the diagonal subalgebra D u if and only if it satisfies a joint properness condition, Definition 3.14. We conclude that D u isalways canonically isomorphic to D r , the diagonal in T λ ( P ), Corollary 3.19. We complete Section 3 byshowing that a proper subset of the relations defining T u ( P ) gives the presentation of Li’s semigroupC ∗ -algebra, Proposition 3.22. This leads to Corollary 3.23 where we show that T u ( P ) is a canonicalquotient of C ∗ s ( P ); the two are isomorphic if and only if P satisfies independence.In Section 4 we give a brief introduction to partial actions and their crossed products and reviewthe reduced partial crossed product realization of T λ ( P ) given by Li in [15]. We do this explicitly interms of constructible right ideals, bypassing the use of inverse semigroups. The main result hereis Theorem 4.7, where we show that T u ( P ) is isomorphic to the full partial crossed product of thepartial action of G on the diagonal algebra. This allows us to verify that the improvements predictedby Li in [15, Remark 5.6.46] for the full partial crossed product are realized by our universal ToeplitzC ∗ -algebra, see Theorem 4.9 and Corollary 4.10.In Section 5 we study conditions that ensure that a representation of the reduced Toeplitz algebra T λ ( P ) is faithful. We notice first that T λ ( P ) has a copy of the reduced crossed product of the diagonalby the restriction of the partial action of G to the group of units P ∗ . The first main result, Theorem 5.1,shows that a representation of T λ ( P ) is faithful if and only if its restriction to the crossed product D r (cid:111) γ,r P ∗ is faithful. Setting P ∗ = { e } recovers [15, Corollary 5.7.3] and assuming P is a rightLCM monoid gives a stronger version of the faithfulness result [3, Theorem 7.4], albeit, under theextra assumption that P embeds in a group. The second main result in this section is Theorem 5.9,where we show that the partial action of G on D r is topologically free if and only if the restrictedaction of P ∗ is topologically free. Using a recent result of Abadie–Abadie [1, Theorem 4.5] we thenconclude that the action of P ∗ is topologically free if and only if every ideal of T u ( P ) that has trivialintersection with D u is contained in the kernel of the canonical map T u ( P ) → T λ ( P ). We also givethere a criterion in terms of constructible ideals to decide whether the action of P ∗ on the diagonalis topologically free. As a consequence, when the action of P ∗ is topologically free, faithfulness ofrepresentations of T λ ( P ) is decided by their restrictions to the diagonal, Corollary 5.10. When wecombine these results with the faithfulness criteria for representations of the diagonal from Section 3,we obtain Theorem 5.11, the third main result of this section, which shows that if T u ( P ) → T λ ( P )is an isomorphism, then T λ ( P ) is the unique C ∗ -algebra generated by a jointly proper semigroup ofisometries satisfying the presentation of T u ( P ).We begin Section 6 by showing that there is a canonical homomorphism of T u ( P ) onto the covariancealgebra C (cid:111) C P P of the canonical product system over P with one-dimensional fibers from [38]. InLemma 6.4 we identify projections in the diagonal that are in the kernel of this map; these leadnaturally to a notion of foundation sets, generalizing those introduced by Sims and Yeend in [39] forquasi-lattice orders. In Corollary 6.6 we arrive at an explicit presentation of the covariance algebrain which the extra relations are a natural augmentation of the presentation of T u ( P ) by the newfoundation sets. This is not a coincidence: the original motivation for our presentation of T u ( P )was the view that the covariance algebra from [38] had to be the universal boundary quotient of anappropriately defined universal Toeplitz algebra. In Proposition 6.9 we show that the presentation of C (cid:111) C P P in terms of generalized foundation sets is maximal in the sense that if one imposes furtherrelations associated to other sets, then the resulting C ∗ -algebra is trivial. The main result of Section 6 MARCELO LACA AND CAMILA F. SEHNEM is Theorem 6.13, where we establish several equivalent descriptions of
C (cid:111) C P P . In particular, weshow that C (cid:111) C P P is isomorphic to C( ∂ Ω P ) (cid:111) G , where ∂ Ω P is the boundary of Ω P as defined in[15, Definition 5.7.8]. At this point we refer to C (cid:111) C P P as the full boundary quotient of T u ( P ). Asimmediate consequences of our analysis of boundary quotients, we derive in Corollary 6.17 a criterionfor when the full boundary quotient is T u ( P ) itself and then we characterize topological freeness ofthe partial action of G on the boundary of the diagonal, Proposition 6.18. Our second main result ofthe section is the characterization of purely infinite simple boundary quotients, Corollary 6.19.In the final four sections we discuss several classes of examples that illustrate the range of applicationof our results. In Section 7 we discuss a specific numerical semigroup studied by Raeburn and Vittadelloin [36], giving a characterization of the left regular C ∗ -algebra in terms of generators and relationsand faithfulness criteria for representations. In Section 8 we adapt first our criteria for topologicalfreeness to the ax + b -monoid of an integral domain, which we then apply in the following section toorders in number fields. Then we give a direct proof that the associated boundary quotient is purelyinfinite simple using Corollary 6.19, a fact that can also be derived from earlier work of Cuntz and Li[13], see also [27]. In Section 9 we discuss the ax + b -monoids of orders in algebraic number fields. Wefirst obtain a uniqueness theorem for their left regular C ∗ -algebras generalizing earlier results for ringsof algebraic integers [14]. We then prove that the independence condition fails for the multiplicativeand ax + b -monoids of all nonmaximal orders, establishing them as a rich source of new examples. Wealso show how our presentation applies in the concrete case of Z [ √− ∗ -algebras [3]. Acknowledgments:
This project was started at the workshop
Cuntz–Pimsner Cross-Pollination at the Lorentz Center in Leiden, and we would like to thank the organizers and the institute forproviding the opportunity and a wonderful environment for creative interaction. We are also verygrateful for the hospitality of the departments of mathematics at Victoria and Florianópolis duringvisits in which part of this research was carried out.2.
Neutral words, quotient sets, and constructible ideals
We recall here the basic facts leading to the constructible right ideals introduced in [28]. Ourapproach and the notation we use are inspired by those of [38]. Let P ⊂ G be a submonoid of agroup G , and for each k ∈ N consider the set of words of length 2 k in P , W ( P ) k := { ( p , p , · · · , p k − , p k ) | p j ∈ P, for j = 1 , , · · · , k } . When the context makes it clear what P is, we write simply W instead of W ( P ). By convention wewrite W = {∅} . Using concatenation of words as composition law, it is easy to see that W k W l = W k + l .We define a generalized (iterated) left quotient map , assigning an element of G to each word α ∈ W k ,by α = ( p , p , · · · , p k − , p k ) ˙ α := p − p · · · p − k − p k . Thus, ˙ W k ⊂ G is the set of products of k left quotients of elements of P . Again by convention,we write ˙ W = { e } , and note that ( αβ )˙ = ˙ α ˙ β . For each α ∈ W k we define the reverse word˜ α := ( p k , p k − , · · · p , p ) ∈ W k . It is easy to see that ˙˜ α = ( ˙ α ) − . We shall say that a word α is neutral if ˙ α = e . Lemma 2.1.
Suppose P is a submonoid of a group G , and assume P generates G . Then { e } ⊆ ˙ W ⊆ · · · ⊆ ˙ W k ⊆ ˙ W k +1 · · · and S k ˙ W k = G .Proof. Since e ∈ P by assumption, for each α ∈ W k the concatenation α ( e, e ) is in W k +1 and satisfies( α ( e, e ))˙ = ˙ α , proving that ˙ W k ⊆ ˙ W k +1 . Suppose now α ∈ W k and β ∈ W l so that ˙ α ∈ ˙ W k and˙ β ∈ ˙ W l . Then ˙ α ˙ β = ( αβ )˙ ∈ ˙ W k + l , and also ˙ α − = ˙˜ α ∈ ˙ W k . This shows that the subset S k ˙ W k of G contains the products and inverses of its elements, hence is a subgroup of G . Since ( e, p )˙ = p forevery p ∈ P we have P ⊆ ˙ W ⊆ S k ˙ W k , and clearly any subgroup of G that contains P must contain S k ˙ W k . (cid:3) OEPLITZ ALGEBRAS OF SEMIGROUPS 5
Lemma 2.2. If ˙ W k = ˙ W k + m for some m ≥ , then ˙ W k = ˙ W k + m for every m ≥ , and ˙ W k is agroup.Proof. By Lemma 2.1 we see immediately that ˙ W k = ˙ W k + j for every j = 1 , , . . . , m . An easyinduction argument shows that the sequence remains constant after k + m too, because ˙ W k + m +1 =˙ W k +1 ˙ W m = ˙ W k ˙ W m = ˙ W k + m = ˙ W k . (cid:3) Remark . Recall the celebrated result of Ore stating that a cancellative monoid P embeds in a group G in such a way that G = P − P if and only if P P − ⊆ P − P , namely if and only if every right quotientcan be written as a left quotient. Such semigroups P are called right reversible. In this case our sequence( ˙ W k ) k ∈ N stabilizes at the first step because ( P − P )( P − P ) = P − ( P P − ) P ⊆ P − ( P − P ) P = P − P .Our choice to start the quotients with P − introduces an asymmetry. Indeed, when P is left reversible,namely, when P − P ⊂ P P − it is clear that G = P P − . But our sequence W k stabilizes at the secondstep because ( P − P )( P − P )( P − P ) = P − ( P P − ) P P − P = P − ( P − P ) P P − P = P − P P − P ,and thus we would write G = P − P P − P . Definition 2.4.
For each word α = ( p , p , . . . , p k ) ∈ W k we define the iterated quotient set of α tobe the set Q ( α ) := { e, p − k p k − , p − k p k − p − k − p k − , . . . , p − k p k − p − k − p k − · · · p − p } , where the last element listed is ˙˜ α = ( ˙ α ) − . We will often drop the ‘iterated’ and simply say‘quotient set’. The apparent reversal of α in defining the iterated quotients may seem unmotivatedat first but is better adapted to the partial actions that will appear later in Section 4. If we needto refer to the analogous iterated quotient set taken from left to right, we will just use Q (˜ α ) := { e, p − p , . . . , p − p · · · p − k − p k } . Here the last element is ˙ α . Lemma 2.5.
Suppose α ∈ W k and β ∈ W l , and, as before, denote by ˜ α the reverse of α and by αβ ∈ W k + l the concatenation of α and β . Then (1) Q (˜ α ) = ˙ αQ ( α ) ; (2) Q ( βα ) = Q ( α ) ∪ ˙˜ αQ ( β ) ; (3) Q (˜ αα ) = Q ( α ) ;in particular, if ˙ α = e , then (4) Q (˜ α ) = Q ( α ) ; and (5) Q ( βα ) = Q ( α ) ∪ Q ( β ) .Proof. Let α = ( p , p , . . . , p k ). Multiplying every element of Q ( α ) = { e, p − k p k − , p − k p k − p − k − p k − , . . . , p − k p k − p − k − p k − · · · p − p , ˙˜ α } on the left by ˙ α = p − p p − p · · · p − k − p k and simplifying each product, gives˙ αQ ( α ) = { ˙ α, p − p · · · p k − , . . . , p − p , e } which is precisely the set Q (˜ α ) listed in reverse. This proves (1).Now let β = ( q , q , . . . , q l ). It is easy to see that the first k iterated quotients in Q ( βα ) areprecisely those of α , with the last one being ˙˜ α , and the following l iterated quotients are those of β multiplied by ˙˜ α on the left. This gives (2).In order to prove (3), set β = ˜ α , then use first (2) and then (1) to compute Q (˜ αα ) = Q ( α ) ∪ ˙˜ αQ (˜ α ) = Q ( α ) ∪ ˙˜ α ˙ αQ ( α )which proves (3) because ˙˜ α ˙ α = e . Assertions (4) and (5) follow immediately from (1) and (2). (cid:3) Notice that neutral words suffice to generate all quotient sets. Indeed, by Lemma 2.5 (3), we maysubstitute α with the neutral word ˜ αα without changing the iterated quotient set.There is a left action of P on words: if p ∈ P and α ∈ W k , then pα := ( pp , pp , . . . pp k ). Thissatisfies ( pα )˙ = ˙ α , ( pα )˜= p ˜ α , and Q ( pα ) = Q ( α ). Following [38], given a finite subset F ⊆ G , we set K F := \ g ∈ F gP. MARCELO LACA AND CAMILA F. SEHNEM
With the usual notation for multiplication of group elements and sets, we have gK F = K gF . We willbe interested in the sets K F arising from taking F to be the iterated quotient set of a word; thisproduces subsets of P because e ∈ Q ( α ) for every α . In order to lighten the notation we will write K ( α ) instead of K Q ( α ) when possible, so that K ( α ) := P ∩ ( p − k p k − ) P ∩ ( p − k p k − p − k − p k − ) P ∩ · · · ∩ ( ˙˜ α ) P. For further reference we list the following properties of K ( α ) in relation to concatenation and reversalof words. Proposition 2.6 (cf. Section 2.1 of [28]) . If α and β are in W , then (1) K (˜ α ) = ˙ αK ( α ) ; (2) K ( βα ) = K ( α ) ∩ ˙˜ αK ( β ) ; (3) K (˜ αα ) = K ( α ) ;in particular, if ˙ α = e , then (4) K (˜ α ) = K ( α ) ; and (5) K ( βα ) = K ( α ) ∩ K ( β ) ;and if, instead, ˙ β = e and p ∈ P , then (6) K (( e, p ) β ( p, e )) = K ( β ( p, e )) = pK ( β )(7) K (( p, e ) β ( e, p )) = K ( β ( e, p )) = P ∩ p − K ( β ) ;Proof. The verification of the first five items is by straightforward application of the correspondingproperties of the iterated quotient sets from Lemma 2.5. We show next how to derive the last two byrepeated applications of item (2). Notice first that K ( β ( p, e )) ⊂ K (( p, e )) = P ∩ e − pP = pP andthat K (( e, p )) = P ∩ p − eP = P . Then K (( e, p ) β ( p, e )) = K ( β ( p, e )) ∩ ( β ( p, e ))˙˜ K (( e, p )) = K ( β ( p, e )) ∩ pP == K ( β ( p, e )) = K ( p, e ) ∩ pK ( β ) = pP ∩ pK ( β ) = pK ( β ) , proving (6). Similarly, K (( p, e ) β ( e, p )) = K ( β ( e, p )) ∩ ( β ( e, p ))˙˜ K (( p, e ) = K ( β ( e, p )) ∩ p − ( pP )= K ( β ( e, p )) = K ( e, p ) ∩ p − K ( β ) = P ∩ p − K ( β ) , proving (7). (cid:3) We record here for later use the following easy consequence of the proposition.
Corollary 2.7.
Suppose α and β are words with β neutral, then K ( αβ ˜ α ) = K (˜ α ) ∩ ˙ αK ( β ) . The map α K ( α ) is far from injective, but it still provides a convenient way to parametrizeconstructible right ideals of P in terms of neutral words. Thus for every monoid P we let J ( P ) := { K ( α ) | α ∈ W ( P ) } , dropping the reference to P and writing simply J when there is no risk of confusion. We see nextthat J is equal to the set of all constructible right ideals as introduced in Section 2.1 of [28], modulohaving to add the empty set, which is not of the form K ( α ) when P is left reversible. The propertiesof K ( α ) listed in Proposition 2.6 allow us to give direct proofs, and to show that J is automaticallyclosed under finite intersections, a fact established in Section 3 of [28]. Proposition 2.8 (cf. [28]) . The collection J = { K ( α ) | α ∈ W} of subsets of P satisfies (1) J = { K ( α ) | α ∈ W , ˙ α = e } = { K (˜ αα ) | α ∈ W} ; (2) P ∈ J ; (3) K ( α ) ∩ K ( β ) ∈ J for every α, β ∈ W ; (4) K ( α ) p ⊂ K ( α ) for every α ∈ W and p ∈ P ( K ( α ) is a right ideal in P ); (5) pK ( α ) ∈ J for every α ∈ W and p ∈ P ; and (6) P ∩ p − K ( α ) ∈ J for every α and p ∈ P .Moreover, J is the smallest collection of subsets of P that contains P and is invariant under the leftactions of P and of P − as in parts (5) and (6) . OEPLITZ ALGEBRAS OF SEMIGROUPS 7
Proof.
Since (˜ αα )˙ = ˙˜ α ˙ α = e it is obvious that J ⊇ { K ( α ) | α ∈ W , ˙ α = e } ⊇ { K (˜ αα ) | α ∈ W} .In order to show that J ⊆ { K (˜ αα ) | α ∈ W} , suppose α ∈ W k and recall that K ( α ) = K (˜ αα ) byProposition 2.6(3). This proves part (1).Part (2) is obvious because P = K (( p, p )) for each p ∈ P , and part (3) follows easily fromProposition 2.6(5) since part (1) allows us to work with neutral words.Part (4) folows from the observation that the factor p is absorbed by P on the right in each termof the intersection that defines K F .Parts (5) and (6) now follow directly from Proposition 2.6(6) and (7), respectively.Up to this point we have verified that J is a collection of right ideals in P that contains P , is closedunder intersections, and is invariant under the left actions by P and P − given in parts (5) and (6).To complete the proof we need to show that J is contained in any collection of subsets of P thatcontains P and is invariant under the left actions by P and P − given in parts (5) and (6). This isdone by an easy induction argument based upon rewriting K ( α ) as K ( α ) = P ∩ p − k ( p k − ( P ∩ p − k − ( p k − ( · · · ( P ∩ p − ( p P )) · · · )))) . (cid:3) We close this section with a brief discussion of what happens when P embeds into different groups;see the argument around equation (37) in Section 3 in [28] and also [38, Lemma 3.9]. One issue is that,in principle, the subset K ( α ) of P could depend on the specific embedding of P in a group. That thisis not the case is implicit in Proposition 2.8, because the original definition of constructible idealsin [28] does not use a group at all. Nevertheless, we wish to give a direct proof of it in the presentcontext of submonoids of groups.Recall first that when P embeds as a submonoid of any group, then there exists a universal group G ( P ) generated by a canonical copy of P . Every embedding P , → G extends to a unique grouphomomorphism γ : G ( P ) → G . Lemma 2.9.
Suppose P is a submonoid of a group G . Let γ : G ( P ) → G be the unique grouphomomorphism extending P , → G . For each α = ( p , . . . , p k ) ∈ W k we have (1) K ( α ) = ∅ whenever γ ( ˙ α ) = e in G and ˙ α = e in G ( P ) ; (2) γK ( α ) = K ( γ ( α )) := P ∩ γ ( p − k p k − ) P ∩ . . . ∩ γ ( p − k p k − p − k − · · · p − p ) P , where we mayidentify P and γ ( P ) .Proof. For the first assertion, suppose that γ ( ˙ α ) = e in G and K ( α ) = ∅ , and let s ∈ K ( α ) = P ∩ p − k p k − P ∩ p − k p k − p − k − p k − P ∩ . . . ∩ p − k p k − p − k − · · · p − p P. In particular, s ∈ ˙˜ αP , so there exists a unique t ∈ P such that ˙˜ αt = s in G ( P ). Applying γ on bothsides of the equality and using γ ( ˙˜ α ) = γ ( ˙ α ) − = e , we obtain γ ( t ) = γ ( ˙˜ α ) γ ( t ) = γ (cid:0) ˙˜ αt (cid:1) = γ ( s ) . This implies that s = t because γ is injective on P , which forces ˙ α = ˙˜ α − = e in G ( P ).For the second assertion, notice first that γ ( K ( α )) ⊆ K ( γ ( α )) is clear. Let r ∈ P be such that γ ( r ) ∈ K ( γ ( α )). There is a unique s ∈ P satisfying γ ( r ) = γ ( p − k p k − ) γ ( s ) = γ ( p − k p k − s ). Thisentails γ ( p k r ) = γ ( p k − s ) and hence r = p − k p k − s in G ( P ) since γ is injective on P . But γ ( r )also lies in γ ( p − k p k − p − k − p k − ) P . So there is a unique s ∈ P so that γ ( r ) = γ ( p − k p k − p − k − p k − ) γ ( s ) . Then γ ( s ) = γ ( p − k − p k − s ) and we conclude as above that s = p − k − p k − s in G ( P ). Thus r = p − k p k − s = p − k p k − p − k − p k − s ∈ p − k p k − p − k − p k − P. Continuing with this procedure, we deduce that r ∈ K ( α ). Therefore K ( γ ( α )) = K ( α ) as asserted. (cid:3) Toeplitz C ∗ -algebras for submonoids of groups For each monoid P that embeds in a group we introduce here a new universal Toeplitz C ∗ -algebra T u ( P ), defined via a conceptually simple set of relations. Our choice of relations implies that the linearcombinations of projections associated to constructible ideals always behave exactly as they do in theC ∗ -algebra T λ ( P ) generated by the left regular representation. This is particularly relevant when theunderlying monoid does not satisfy independence. When we remove some of the relations defining MARCELO LACA AND CAMILA F. SEHNEM T u ( P ) we obtain a new presentation of the C ∗ -algebra C ∗ s ( P ) from [28, Definition 3.2]. Hence T u ( P )is canonically a quotient of C ∗ s ( P ), and we will see that the two coincide if and only if P satisfiesindependence.3.1. Presentation of a universal Toeplitz C ∗ -algebra. The following notation will be very usefulthroughout.
Notation 3.1.
When w : P → B is a map from P to a C ∗ -algebra B and α = ( p , p , . . . , p k − , p k )is a word in W , we will denote by ˙ w α the product˙ w α := w ∗ p w p . . . w ∗ p k − w p k ∈ B. It is easy to see that ˙ w α ˙ w β = ˙ w αβ and ( ˙ w α ) ∗ = ˙ w ˜ α . Recall that when α ∈ W and ˙ α = e we say α is aneutral word.Next we draw some consequences from assuming that the alternating products factor through theconstructible ideals. Proposition 3.2.
Let P be a submonoid of a group G . Suppose w : P → B is a map from P to a C ∗ -algebra B such that (T1) w e = 1 ; (T2) ˙ w α = 0 if K ( α ) = ∅ with ˙ α = e ; and (T3) ˙ w α = ˙ w β if α and β are neutral words in W such that K ( α ) = K ( β ) .Then w also has the following properties: (1) w p is an isometry for all p ∈ P ; (2) w p w q = w pq for all p, q ∈ P ; (3) the set { ˙ w α | α ∈ W , ˙ α = e } does not depend on the embedding P , → G and is a commutingfamily of projections that is closed under multiplication and contains the identity; and (4) if F is a finite set of neutral words in W with S β ∈ F K ( β ) = K ( α ) for some neutral word α ∈ W such that K ( α ) ∈ { K ( β ) | β ∈ F } , then ˙ w α = X ∅6 = A ⊂ F ( − | A | +1 Y β ∈ A ˙ w β . (3.3) Proof.
We remark that the key ideas in the proof appear in [28], especially equations (32) and (33).Property (1) holds because K (( p, p )) = P ∩ p − pP = P = K (( e, e )), so w ∗ p w p = w ∗ e w e = 1 byconditions (T3) and (T1). In order to prove property (2), notice that( w p w q − w pq ) ∗ ( w p w q − w pq ) = w ∗ q w ∗ p w p w q − w ∗ q w ∗ p w pq − w ∗ pq w p w q + w ∗ pq w pq = 2 − w ∗ q w ∗ p w pq − w ∗ pq w p w q . The product w ∗ q w ∗ p w pq is associated to the word α = ( q, e, p, pq ), which satisfies ˙ α = e and Q ( α ) = { e, ( pq ) − p, ( pq ) − pe − q ) } = { e, q − } , so that K (( q, e, p, pq )) = P ∩ q − P = P = K ( e, e ). Hence w ∗ q w ∗ p w pq = w ∗ e w e = 1 by conditions (T3) and (T1); similarly w ∗ pq w p w q = 1, using ˜ α . It follows that w p w q − w pq = 0, proving property (2).The set { ˙ w α | α ∈ W , ˙ α = e } does not depend on the embedding P , → G because of Lemma 2.9(1)and condition (T2). It is also obviously closed under multiplication because ˙ w α ˙ w β = ˙ w αβ and αβ isneutral whenever α and β are, and it certainly contains 1 = ˙ w ( e,e ) . In order to prove the remainder ofproperty (3), let α and β be neutral words in W and recall that K ( αβ ) = K ( α ) ∩ K ( β ) = K ( βα ) byProposition 2.6(5). Then condition (T3) applied to the neutral words αβ and βα yields˙ w α ˙ w β = ˙ w αβ = ˙ w βα = ˙ w β ˙ w α . To see that ˙ w α is a projection for each neutral word α , recall that ˙ w ∗ α = ˙ w ˜ α and that K (˜ αα ) = K ( α )by Proposition 2.6(3). Hence condition (T3) yields˙ w ∗ α ˙ w α = ˙ w ˜ α ˙ w α = ˙ w ˜ αα = ˙ w α , which implies that each ˙ w α is a projection, completing the proof of property (3).Assume now that F is a finite set of neutral words in W with S β ∈ F K ( β ) = K ( α ) for some neutralword α ∈ W such that K ( α ) ∈ { K ( β ) | β ∈ F } . Then ˙ w α ˙ w β = ˙ w αβ = ˙ w β = ˙ w β ˙ w α for every β ∈ F OEPLITZ ALGEBRAS OF SEMIGROUPS 9 because K ( αβ ) = K ( α ) ∩ K ( β ) = K ( β ). Since ˙ w α − ˙ w β = 0 for some β ∈ F by condition (T3), aneasy expansion yields 0 = Y β ∈ F ( ˙ w α − ˙ w β ) = ˙ w α + X ∅6 = A ⊂ F ( − | A | Y β ∈ A ˙ w β (3.4)where the products over all subsets A of F can be taken in any order because of property (3). Thisproves equation (3.3). (cid:3) Remark . Proposition 3.2 highlights the significance of Li’s constructible right ideals. If we view(T1) and (T2) as ‘calibrations’, we are essentially just asking that the map α ˙ w α factor through α K ( α ), and yet we obtain a very strong algebraic structure for the collection { w p | p ∈ P } as a consequence. We point out that the order in which the projections are multiplied or thewords concatenated does not affect the formulas in (3.4) because of part (3) and condition (T3) ofProposition 3.2. Definition 3.6.
Let P be a submonoid of a group G . We define the universal Toeplitz algebra of P ,denoted by T u ( P ), to be the universal C ∗ -algebra with generators { t p : p ∈ P } subject to the relations(T1) t e = 1;(T2) ˙ t α = 0 if K ( α ) = ∅ with ˙ α = e ;(T3) ˙ t α = ˙ t β if α and β are neutral words such that K ( α ) = K ( β );(T4) Q β ∈ F ( ˙ t α − ˙ t β ) = 0 if F is a finite set of neutral words such that K ( α ) = S β ∈ F K ( β ) forsome neutral word α . Remark . We would like to make a few comments concerning Definition 3.6.(1) Relation (T3) is simply the special case of (T4) for | F | = 1. Further, with the convention thatthe union over the empty set is empty, if we interpret an empty product in (T4) as beingequal to ˙ t α then we can also derive (T2) from (T4). We have chosen to include (T2) and (T3)explicitly here mainly for clarity and to facilitate the comparison with further constructions.It seems also plausible that the verification of specific cases would probably have to go through(T2) and (T3) anyway.(2) Notice that if P is left reversible then K ( α ) is never empty, making (T2) vacuous. So (T2)only applies when P is not left reversible.(3) Further insight into relation (T4) is gained from noticing its relation to independence. Recallfrom [28] that the semigroup P is said to satisfy the independence condition if the union S β ∈ F K ( β ) of constructible ideals in P is a constructible ideal itself only when S β ∈ F K ( β ) = K ( α ) for some neutral word α ∈ W such that K ( α ) ∈ { K ( β ) | β ∈ F } , equivalently, whenthe characteristic functions of constructible right ideals are linearly independent. This, inturn, implies that the product in (T4) vanishes for every w satisfying (T3), because one of thefactors is zero. Thus, in particular, semigroups that satisfy independence and (T3) also satisfy(T4) automatically. The full implication of this observation is spelled out in Corollary 3.23below.(4) It is also helpful to notice that in the particular case when the ideals K ( β ) happen to bemutually disjoint, the only nonzero terms in equation (3.4) are those for which the subset A of F is a singleton. In this case (T4) simply reduces to the familiar relation ˙ t α = L β ∈ F ˙ t β involving a sum of mutually orthogonal projections.We list next several equivalent formulations of relation (T4) that are helpful to understand betterits meaning and motivation. In particular, relation (T4) is a stronger version of the property stated inProposition 3.2(4) because it applies whenever the union of a finite collection of constructible ideals isa constructible ideal, regardless of whether this ideal is a member of the collection or not. Lemma 3.8.
Suppose B is a unital C ∗ -algebra and t : P → B is a map that satisfies relations (T1)–(T3) . The following are equivalent: (1) relation (T4) ; (2) relation (T4) restricted to the special cases where independence fails, that is, for neutral words α ∈ W and finite sets F ⊂ W such that K ( α ) = S β ∈ F K ( β ) and K ( α ) = K ( β ) for all β ∈ F ; (3) the expanded version of relation (T4) , ˙ t α = X ∅6 = A ⊂ F ( − | A | +1 Y β ∈ A ˙ t β (3.9) if F ⊂ W is a finite set of neutral words and α ∈ W is a neutral word that satisfies K ( α ) = S β ∈ F K ( β ) , equivalently, restricted to special cases where independence fails. (4) the expanded version of relation (T4) written in terms of concatenation of words: ˙ t α = X ∅6 = A ⊂ F ( − | A | +1 ˙ t Q β ∈ A β (3.10) f or F and α as in (T4) , equivalently, restricted to the special cases where independence fails.Proof. That (1) = ⇒ (2) is immediate. To see that (2) = ⇒ (1), we only need to notice that in thecases that are left out of (2), that is to say, for each neutral word α and finite set F of neutral wordssuch that K ( α ) = S β ∈ F K ( β ) = K ( β ) for some β ∈ F , the product in (T4) clearly vanishes becauseit includes the factor ( ˙ t α − ˙ t β ), which is trivial by (T3). The remaining equivalences are easy to seefrom the expansion in equation (3.4). (cid:3) The left regular representation.
The (reduced) Toeplitz C ∗ -algebra T λ ( P ) is, by definition,the C ∗ -algebra generated by the image of the left regular representation L of P on ‘ ( P ), which isgiven by L p δ q = δ pq on the standard orthonormal basis elements. The range projection L p L ∗ p of thegenerating isometry L p is the multiplication operator by the characteristic function pP of the set pP . Since the representation of ‘ ∞ ( P ) as multiplication operators on ‘ ( P ) is isometric, there is noharm in confusing these multiplication operators with the functions they represent. We will use thefollowing basic observation from [28]. Lemma 3.11. [28, Definition 2.12 and Lemma 3.1]
Let P be a submonoid of a group G , and let K ( α ) ∈ ‘ ∞ ( P ) denote the characteristic function of K ( α ) . Then ˙ L α = K ( α ) for every neutral word α and D r := span { K ( α ) ∈ T λ ( P ) | ˙ α = e } is a C ∗ -subalgebra of ‘ ∞ ( P ) , which we call the reduceddiagonal . There is also a full diagonal subalgebra D u := span { ˙ t α ∈ T u ( P ) | ˙ α = e } at the level of the universalToeplitz algebra. This does not depend on the embedding P , → G and is a commutative unitalC ∗ -subalgebra of T u ( P ) by Proposition 3.2(3). Proposition 3.12.
Let P be a submonoid of a group G , and denote by { t p | p ∈ P } the set ofcanonical generators for T u ( P ) . Then the map t p L p extends to a surjective ∗ -homomorphism λ + : T u ( P ) → T λ ( P ) .Proof. Since ˙ L α = K ( α ) it is easy to see that relations (T2) and (T3) are satisfied by L , and relation(T1) is obvious for L . In order to prove that L also satisfies (T4), let F be a finite set of neutral wordssuch that the union of their corresponding right ideals is a constructible right ideal, that is, such that S β ∈ F K ( β ) = K ( α ) for some neutral word α . Then ∅ = T β ∈ F ( K ( α ) \ K ( β )), and hence0 = Y β ∈ F ( K ( α ) − K ( β ) ) , which shows that L satisfies relation (T4). The resulting canonical homomorphism λ + : T u ( P )
7→ T λ ( P )is surjective because the image contains the generating isometries L p for p ∈ P . (cid:3) It is clear from Proposition 3.12 that the restriction λ + (cid:22) D u maps D u onto D r and we would liketo show next that this restriction is in fact an isomorphism. We will obtain this as a corollary ofour characterization of the representations of T u ( P ) that are faithful on D u . The following fact isprobably well known, but we have not been able to find a general reference. We include the concretestatement here for completeness and to set the notation. Lemma 3.13 (cf. Lemma 1.4 of [25]) . Suppose F is a finite set of mutually commuting projectionsin a unital C ∗ -algebra and let λ X ∈ C for each X ∈ F . For every subset A of F define Q A := Y X ∈ A X Y X ∈ F \ A (1 − X ) , OEPLITZ ALGEBRAS OF SEMIGROUPS 11 which includes the extreme cases Q ∅ = Q F (1 − X ) and Q F = Q F X .Then P A ⊂ F Q A is a decomposition of the identity into mutually orthogonal projections, X X ∈ F λ X X = X ∅6 = A ⊂ F ( P X ∈ A λ X ) Q A and (cid:13)(cid:13)(cid:13) X X ∈ F λ X X (cid:13)(cid:13)(cid:13) = max (cid:8)(cid:12)(cid:12)P X ∈ A λ X (cid:12)(cid:12) | ∅ 6 = A ⊂ F, Q A = 0 (cid:9) Proof.
The proof follows, mutatis mutandis, from the proof of [25, Lemma 1.4], keeping the product Q X ∈ A X instead of substituting it by X σA . (cid:3) Definition 3.14.
A map w : P → B satisfying (T1)–(T4) of Definition 3.6, is said to be jointly proper if Y α ∈ F (1 − ˙ w α ) = 0 (3.15)for every finite collection F of neutral words such that K ( α ) is a proper constructible right ideal foreach α ∈ F . By extension, we will also say that a unital ∗ -homomorphism ρ of T u ( P ) into a C ∗ -algebra B is jointly proper if Y α ∈ F (1 − ρ ( ˙ t α )) = 0for every finite collection F as above. Because of (T3), it suffices to verify that (3.15) holds for words α in a collection large enough to generate all constructible ideals. Example 3.16.
Recall that an isometry V is called proper if V V ∗ = 1. So Coburn’s theorem [7] canbe rephrased by saying that the C ∗ -algebra generated by a proper isometry is canonically unique. Inorder to see why the stronger condition of joint properness might be necessary for a uniqueness result,let S be the unilateral shift on ‘ ( N ) and consider the two isometries V := S ⊕ W := 1 ⊕ S defined on ‘ ( N ) ⊕ ‘ ( N ). Then V and W are proper isometries that ∗ -commute. Thus the isometricrepresentation T : N → B ( ‘ ( N ) ⊕ ‘ ( N )) defined by T (1 , := V and T (0 , := W is covariant inthe sense of Nica, equivalently, satisfies (T3), but is not jointly proper. Denoting as usual by c theC ∗ -algebra of convergent sequences, we see that the diagonal in C ∗ ( T N ) is isomorphic to c ⊕ c whilethe diagonal in T u ( N ) is c ⊗ c .Other familiar examples of proper isometries that are not jointly proper arise from isometricrepresentations V : F + n → H with n < ∞ . In this case Nica-covariance means that the generatingisometries V j have mutually orthogonal ranges. If P j V j V ∗ j = 1, or, equivalently, if Q j (1 − V j V ∗ j ) = 0,then the resulting representation of TO n is not jointly proper. In fact, such a representation factorsthrough O n , whose diagonal is isomorphic to the continuous functions on infinite path space, whilethe diagonal in TO n itself is isomorphic to the continuous functions on the space of finite and infinitepaths. Remark . If R and S are constructible ideals with R ⊂ S , then R ≤ S and hence − S ≤ − R .Hence the verification of whether a representation is jointly proper can sometimes be reduced to finitesubsets of larger ideals, which are generally associated to shorter words.Definition 3.14 generalizes to constructible right ideals the condition used in [25, Proposition 2.3] tocharacterize representations that are faithful on the diagonal algebra when P is the positive cone in aquasi-lattice ordered group. So it should not be too much of a surprise that such a characterization ispossible for general submonoids of groups too. Proposition 3.18.
Suppose P is a submonoid of the group G . A ∗ -homomorphism ρ : T u ( P ) → B isfaithful on the full diagonal subalgebra D u if and only if it is jointly proper.Proof. Evaluation at e ∈ P shows that Q α ∈ F ( − K ( α ) ) = 0 in D r ; so the representation λ + arisingfrom the map L : P → T λ ( P ) is jointly proper. Hence the same is true for the identity representationof T u ( P ) arising from the universal map t : P → T u ( P ), so the condition is necessary.For sufficiency, suppose ρ is jointly proper and let a = 0 be an element of the form a := X α ∈ F λ α ˙ t α ∈ D u , where F ⊂ W is a finite collection of neutral words. We will show that ρ ( a ) = 0 in B . Since a is alinear combination of commuting projections, Lemma 3.13 gives a nonempty subset A of F such thatthe projection Q A := Y α ∈ A ˙ t α Y β ∈ F \ A ( − ˙ t β )is nonzero and satisfies k Q A a k = k a k = | P α ∈ A λ α | .If the set T α ∈ A K ( α ) \ S β ∈ F \ A K ( β ) were empty, then we would have \ α ∈ A K ( α ) = [ β ∈ F \ A (cid:16) K ( β ) ∩ \ α ∈ A K ( α ) (cid:17) , and equation (3.10) would realize Q A ˙ t α = ˙ t T α ∈ A K ( α ) as a linear combination of subprojections ofthe projections ˙ t β for β ∈ F \ A . This would force Q A = 0, contradicting the choice of Q A . So thereexists p ∈ T α ∈ A K ( α ) \ S β ∈ F \ A K ( β ), and we may define a projection by Q := t p t ∗ p Q A t p t ∗ p = t p Y β ∈ F \ A ( − ( t ∗ p ˙ t β t p )) t ∗ p = t p (cid:16) Y β ∈ F \ A ( − ˙ t ( p,e ) β ( e,p ) ) (cid:17) t ∗ p . Clearly Q is a subprojection of Q A and Q = 0 because through the left regular representation we have λ + ( Q )( p ) = Q β ∈ F \ A (cid:0) − K ( β ) ∩ pP (cid:1) ( p ) = 1. Hence aQ = (cid:0) P α ∈ A λ α (cid:1) Q , and thus k aQ k = k a k = | P α ∈ A λ α | .Passing to the representation ρ , we get ρ ( Q ) = ρ ( t p ) (cid:16) Y β ∈ F \ A ( − ρ ( ˙ t ( p,e ) β ( e,p ) ) (cid:17) ρ ( t p ) ∗ , where the middle factor is of the type that appears in the joint properness condition (3.15). FromProposition 2.6(6) we know that K (( p, e ) β ( e, p )) = P ∩ p − K ( β )and since p / ∈ K ( β ) by construction, we conclude that the ideal K (( p, e ) β ( e, p )) is proper for each β ∈ F \ A . Thus, our assumption (3.15), together with the fact that ρ ( t p ) is an isometry, imply that ρ ( Q ) = 0. But since ρ ( a ) ρ ( Q ) = ρ ( aQ ) = (cid:0) P α ∈ A λ α (cid:1) ρ ( Q ) and (cid:12)(cid:12)(cid:0) P α ∈ A λ α (cid:1)(cid:12)(cid:12) = k a k 6 = 0, we musthave ρ ( a ) = 0 as wanted.To establish the proposition, observe that for each ∩ -closed finite subcollection C of constructibleideals, A ( C ) := span { ˙ t α | α ∈ W , K ( α ) ∈ C , ˙ α = e } is a finite dimensional C ∗ -subalgebra of D u spanned by a finite set of projections arising fromconstructible right ideals. To see this, for each S ∈ C , choose a neutral word α S with K ( α S ) = S .Then it follows from relation (T3) of Definition 3.6 that A ( C ) = span { ˙ t α S | S ∈ C} as claimed. Nownotice that D u = lim C A ( C ), with the limit taken over the ∩ -closed finite subcollections of constructibleideals directed by inclusion. From the above we deduce that the representation ρ is faithful on each A ( C ), and so it is also faithful on D u . (cid:3) Corollary 3.19.
The restriction of λ + to the full diagonal D u gives a canonical isomorphism D u ∼ = D r .A representation of T λ ( P ) is faithful on D r if and only if it is jointly proper.Proof. The first line of the proof of Proposition 3.18 verifies that λ + is jointly proper, so the firstassertion follows by Proposition 3.18. Once we know that D u is canonically isomorphic to D r , thesecond assertion follows directly also from Proposition 3.18. (cid:3) Let E r : T λ ( P ) → D r be the restriction of the canonical diagonal conditional expectation from B ( ‘ ( P )) onto ‘ ∞ ( P ). It is determined by h E r ( b ) δ p | δ q i = ( h bδ p | δ p i if p = q, b ∈ T λ ( P ) and is a faithful conditional expectation. OEPLITZ ALGEBRAS OF SEMIGROUPS 13
Corollary 3.20.
Let P be a submonoid of a group. Then E u := ( λ + (cid:22) D u ) − ◦ E r ◦ λ + is a conditionalexpectation from T u ( P ) onto D u that vanishes on the subspace B g := span { ˙ t α | α ∈ W , ˙ α = g } ⊂ T u ( P ) whenever g = e . Moreover, ker λ + = { b ∈ T u ( P ) | E u ( b ∗ b ) = 0 } . Proof.
The first assertion is clearly true. For the last one, notice that E u ( b ∗ b ) = 0 if and onlyif λ + ( b ∗ b ) = 0 because E r is faithful. Now the result follows because λ + ( b ∗ b ) = 0 if and only if λ + ( b ) = 0. (cid:3) Li’s semigroup C ∗ -algebra. Next we wish to compare T u ( P ) with the full semigroup C ∗ -algebraof a submonoid of a group, denoted by C ∗ s ( P ) in [28]. We begin by recalling that definition. Definition 3.21. (cf.[28, Definition 3.2]) Let P be a submonoid of a group G . The semigroup C ∗ -algebra of P , denoted by C ∗ s ( P ), is the universal C ∗ -algebra generated by a family of isometries { v p | p ∈ P } and projections { e S | S ∈ J ∪ {∅}} such that(i) v p v q = v pq whenever p, q ∈ P ;(ii) e ∅ = 0;(iii) ˙ v α = e S whenever S ∈ J and α ∈ W satisfy ˙ α = e and K ( α ) = S .The family { e S | S ∈ J ∪ {∅}} replicates the semilattice structure of the corresponding family ofsubsets of P . This property is included as part of [28, Definition 2.2] for general semigroups, but forsubmonoids of groups it is a consequence of the relations listed above. Perhaps surprisingly, one canachieve the same effect by requiring a lot less, as shown in the next proposition. Proposition 3.22.
Let P be a submonoid of a group and let C ∗ ( P ) be the universal C ∗ -algebragenerated by a family of elements { w p | p ∈ P } subject to the relations (T1)–(T3) of Proposition .Then C ∗ ( P ) is canonically isomorphic to C ∗ s ( P ) , i.e. there is an isomorphism that maps w p to v p .Proof. We know from Proposition 3.2 that any family { w p | p ∈ P } satisfying (T1) and (T3) consistsof a semigroup of isometries, so there is no concern about existence of the universal object C ∗ ( P ).Denote by v p and e S the generating isometries and projections of C ∗ s ( P ). Since 1 = v ∗ e v e = v ∗ e v e v e = v e by the first relation in Definition 3.21 we see that (T1) holds in C ∗ s ( P ). If α ∈ W is a neutral word with K ( α ) = ∅ , then ˙ v α = e K ( α ) = 0, from which it follows that relation (T2) in the presentation of C ∗ ( P )holds in C ∗ s ( P ). A similar argument shows that (T3) also holds. Hence there is a ∗ -homomorphismC ∗ ( P ) → C ∗ s ( P ) that maps the element w p ∈ C ∗ ( P ) to the corresponding isometry v p ∈ C ∗ s ( P ). Thishomomorphism is surjective by Corollary 2.10 and Lemma 3.3 of [28].To obtain the inverse map, recall that the family of constructible right ideals of P is given by (cid:8) K ( α ) | α ∈ W , ˙ α = e } by Proposition 2.8. So for each S ∈ J we may choose α S = ( p , p , . . . , p k ) with ˙ α S = e such that S = K ( α S ). Then ˙ w α S = w ∗ p w p · · · w p k − w ∗ p k − w p k is a projection by Proposition 3.2(3). Because of relation (T3), the projection ˙ w α S in C ∗ ( P ) doesnot depend on the choice of the neutral word α S representing the right ideal S . It follows from thedefinition of the ˙ w α S ’s and Proposition 3.2 that the family of projections { ˙ w α S | S ∈ J } in C ∗ ( P ),together with the family of isometries { w p | p ∈ P } , satisfy the conditions of Definition 3.21. Hencethe maps v p w p , e S ˙ w α S and e ∅ ∗ s ( P ) → C ∗ ( P ), which isobviously the inverse of the one determined above by w p v p . (cid:3) Corollary 3.23.
Suppose P is a submonoid of a group. Then the map v p t p extends to a canonicalsurjective ∗ -homomorphism λ : C ∗ s ( P ) → T u ( P ) , which is an isomorphism if and only if P satisfiesindependence. Proof.
That v p t p extends to a surjective ∗ -homomorphism λ : C ∗ s ( P ) → T u ( P ) follows fromProposition 3.22 since the relations defining T u ( P ) include those defining C ∗ ( P ). Suppose that λ isan isomorphism. Then the restriction of λ to the diagonal subalgebra D s = span { e S | S ∈ J } is faithful. Hence ( λ + ◦ λ ) (cid:22) D s : D s → D r is an isomorphism by Proposition 3.12. But λ + ◦ λ is precisely the left regular representation λ : C ∗ s ( P ) → T λ ( P ). So P satisfies independence by[28, Corollary 2.27].From Proposition 3.22 we know that (T1)–(T3) hold in C ∗ s ( P ) for general P . When P satisfiesindependence, condition (2) of Lemma 3.8 is void, hence automatically satisfied. Thus Lemma 3.8implies that (T4) holds in C ∗ s ( P ). So by the universal property of T u ( P ) the map that sends t p to v p for each p ∈ P extends to a homomorphism T u ( P ) → C ∗ s ( P ), which is the inverse of λ . (cid:3) Another semigroup C ∗ -algebra, denoted C ∗ s ( ∪ ) ( P ), is mentioned in passing, right before Subsection3.1 of [28]. Its presentation is not given explicitly, but the notation makes it clear that C ∗ s ( ∪ ) ( P ) ismeant to be the quotient of the C ∗ -algebra C ∗ ( ∪ ) ( P ) from [28, Definition 2.4] by the ideal generatedby the relation III G in [28, Definition 3.2]. We can deduce from [28, Lemma 3.3] that C ∗ s ( ∪ ) ( P )coincides with the C ∗ -algebra with the same defining relations as C ∗ s ( P ) but with extra generators inthe presentation. These come from the projections { e X | X ∈ J ( ∪ ) } , where J ( ∪ ) = (cid:8) [ R ∈C R | ∅ 6 = C ⊂ J , |C| < ∞ (cid:9) . Our (T4) implies relation II ( ∪ ) (iv) in [28, Definition 2.4] in the special case when X and Y areconstructible ideals whose union is also a constructible ideal. By [28, Proposition 2.24], it is alsoreasonable to expect that C ∗ s ( ∪ ) ( P ) should have the same property as our T u ( P ), of being a quotientof C ∗ s ( P ) that is isomorphic to it when independence holds. Thus, it is natural to wonder whetherC ∗ s ( ∪ ) ( P ) and T u ( P ) are one and the same. On the path to decide this question we establish nextsome equivalent forms of relation (T4). Lemma 3.24.
Suppose that w : P → B is a map of P into a C ∗ -algebra B that satisfies relations (T1) and (T2) . The following are equivalent: (1) w : P → B satisfies relation (T4) ; (2) X β ∈ F λ β ˙ w β = 0 whenever F is a finite set of neutral words in W and the linear combination X β ∈ F λ β K ( β ) vanishes in D r ; (3) X ∅6 = B ⊂ F ( − | B | Y β ∈ B ˙ w β = X ∅6 = C ⊂ H ( − | C | Y γ ∈ C ˙ w γ whenever F and H are finite sets of neutralwords in W such that S β ∈ F K ( β ) = S γ ∈ H K ( γ ) ; and (4) Y β ∈ F (1 − ˙ w β ) = Y γ ∈ H (1 − ˙ w γ ) whenever F and H are finite sets of neutral words in W suchthat S β ∈ F K ( β ) = S γ ∈ H K ( γ ) .Proof. Notice first that each one of the conditions (1)–(4) imply that (T3) holds. We will prove(3) = ⇒ (1) = ⇒ (2) = ⇒ (3) ⇐⇒ (4). Setting F = { α } in (3) gives (1), and a standard application ofthe expansion in (3.4) shows that (3) and (4) are equivalent. This takes care of the first and the last(double) implications.Assume now (2) holds and S β ∈ F K ( β ) = S γ ∈ H K ( γ ). Then X ∅6 = B ⊂ F ( − | B | +1 K ( Q i ∈ B β ) = X ∅6 = C ⊂ H ( − | C | +1 K ( Q k ∈ C γ ) in D r . So by condition (2) the corresponding equality will hold in B with ˙ w K ( Q B β ) in place of K ( Q B β ) and ˙ w K ( Q C γ ) in place of K ( Q C γ ) . This gives (3) and establishes (2) = ⇒ (3). OEPLITZ ALGEBRAS OF SEMIGROUPS 15
In order to see that (1) = ⇒ (2), assume that w satisfies (T4), and let ρ w be the resultingrepresentation of T u ( P ). If P β ∈ F λ β K ( β ) = 0 in D r , then P β ∈ F λ β ˙ t β = 0 in D u because ofCorollary 3.19, so necessarily X β ∈ F λ β ˙ w β = ρ w (cid:0) X β ∈ F λ β ˙ t β (cid:1) = 0 . (cid:3) Proposition 3.25.
The following C ∗ -algebras are canonically isomorphic: (1) the universal Toeplitz C ∗ -algebra T u ( P ) ; (2) the C ∗ -algebra with presentation (T1)–(T3) , and (T4lc): P α ∈ F λ α ˙ t α = 0 whenever P α ∈ F λ α K ( α ) vanishes in ‘ ∞ ( P ) ; (3) the C ∗ -algebra C ∗ s ( ∪ ) ( P ) from [28, Section 3] defined as the quotient of C ∗ ( ∪ ) ( P ) by the ideal h ˙ v α − e K ( α ) | α ∈ W , ˙ α = e i from relation III G in [28, Definition 3.2] .Proof. That T u ( P ) is canonically isomorphic to the C ∗ -algebra with the presentation (T1)–(T3) and(T4lc) given in item (2) follows from the equivalence of conditions (1) and (2) in Lemma 3.24. FromProposition 3.22 we know that (T1)–(T3) hold for the isometries in C ∗ s ( ∪ ) ( P ). By [28, Corollary 2.22] themap D ( ∪ ) → D r that sends a generating projection to the characteristic function of the correspondingright ideal is an isomorphism. So condition (2) from Lemma 3.24 also holds in C ∗ s ( ∪ ) ( P ). Thus, themap t p v p extends to a canonical ∗ -homomorphism of T u ( P ) to C ∗ s ( ∪ ) ( P ).For the inverse of the above ∗ -homomorphism, let { t p | p ∈ P } be the canonical generating elementsof T u ( P ). If X = S β ∈ F K ( β ) ∈ J ( ∪ ) , we may define (cid:15) X := X ∅6 = B ⊂ F ( − | B | +1 Y β ∈ B ˙ t β = 1 − Y β ∈ F (1 − ˙ t β )because the right hand side above only depends on X by parts (3) and (4) of Lemma 3.24. It is noweasy to verify that the pair of maps ( t, (cid:15) ) satisfies the relations defining C ∗ s ( ∪ ) ( P ). Relation III G issimply the definition of (cid:15) X with X = K ( α ). The relation t p (cid:15) X t ∗ p = (cid:15) pX can be derived from relationIII G since it holds when X is in J (see [28, Lemma 3.3]). The remaining relations to be verified onlyinvolve the (cid:15) X ’s and come from the presentation of C ∗ ( ∪ ) ( P ). These relations are satisfied in D r asobserved right after [28, Definition 2.4]. But we know from Corollary 3.19 that there is a canonicalisomorphism D u ∼ = D r , so the relations are satisfied by the set of projections { (cid:15) X | X ∈ J ( ∪ ) } in D u as well. Hence T u ( P ) and C ∗ s ( ∪ ) ( P ) are canonically isomorphic. (cid:3) Semigroup C ∗ -algebras as partial crossed products A partial action of G on D r is constructed in [15, Section 5.5.2], and the corresponding reducedpartial crossed product is shown to be isomorphic to T λ ( P ), [15, Theorem 5.6.41]. Here we aim to showthat the full partial crossed product of that action is always canonically isomorphic to our T u ( P ). Forease of reference and to establish our notation we describe the partial action of G on D r in terms ofthe constructible right ideals of P , without passing through the inverse semigroup of partial bijections I l ( P ) associated to P . This makes our study of faithful representations for T λ ( P ) and of simplicity ofthe boundary quotient more accessible. We point out, nevertheless, that many of the results of thissection could also be extracted from [15, Section 5].4.1. Partial action basics.
We begin with some basic facts concerning partial actions and partialcrossed products.
Definition 4.1 ([20, Definition 11.4]) . A partial action of a discrete group G on a C ∗ -algebra A isa pair γ = ( { A g } g ∈ G , { γ g } g ∈ G ), where { A g } g ∈ G is a collection of closed two-sided ideals of A and γ g : A g − → A g is a ∗ -isomorphism for each g ∈ G , such that for all g, h ∈ G (1) A e = A and γ e is the identity on A ;(2) γ g ( A g − ∩ A h ) ⊆ A gh ;(3) γ g ◦ γ h = γ gh on A h − ∩ A ( gh ) − . We recall the construction of full and reduced partial crossed products based on full and reduced cross-sectional C ∗ -algebras of Fell bundles. Further details can be found in [20]. Let γ = ( { A g } g ∈ G , { γ g } g ∈ G )be a partial action of G on A . We build a Fell bundle B γ = ( B γ g ) g ∈ G over G as follows. We set B γ g := A g as a complex Banach space. For a ∈ A g , we write aδ g to identify the element in B γ g corresponding to a . The multiplication map B γ g × B γ h → B γ gh is then given by ( aδ g ) · ( bδ h ) := γ g ( γ g − ( a ) b ) δ gh , a ∈ A g , b ∈ A h , g, h ∈ G. (4.2)This is well defined by condition (2) of Definition 4.1. The resulting multiplication operation on B γ isassociative. For each g ∈ G , we define an involution ∗ : B γ g → B α g − by( aδ g ) ∗ := γ g − ( a ∗ ) δ g − , a ∈ A g . Then B γ = ( B γ g ) g ∈ G is a Fell bundle whose unit fiber algebra is A (see [20, Proposition 16.6]). Definition 4.3.
The Fell bundle B γ = ( B γ g ) g ∈ G constructed above is called the semidirect prod-uct bundle relative to γ = ( { A g } g ∈ G , { γ g } g ∈ G ). The partial crossed product of A by G under( { A g } g ∈ G , { γ g } g ∈ G ), denoted by A (cid:111) γ G , is the (full) cross-sectional C ∗ -algebra of B γ = ( B γ g ) g ∈ G .The reduced partial crossed product A (cid:111) γ,r G is defined to be the reduced cross-sectional C ∗ -algebraof B γ .Recall that a map v : G → B from G to a unital C ∗ -algebra B is said to be a ∗ -partial representation of G in B if v g is a partial isometry for each g ∈ G with v e = 1, and the set of partial isometries { v g | g ∈ G } satisfies the relations v ∗ g = v g − and v g v h v h − = v gh v h − , for all g, h ∈ G . Let γ = ( { A g } g ∈ G , { γ g } g ∈ G ) be a partial action. A covariant representation of( { A g } g ∈ G , { γ g } g ∈ G ) in B is a pair ( π, v ), where v : G → B is a ∗ -partial representation and π : A → B is a ∗ -homomorphism, such that for all g ∈ G and a ∈ A g − , v g π ( a ) v g − = π ( γ g ( a )) . A covariant representation ( π, v ) of ( { A g } g ∈ G , { γ g } g ∈ G ) yields a representation π × v : A (cid:111) γ G → B induced by the formula ( π × v )( aδ g ) = π ( a ) v g , for g ∈ G and a ∈ A g . By [20, Theorem 13.2], the map ( π, v ) π × v gives a one-to-one correspondencebetween nondegenerate covariant representations of ( { A g } g ∈ G , { γ g } g ∈ G ) on H such that v g v g − is theorthogonal projection onto π ( A g ) H = span { π ( a ) ξ | ξ ∈ H , a ∈ A g } and nondegenerate representationsof the partial crossed product A (cid:111) α G on H .4.2. Toeplitz algebras as partial crossed products.
Suppose that P is a submonoid of a group G .By [15, Section 5.5.2] there is a partial action of G on D r with D r (cid:111) r G ∼ = T λ ( P ). We wish to describethis partial action explicitly in terms of words and their constructible ideals. For each g ∈ G , let A g − := span { K ( α ) | α ∈ W , ˙ α = g } ⊆ D r . So A e = D r by Lemma 3.11. Also, notice that A g − is precisely the ideal D g − defined in [15, p. 188]because in T λ ( P ) K ( α ) = K (˜ αα ) = ˙ L ˜ α ˙ L α = ˙ L ∗ α ˙ L α . Proposition 4.4 (cf. [15, Section 5.5.2]) . Let P be a submonoid of a group G . For each g ∈ G , thereis a unique ∗ -isomorphism γ g : A g − → A g given on a projection K ( α ) ∈ A g − by γ g ( K ( α ) ) = gK ( α ) = K (˜ α ) . Moreover, γ = ( { A g } g ∈ G , { γ g } g ∈ G ) is a partial action of G on D r . OEPLITZ ALGEBRAS OF SEMIGROUPS 17
Proof.
We view B ( ‘ ( P )) as a closed C ∗ -subalgebra of B ( ‘ ( G )) using the canonical embedding of ‘ ( P )as a closed subspace of ‘ ( G ). We claim that λ g K ( α ) λ g − = K (˜ α ) , where λ : G → B ( ‘ ( G )) is the left regular representation of G . To show this, let h ∈ G . Since λ g − ( δ h ) = δ g − h , it follows that( λ g K ( α ) λ g − )( δ h ) = ( δ h if g − h ∈ K ( α ) , . Since g = ˙ α , Proposition 2.6 yields gK ( α ) = K (˜ α ). Hence the automorphism X λ g Xλ g − of B ( ‘ ( G )) restricts to a ∗ -isomorphism γ g : A g − → A g determined by γ g ( K ( α ) ) = K (˜ α ) .To see that A g − is an ideal of D r , let α = ( p , . . . , p k ) be such that ˙ α = g . Let β = ( q , . . . , q l ) ∈W with ˙ β = e . It follows from Proposition 2.6(5) with the roles of α and β exchanged that K ( α ) K ( β ) = K ( β ) K ( α ) = K ( β ) ∩ K ( α ) = K ( αβ ) . This lies in A g − because ˙ αβ = ˙ α ˙ β = ˙ α = g. Let us now prove that ( { A g } g ∈ G , { γ g } g ∈ G ) satisfies axiom (ii) of Definition 4.1. That is, for all g, h ∈ G , one has γ g ( A g − ∩ A h ) ⊆ A gh . Let α = ( p , . . . , p k ) and β = ( q , . . . , q l ) be words in P with˙ α = g and ˙ β = h − , so that K ( α ) ∈ A g − and K ( β ) ∈ A h . Again we view B ( ‘ ( P )) as a C ∗ -subalgebra of B ( ‘ ( G )) usingthe canonical inclusion ‘ ( P ) , → ‘ ( G ). Thus for all k ∈ G , γ g ( K ( α ) K ( β ) )( δ k ) = ( λ g K ( α ) K ( β ) λ ∗ g )( δ k ) = ( δ k if g − k ∈ K ( α ) ∩ K ( β ) , . Now we compute g ( K ( α ) ∩ K ( β )) = gK ( α ) ∩ gK ( β ) = K (˜ α ) ∩ gK ( β ) . Replacing g by ˙ α in the above and using Proposition 2.6, we deduce that g ( K ( α ) ∩ K ( β )) = K ( β ˜ α ) . Hence γ g ( K ( α ) K ( β ) ) = K ( β ˜ α ) . Since˙ β ˜ α = ˙ β ˙˜ α = h − g − = ( gh ) − , it follows that γ g ( K ( α ) K ( β ) ) ∈ A gh and so γ g ( A g − ∩ A h ) ⊆ A gh as desired. Axiom (iii) of Definition4.1 follows from the computation γ gh ( b ) = λ gh bλ ∗ gh = λ g λ h bλ ∗ h λ ∗ g = λ g γ h ( b ) λ ∗ g = γ g ( γ h ( b ))for all b ∈ A g − ∩ A h . We then conclude that ( { A g } g ∈ G , { γ g } g ∈ G ) is a partial action of G on D r . (cid:3) Remark . Observe that A g − = { } if and only if g − P ∩ P = ∅ . One direction is obvious because K ( α ) ⊂ g − P ∩ P whenever ˙ α = g . For the converse assume g − P ∩ P = ∅ and take p, q ∈ P suchthat g − q = p ; then α = ( e, q, p, e ) satisfies ˙ α = qp − = g and K ( α ) = pP ∈ A g − . Also notice thatif p ∈ P , then the ideal A p is the corner determined by the projection pP and A p − = D r . Lemma 4.6.
Let P be a submonoid of a group G . Let ( { A g } g ∈ G , { γ g } g ∈ G ) be the partial action of G on D r from Proposition . Let α = ( p , p , . . . , p k ) ∈ W k with ˙ α = g − . Then K ( α ) δ g = ˙ δ ˜ α := δ p − k p k − P δ p k − . . . δ p − p P δ p . In particular, for every g ∈ G , one has B γ g = span { ˙ δ β | β ∈ W , ˙ β = g } and the full and reduced partial crossed products D r (cid:111) γ G and D r (cid:111) γ,r G are generated as C ∗ -algebrasby the semigroup of isometries { pP δ p | p ∈ P } . Proof.
Let ˜ π : D r (cid:111) γ G → B ( H ) be a nondegenerate representation of D r (cid:111) γ G on a Hilbert space H .Let ( π, w ) be the unique nondegenerate covariant representation of ( { A g } g ∈ G , { γ g } g ∈ G ) on H suchthat w g w g − is the orthogonal projection onto π ( A g ) H and ˜ π = π × w . We will prove by inductionon k that ˜ π ( K ( α ) δ g ) = π ( K ( α ) ) w g = w p − k w p k − . . . w p − w p = ˙ w ∗ α for all α = ( p , p , . . . , p k ) ∈ W k and g = ˙˜ α .The base case k = 0 only occurs if g = e , and clearly π ( K ( α ) ) w e = π ( ) w e = w e in this case.Suppose k = 1, so that α = ( p , p ). Then π ( K ( α ) ) w p − p = π ( γ p − ( p P p P )) w p − p = w p − π ( p P p P ) w p w p − p = w p − π ( p P ) π ( p P ) w p = w p − w p . We used above that w : G → B ( H ) is a ∗ -partial representation and pP is the unit of the ideal A p , sothat w p w p − = π ( pP ) for all p ∈ P .Now fix k > π ( K ( β ) δ h ) = π ( K ( β ) ) w h = ˙ w ∗ β for all β ∈ W k − and h = ˙˜ β . Let α ∈ W k and g = ˙˜ α . Set α = ( p , p , . . . , p k − , e ) ∈ W k . Noticethat K ( α ) = γ p − k ( K ( α ) p k P ) . Hence ˜ π ( K ( α ) δ g ) = π ( K ( α ) ) w g = w p − k π ( K ( α ) ) π ( p k P ) w p k w g = w p − k π ( K ( α ) ) w p k g = w p − k π ( K ( α ) ) w ˙˜ α . Observe that we still have α ∈ W k . Put α := ( p , p , . . . , p k − , p k − ). So α ∈ W k − . Also, π ( K ( α ) ) = π ( γ p k − ( K ( α ) )) = w p k − π ( K ( α ) ) w p − k − . Therefore w p − k π ( K ( α ) ) w ˙˜ α = w p − k w p k − π ( K ( α ) ) w p − k − w ˙˜ α = w p − k w p k − π ( K ( α ) ) w p − k − w p k − w ˙˜ α = w p − k w p k − π ( K ( α ) ) w ˙˜ α . We can now apply our induction hypothesis to α to conclude that˜ π ( K ( α ) δ g ) = π ( K ( α ) ) w g = w p − k w p k − . . . w p − w p as asserted. Since we can always take a covariant pair ( π, w ) such that π is a faithful representationof D r and, for all p ∈ P , one has w p − = ( π × w )( δ p − ) and w p = ( π × w )( pP δ p ), we deduce that K ( α ) δ g = δ p − k p k − P δ p k − . . . δ p − p P δ p in B γ g = A g δ g . Hence B γ g = span { δ q − q P δ q . . . δ q − l − q l P δ q l | l ≥ , q − q · · · q − l − q l = g } = span { ˙ δ β | β ∈ W , ˙ β = g } for all g ∈ G . The last assertion in the statement follows because L g ∈ G B γ g is dense in D r (cid:111) γ G and D r (cid:111) γ,r G . This finishes the proof of the lemma. (cid:3) By [15, Theorem 5.6.41] the reduced Toeplitz C ∗ -algebra T λ ( P ) is canonically isomorphic to thereduced partial crossed product D r (cid:111) γ,r G . We show next that the full version of this isomorphismholds for T u ( P ). As a byproduct of our construction, we also recover the reduced result. OEPLITZ ALGEBRAS OF SEMIGROUPS 19
Theorem 4.7.
Let P be a submonoid of a group G . Let ( { A g } g ∈ G , { γ g } g ∈ G ) be the partial actionof G on D r from Proposition . Then the map t p pP δ p induces an isomorphism T u ( P ) ∼ = D r (cid:111) γ G. In addition, L p pP δ p gives rise to an isomorphism between reduced C ∗ -algebras T λ ( P ) ∼ = D r (cid:111) γ,r G. Proof.
To see that the map that sends t p to pP δ p induces a surjective ∗ -homomorphism ψ : T u ( P ) → D r (cid:111) γ G , notice that K ( α ) δ e = K (˜ α ) δ e = δ p − p P δ p . . . δ p − k − p k P δ p k = ˙ δ α in D r (cid:111) γ G whenever α = ( p , . . . , p k − , p k ) ∈ W k satisfies ˙ α = e . Hence the defining relations (T1)–(T4) of Definition 3.6 are satisfied in D r (cid:111) γ G and so the map t p pP δ p extends to a ∗ -homomorhism ψ : T u ( P ) → D r (cid:111) γ G . This is surjective by Lemma 4.6.In order to construct an inverse for ψ , for each g ∈ G , consider the subspace of T u ( P ) given by B g = span { ˙ t α | α ∈ W , ˙ α = g } . Then ψ : T u ( P ) → D r (cid:111) γ G is faithful when restricted to B g for all g ∈ G , because it is so on B e = D u and b ∗ b ∈ B e for all b ∈ B g . Put ψ g := ψ (cid:22) Bg . Then ψ g : B g → A g δ g is an isomorphism by Lemma 4.6.Thus we can define a representation ψ of ( { A g } g ∈ G , { γ g } g ∈ G ) in T u ( P ) by ψ ( aδ g ) = ψ − g ( aδ g )for all g ∈ G and a ∈ A g . Since B g B h ⊆ B gh and B ∗ g = B g − , it follows that ψ − g ( b ) ∗ = ψ − g − ( b ∗ ) and ψ − g ( b ) ψ − h ( c ) = ψ − gh ( bc ) for all g, h ∈ G , a ∈ λ ( B g ), b ∈ λ ( B h ). Thus ψ ( aδ g ) ψ ( bδ h ) = ψ − g ( aδ g ) λ − h ( bδ h )= ψ − gh ( aδ g · bδ h )= ψ ( aδ g · bδ h ) . Similarly, one can show that ψ preserves the involution operation ∗ : A g δ g → A g − δ g − . Hence it givesrise to a ∗ -homomorphism ˜ ψ : D r (cid:111) γ G → T u ( P ) by universal property of D r (cid:111) γ G . Since D r (cid:111) γ G isgenerated as a C ∗ -algebra by the set of isometries { pP δ p | p ∈ P } , we see that ˜ ψ is the inverse of ψ as desired.It remains to establish the isomorphism T λ ( P ) ∼ = D r (cid:111) γ,r G . Let Λ : D r (cid:111) γ G → D r (cid:111) γ,r G be the leftregular representation associated to the partial action ( { A g } g ∈ G , { γ g } g ∈ G ) and let E Λ : D r (cid:111) γ G → D r δ e be the corresponding conditional expectation. By [20, Proposition 19.7],ker Λ = { c ∈ D r (cid:111) γ G | E Λ ( c ∗ c ) = 0 } . Hence Corollary 3.20 and the commutativity of the diagram T u ( P ) ψ (cid:47) (cid:47) E u (cid:15) (cid:15) D r (cid:111) γ G E Λ (cid:15) (cid:15) D u ψ (cid:22) Du (cid:47) (cid:47) D r δ e yield ψ (ker λ + ) = ker Λ. Thus T λ ( P ) ∼ = D r (cid:111) γ,r G via an isomorphism that identifies the canonicalgenerating elements. (cid:3) Remark . It is pointed out in [15, Remark 5.6.46] that instead of defining the full semigroupC ∗ -algebra as an inverse semigroup C ∗ -algebra, one could focus on the full C ∗ -algebra of the partialtransformation groupoid G (cid:110) Ω P or, equivalently, the full partial crossed product D r (cid:111) γ G . When wecombine Theorem 4.7 with Proposition 3.25, we see that the latter C ∗ -algebra is canonically isomorphicto the C ∗ -algebra C ∗ s ( ∪ ) ( P ) mentioned in [28, Section 3]. The reason for the isomorphism is that bothcoincide with our T u ( P ). As suggested also in [15, Remark 5.6.46], the full partial crossed productversion would yield stronger, ‘independence-free’, versions of [15, Theorem 5.6.44] and [15, Corollary5.6.45], see below. Arguably, choosing such a path is more justified and the stronger results are moreappealing now that we have introduced the C ∗ -algebra T u ( P ) via a transparent presentation. Theorem 4.9 (cf. [15, Theorem 5.6.44]) . Suppose P is a submonoid of a group G and consider thefollowing conditions: (1) T u ( P ) is nuclear; (2) T λ ( P ) is nuclear; (3) the groupoid G (cid:110) Ω P is amenable; (4) the left regular representation λ + : T u ( P ) → T λ ( P ) is faithful.Then (1) ⇐⇒ (2) ⇐⇒ (3) = ⇒ (4). If G is exact, then (4) = ⇒ (1), and all conditions areequivalent.Proof. By Theorem 4.7, T u ( P ) and T λ ( P ) are the full and the reduced crossed product of the partialaction of G on Ω P , and hence are respectively isomorphic to the full and reduced groupoid C ∗ -algebrasof the partial transformation groupoid G (cid:110) Ω P . So the first statement follows directly from [15, Theorem5.6.7]. If G is exact and P satisfies independence, the implication (4) = ⇒ (1) has already been obtained,for C ∗ s ( P ), in [6, Corollary 5.5]; for general P we use Theorem 4.7 and [6, Theorem 4.10]. (cid:3) Corollary 4.10 (cf. [15, Corollary 5.6.45]) . If the monoid P embeds in an amenable group G , thenall the conditions in Theorem hold.Proof. Since G is amenable, all the conditions in Theorem 4.9 are equivalent. As indicated in theproof of [15, Corollary 5.6.45], it also follows that the groupoid G (cid:110) Ω P is amenable by Theorem 20.7and Theorem 20.10 in [20]. This proves the corollary. (cid:3) Remark . We can also see now that the conclusion of [15, Theorem 5.6.42] holds, without theassumption of independence, for the C ∗ -algebra T u ( P ) instead of C ∗ s ( P ). The proof goes along thesame lines, but relies on our Theorem 4.9 instead of [15, Theorem 5.6.44].5. Faithful representations of T λ ( P )Our main purpose in this section is to study faithfulness of representations of T λ ( P ), for which weuse the partial crossed product picture of T λ ( P ) as described in Section 4. The first result reducesthe question of whether a representation of T λ ( P ) is faithful to whether its restriction to the crossedproduct of D r by the action of the group of units is faithful. This generalizes earlier results, from [15],valid for trivial unit group and from [3] about right LCM monoids. It turns out that topologicalfreeness of the partial action of G is equivalent to that of its restriction to the group of units, and wecharacterize this in terms of the action of units on constructible right ideals. We finish the sectionby deriving a general uniqueness theorem for the C ∗ -algebra generated by a collection of elementssatisfying the presentation of T u ( P ).5.1. A characterization of faithful representations. If P is embedded as a submonoid in agroup G , then P ∗ = P ∩ P − is a subgroup of G . The partial action of G restricts to an action of P ∗ on the diagonal subalgebra D r , and the crossed product D r (cid:111) γ,r P ∗ embeds canonically in thepartial crossed product T λ ( P ) ∼ = D r (cid:111) γ,r G . This observation plays a crucial role in the followingcharacterization of faithful representations of T λ ( P ). Theorem 5.1.
Every nontrivial ideal of T λ ( P ) ∼ = D r (cid:111) γ,r G has nontrivial intersection with thesubalgebra D r (cid:111) γ,r P ∗ . In other words, a representation of T λ ( P ) is faithful if and only if it is faithfulon D r (cid:111) γ,r P ∗ .Proof. Let E r : T λ ( P ) → D r be the canonical faithful conditional expectation of T λ ( P ) onto thediagonal subalgebra. In order to prove the theorem, it suffices to show that if a representation ρ : T λ ( P ) → B ( H ) is faithful on the reduced crossed product D r (cid:111) γ,r P ∗ , then there is a conditional OEPLITZ ALGEBRAS OF SEMIGROUPS 21 expectation ϕ ρ , defined on the image of ρ and having range ρ ( D r ), so that the square T λ ( P ) ∼ = D r (cid:111) γ,r G ρ ( T λ ( P )) D r ρ ( D r ) E r ϕ ρ ρρ (cid:22) D r (5.2)commutes. The usual argument then completes the proof: if ρ ( b ) = 0, then ( ϕ ρ ◦ ρ )( b ∗ b ) = 0, andhence ( ρ (cid:22) D r ◦ E r )( b ∗ b ) = 0. Since E r ( b ∗ b ) ∈ D r , this implies that E r ( b ∗ b ) = 0. Thus b ∗ b = 0 and b = 0 because E r is faithful.We denote by A cg the dense ∗ -subalgebra of A g spanned by the set { K ( α ) | ˙ α = g − } . Thus L g ∈ G A cg δ g is a dense ∗ -subalgebra of D r (cid:111) γ,r G . In order to show that the conditional expectation ϕ ρ exists we show that for each (finite) linear combination P g ∈ F a g δ g in L g ∈ G A cg δ g (in which we mayassume there is a term a e by setting it to be zero if necessary), there exist an element p ∈ P and aprojection Q ∈ D r such that(1) | a e ( p ) | = k a e k ;(2) Q ( p ) = 1;(3) Qa g δ g Q = 0 for every g ∈ F \ pP ∗ p − ; and(4) pP Q = Q = Q pP .We relegate the proof of existence of p and Q to Lemma 5.4 below. Supposing for now that p and Q are as above, we have the following estimate. (cid:13)(cid:13)(cid:13) ρ (cid:0) X g ∈ F a g δ g (cid:1)(cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13) ρ ( Q ) X g ∈ F ρ (cid:0) a g δ g (cid:1) ρ ( Q ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) X g ∈ F ∩ pP ∗ p − ρ (cid:0) pP a g Qδ g Q pP (cid:1)(cid:13)(cid:13)(cid:13) because of (3)= (cid:13)(cid:13)(cid:13) X g ∈ F ∩ pP ∗ p − ρ (cid:0) pP δ p δ p − a g Qδ g Q pP δ p δ p − (cid:1)(cid:13)(cid:13)(cid:13) We continue by changing the summation index from g ∈ F ∩ pP ∗ p − to u := p − gp ∈ p − F p ∩ P ∗ and using the multiplication rule (4.2) for generators of the partial crossed product. (cid:13)(cid:13)(cid:13) ρ (cid:0) X g ∈ F a g δ g (cid:1)(cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13) X u ∈ p − F p ∩ P ∗ ρ (cid:0) pP δ p δ p − a pup − Qδ pup − Q pP δ p δ p − (cid:1)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) X u ∈ p − F p ∩ P ∗ ρ ( pP δ p ) ρ (cid:0) δ p − a pup − Qδ pup − Q pP δ p (cid:1) ρ ( δ p − ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) X u ∈ p − F p ∩ P ∗ ρ ( pP δ p ) ρ (cid:0) ( γ p − ( pP a pup − Q ) δ up − ) Q pP δ p (cid:1) ρ ( δ p − ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) X u ∈ p − F p ∩ P ∗ ρ ( pP δ p ) ρ (cid:0) γ up − ( γ pu − ( γ p − ( pP a pup − Q )) Q pP ) δ u (cid:1) ρ ( δ p − ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) X u ∈ p − F p ∩ P ∗ ρ (cid:0) γ up − ( γ pu − ( γ p − ( pP a pup − Q )) Q pP ) δ u (cid:1)(cid:13)(cid:13)(cid:13) . Since this sum is in the crossed product by the action of P ∗ and ρ is assumed to be faithful there, (cid:13)(cid:13)(cid:13) ρ (cid:0) X g ∈ F a g δ g (cid:1)(cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13) X u ∈ p − F p ∩ P ∗ γ up − ( γ pu − ( γ p − ( pP a pup − Q )) Q pP ) δ u (cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13) γ p − ( pP a e Q pP ) (cid:13)(cid:13)(cid:13) because E r is contractive,= (cid:13)(cid:13)(cid:13) pP a e Q pP (cid:13)(cid:13)(cid:13) = k a e k because of (4) and (1),= k ρ ( a e ) k because ρ is faithful on D r . Thus, the map P g ∈ F ρ ( a g δ g ) ρ ( a e ) is well defined and contractive on a dense ∗ -subalgebra of ρ ( T λ ( P )). So it extends uniquely by continuity to give a conditional expectation ϕ ρ : ρ ( T λ ( P )) → ρ ( D r )such that the diagram (5.2) commutes. (cid:3) The following lemma can be extracted from the proof of [15, Theorem 5.7.2]; we formulate itexplicitly because it is useful in a couple of places.
Lemma 5.3.
Let g ∈ G and p ∈ P . The following are equivalent. (1) gpP = pP ; (2) g ∈ pP ∗ p − ; (3) gp ∈ pP ∗ ;Proof. Suppose that gpP = pP and take x, y ∈ P such that gp = px and p = gpy . Multiplying thefirst identity on the right by y , we obtain gpy = pxy and so xy = e . Since P is contained in a group,we deduce that x and y are invertible, that is, x, y ∈ P ∗ . Thus g = pxp − ∈ pP ∗ p − . This proves that(1) = ⇒ (2). The converse holds because if g = pxp − with x ∈ P ∗ , then gpP = pxP = pP . Clearly(3) is just a reformulation of (2). (cid:3) Lemma 5.4.
Let P be a submonoid of a group G . Let F ⊆ G be a finite set and let a = P g ∈ F a g δ g be an element of the dense ∗ -subalgebra L g ∈ G A cg δ g of D r (cid:111) γ,r G . Then there exist a point p ∈ P anda projection Q ∈ D r with the properties (1) – (4) listed in the proof of Theorem .Proof. The proof is an adaptation of the strategy of [25, Lemma 3.2] combined with the main idea ofthe proof of [15, Theorem 5.7.2]. Let A ⊆ J be a finite collection of constructible right ideals and let { λ S | S ∈ A } ⊂ C be scalars such that a e = P S ∈ A λ S S . Because a e is a finite linear combination ofprojections in ‘ ∞ ( P ), there is p ∈ P such that k a e k = | a e ( p ) | . Consider the subset of A given by F p := { S ∈ A | p ∈ S } and put Q F p := Y S ∈ F p S Y S ∈ A \ F p ( − S ) . Note that Q ( p ) = 1 because S ∈ A \ F p implies ( − S )( p ) = 1. We are going to modify Q F p bytaking a subprojection Q ≤ Q F p with Q ( p ) = 1 and Qa g δ g Q = 0 for all g ∈ F \ pP ∗ p − . To do so, weneed to find, in the context of a general submonoid of a group, the correct analogues of the elements ad x,y used in the setting of quasi-lattice orders to define a projection Q right after [25, Equation (3.6)].Let a = P g ∈ F a g δ g be as in the statement of the lemma. Take g ∈ F \ pP ∗ p − . By Lemma 5.3, gpP = pP so we have either gpP ∩ pP (cid:40) pP , or gpP ∩ pP (cid:40) gpP . This latter situation is equivalentto pP ∩ g − pP (cid:40) pP . Now since a g lies in the linear span { K ( α ) | ˙ α = g − } , we can find m ∈ N and words α , . . . , α m ∈ W with ˙ α i = g − such that a g = P mi =1 λ i K ( α i ) , where λ i ∈ C for all i ∈ { , . . . , m } . For each i = 1 , . . . , m , we define d pα i = ( K (( p,e ) α i ) if gpP ∩ pP (cid:40) pP, K (( p,e ) ˜ α i ) if g − pP ∩ pP (cid:40) pP. OEPLITZ ALGEBRAS OF SEMIGROUPS 23
We claim that d pα i ( p ) = 0 for all i ∈ { , . . . , m } . Indeed, if gpP ∩ pP (cid:40) pP , we have that d pα i ( p ) = 0for all i = 1 , . . . , m because p gpP and K (( p, e ) α i ) ⊆ gpP ∩ P . In case g − pP ∩ pP (cid:40) pP , we seethat d pα i ( p ) = 0 because K (( p, e ) ˜ α i ) ⊆ g − pP ∩ P and p g − pP .We set Q g := Q mi =1 ( − d pα i ). We pause here to show that Q g pP a g δ g pP Q g = 0. Suppose that gpP ∩ pP (cid:40) pP . Using the multiplication rule (4.2) in the partial crossed product D r (cid:111) γ,r G , wecompute Q g pP K ( α i ) δ g pP Q g = Q g pP γ g ( γ g − ( K ( α i ) ) pP ) δ g Q g = pP Q g γ g ( K ( ˜ α i ) K ( e,p,p,e ) ) δ g Q g = pP Q g γ g ( K ( ˜ α i ( e,p,p,e )) ) δ g Q g = pP Q g K (( e,p,p,e ) α i ) δ g Q g = pP Q g K (( p,e ) α i ) δ g Q g . Since Q g has a factor − K (( p,e ) α i ) , we deduce that Q g pP K ( α i ) δ g pP Q g = 0.Assume we are in the case g − P ∩ pP (cid:40) pP . Then Q g pP K ( α i ) δ g pP Q g = Q g K ( α i ( e,p,p,e )) δ g pP Q g = Q g K ( α i ( e,p,p,e )) K ( α i ( e,p,p,e )) δ g pP Q g = Q g K ( α i ( e,p,p,e )) δ g γ g − ( K ( α i ( e,p,p,e )) ) Q g pP = Q g K ( α i ( e,p,p,e )) δ g K (( e,p,p,e ) ˜ α i ) Q g pP = Q g K ( α i ( e,p,p,e )) δ g K (( p,e ) ˜ α i ) Q g pP . Again this is zero because Q g has a factor − K (( p,e ) ˜ α i ) . Hence Q g pP a g δ g pP Q g = m X i =1 λ i Q g pP K ( α i ) δ g pP Q g = 0 . Finally, we set Q := Q F p · pP · Y g ∈ F \ ( pP ∗ p − ) Q g . Then Q is projection in D r since it is a finite product of projections in D r . Also, Q is a subprojectionof Q F p with Q ( p ) = 1 and so k Qa e Q k = k a e k . That Qa g δ g Q = 0 for all g ∈ F \ pP ∗ p − follows fromthe computation above. This completes the proof of the lemma. (cid:3) Remark . When we apply Theorem 5.1 to a right LCM monoid P that embeds in a group, werecover the group embeddable case of [3, Theorem 7.4] without having to assume condition (C1) of[3, Definition 2.6]. Our result is about the reduced Toeplitz C ∗ -algebra T λ ( P ), but so is [3, Theorem 7.4],in view of [3, Corollary 7.5]. The necessary and sufficient conditions match because the inner core C I is naturally isomorphic to D r (cid:111) r P ∗ by a standard argument using the assumed faithful conditionalexpectation from C I onto D r .In the right LCM case, condition (C1) and the extra assumptions on the quotient semigroup P/P ∗ are needed to produce a conditional expectation in the proof of [3, Theorem 7.4]. Thus, it is naturalto wonder whether a faithful conditional expectation from T λ ( P ) to D r (cid:111) r P ∗ always exists when P isa submonoid of a group. We show next that this is indeed the case. Proposition 5.6.
Let P be a submonoid of a group G . View T λ ( P ) as the reduced partial crossedproduct D r (cid:111) γ,r G . Then the map aδ g ( aδ g if g ∈ P ∗ , otherwise,extends, by linearity and continuity, to a faithful conditional expectation of D r (cid:111) γ,r G onto D r (cid:111) γ,r P ∗ .Proof. Let λ : G → B ( ‘ ( G )) be the left regular representation of G . Let Q ∗ ∈ B ( ‘ ( G )) denote theorthogonal projection of ‘ ( G ) onto ‘ ( P ∗ ). Because P ∗ is a group, we have Q ∗ λ g Q ∗ = ( λ P ∗ g if g ∈ P ∗ , where λ P ∗ : P ∗ → B ( ‘ ( P ∗ )) denotes the left regular representation of P ∗ .Now let ρ : D r (cid:111) γ,r G → B ( H ) be a faithful representation of D r (cid:111) γ,r G on a Hilbert space H .Observe that (1 ⊗ Q ∗ )( ρ ( aδ g ) ⊗ λ g )(1 ⊗ Q ∗ ) = ( ρ ( aδ g ) ⊗ λ P ∗ g if g ∈ P ∗ , aδ g ∈ A g δ g ρ ( aδ g ) ⊗ λ g ∈ B ( H ⊗ ‘ ( G ))yields a representation of γ = ( { A g } g ∈ G , { γ g } g ∈ G ) whose integrated form factors through the reducedpartial crossed product D r (cid:111) γ,r G . Let ˜ ρ : D r (cid:111) γ,r G → B ( H ⊗ ‘ ( G )) be the induced homomorphism.Then ˜ ρ is faithful because ρ is injective on D r . A similar reasoning shows that the map aδ u ∈ D r (cid:111) γ,r P ∗ ρ ( aδ u ) ⊗ λ P ∗ u ∈ B ( H ⊗ ‘ ( P ∗ ))induces a faithful representation of D r (cid:111) γ,r P ∗ on H ⊗ ‘ ( P ∗ ). We obtain a map E ∗ : D r (cid:111) γ,r G → D r (cid:111) γ,r P ∗ by sending an element b to (1 ⊗ Q ∗ )˜ ρ ( b )(1 ⊗ Q ∗ ) and then identifying the result withthe corresponding element of D r (cid:111) γ,r P ∗ . This is the desired conditional expectation. It is faithfulbecause the map obtained by composing E ∗ with the canonical conditional expectation of D r (cid:111) γ,r P ∗ onto D r is precisely the usual diagonal conditional expectation E r : T λ ( P ) → D r . (cid:3) The action of P ∗ on the spectrum. The underlying reason why the strategy of [25, Section 3]works here is that the set of characters determined by evaluation at points in P is dense in Ω P .However, a quick comparison to the original result from [25] reveals a modification; indeed, for generalsubmonoids of groups, Theorem 5.1 only reduces faithfulness of representations of T λ ( P ) to faithfulnesson the subalgebra D r (cid:111) γ,r P ∗ instead of on D r . If we still wish to know whether a representationof T λ ( P ) is faithful by looking at its restriction to D r , we must rely on the topological freeness ofthe action of P ∗ . Recall that the action of the discrete group P ∗ on the compact space Ω P is said tobe topologically free if for every element e = x ∈ P ∗ , the set of fixed points { χ ∈ Ω P | x · χ = χ } hasempty interior. Topological freeness for partial actions is defined similarly in [21].In order to decide whether the action of P ∗ on Ω P is topologically free it is helpful to review firstthe description of the spectrum Ω P of the diagonal D r given in [15, Corollary 5.6.28]. View J as asemilattice with multiplication given by intersections of constructible ideals. The space ˆ J of characterson J is described in [15, p. 184], following a general construction for the semilattice of idempotents inan inverse semigroup [18, 19]: it consists of nonzero functions J → { , } that are compatible withthe semilattice structure of J . That is, a function χ : J → { , } belongs to ˆ J if it is not identically 0and χ ( R ∩ S ) = χ ( R ) χ ( S ) for all S, R ∈ J , where the multiplication in { , } is inherited from themultiplication in C . In the case that ∅ ∈ J we require χ ( ∅ ) = 0. The topology on ˆ J is the one inducedby pointwise convergence, so ˆ J is a compact Hausdorff spaceBy [15, Corollary 5.6.28], the spectrum of D r is the subspace Ω P of ˆ J given by characters χ : J → { , } satisfying the following additional property: if χ ( S ) = 1 and { S i | i = 1 , . . . , n } ⊂ J are such that S = S ni =1 S i , then there is i ∈ { , . . . , n } with χ ( S i ) = 1. Equivalently, Ω P is the setof points χ ∈ ˆ J such that the relation (T4) holds at χ , namely, such that Q i (cid:0) χ ( S ) − χ ( S i ) (cid:1) = 0whenever the constructible ideals S and S i with i = 1 , . . . , n satisfy S = S ni =1 S i . Notice that Ω P isclosed in ˆ J , hence compact and that if we define ω p ( S ) = S ( p ) for S ∈ J , then { ω p | p ∈ P } is adense subset of Ω P [15, Lemma 5.7.1].We will also need the basis for the topology on Ω P described in equation (4) in [29]. This basisis also mentioned in [15, p. 199], where we believe there is a typo in the negated inequality, whichshould read e e i for the basic open set to contain the point χ e . Lemma 5.7.
Let P be a submonoid of a group. For each nonempty constructible right ideal S ∈ J and each finite (possibly empty) collection C ⊂ J of nonempty constructible right ideals such that S S R ∈C R , let V ( S ; C ) := { χ ∈ Ω P | χ ( S ) = 1; χ ( R ) = 0 for R ∈ C} . Then the collection { V ( S ; C ) } indexed by the pairs ( S, C ) is a basis for the topology of Ω P consistingof nonempty open sets. When P is not left reversible, we may assume C to be nonempty. OEPLITZ ALGEBRAS OF SEMIGROUPS 25
Proof.
View Ω P as a closed subspace of { , } J with the relative topology of pointwise convergence.A basis for this topology is given by the open sets N ( A, B ) indexed by disjoint pairs of finite subsets A and B of J and defined by N ( A, B ) := { ω ∈ Ω P | ω ( S ) = 1 for all S ∈ A and ω ( R ) = 0 for all R ∈ B } . If we let S A := T S ∈ A S , with S A = P for A = ∅ , then ω ( S A ) = Q A ω ( S ), so we may rewrite N ( A, B ) = { ω ∈ Ω P | ω ( S A ) = 1 and ω ( R ) = 0 for R ∈ B } . This shows that N ( A, B ) = V ( S A ; B ). Now when S A ⊂ S R ∈ B R , we have S A = S R ∈ B ( R ∩ S A ).Since the corresponding relation (T4) holds at every ω ∈ Ω P , that is, Q R ∈ B (cid:0) ω ( S A ) − ω ( R ∩ S A ) (cid:1) = 0,we see that N ( A, B ) = ∅ in this case. On the other hand, when S A S R ∈ B R we may choose p ∈ S A \ S R ∈ B R , in which case ω p ∈ N ( A, B ) and thus N ( A, B ) = V ( S A ; B ) is nonempty. (cid:3) The partial action γ = ( { A g } g ∈ G , { γ g } g ∈ G ) of G on D r from Proposition 4.4 induces a partialaction of G by partial homeomorphisms of Ω P , which we now describe. Following [15], for each g ∈ G ,we identify the spectrum of A g − with the subspace of Ω P given by U g − = { χ ∈ Ω P | χ ( K ( α )) = 1 for some α ∈ W , with ˙ α = g } , see [15, p. 189] and [15, Lemma 5.6.40]. Then { U g } g ∈ G is a family of open subspaces of Ω P .By abuse of notation we also denote by γ g the bijection from { K ( α ) | α ∈ W , ˙ α = g } ⊂ J onto { K ( β ) | β ∈ W , ˙ β = g − } ⊂ J that sends K ( α ) to K (˜ α ). Define a map ˆ γ g : U g − → U g byˆ γ g ( χ ) = χ ◦ γ g − . Then ˆ γ = ( { U g } g ∈ G , { ˆ γ g } g ∈ G ) is a partial action of G on Ω P . This gives rise to thetranspose partial action ˆ γ ∗ = ( { C ( U g ) } g ∈ G , { ˆ γ ∗ g } g ∈ G ) on C(Ω P ), where ˆ γ ∗ g is given by f ∈ C ( U g − ) f ◦ ˆ γ g − ∈ C ( U g ) . (5.8)It is then clear that the Gelfand transform D r ∼ = C(Ω P ) intertwines γ and ˆ γ ∗ . Theorem 5.9.
Let P be a submonoid of a group G . The following are equivalent: (1) the partial action of G on Ω P is topologically free; (2) the action of P ∗ on Ω P is topologically free; (3) if u ∈ P ∗ \ { e } and C is a finite collection of proper constructible right ideals, then there exists t ∈ P \ S R ∈C R such that utP = tP (or, equivalently, such that ut / ∈ tP ∗ ); (4) every ideal of T u ( P ) that has trivial intersection with D u is contained in the kernel of the leftregular representation.Proof. The equivalence (1) ⇐⇒ (4) is from [1, Theorem 4.5], and the implication (1) = ⇒ (2) isobvious. In order to prove that (2) = ⇒ (3), assume that the action of P ∗ is topologically free and let u ∈ P ∗ \ { e } be a nontrivial unit. Let C be a finite collection of proper constructible ideals. Sincethe action of P ∗ is topologically free, the nonempty basic open set V ( P ; C ) must contain a point χ that is not fixed by ˆ γ u . By density, we may assume that such a point is of the form ω t for some t ∈ P \ S R ∈C R . Then ω ut = ˆ γ u ( ω t ) = ω t , which means that there is a constructible ideal S thatcontains one of tP and utP but not both, and this translates into utP = tP . That utP = tP isequivalent to ut / ∈ tP ∗ is Lemma 5.3.We finish the proof by showing that (3) = ⇒ (1). Suppose g ∈ G \ { e } . It suffices to show thatevery nonempty basic open subset V ( S ; C ) contained in U g − contains a point that is not fixedby ˆ γ g . Since V ( S ; C ) ⊂ U g − , we know that S \ S R ∈C R ⊂ g − P . Choose q ∈ S \ S R ∈C R ; then ω q ∈ V ( qP ; C ) ⊂ V ( S ; C ). If ˆ γ g ( ω q ) = ω q we are done. If ˆ γ g ( ω q ) = ω q , then gqP = qP and Lemma 5.3shows that g = quq − for some nontrivial unit u ∈ P ∗ \ { e } . Since the ideal ( q − R ) ∩ P is properfor each R ∈ C , we may apply condition (3) to u and the collection C = { ( q − R ) ∩ P | R ∈ C} toget t ∈ P \ S R ∈C ( q − R ) ∩ P with utP = tP . This means that ( q − gq ) tP = tP , or g ( qtP ) = qtP .Since q ∈ g − P , we have ˆ γ g ( ω qt ) = ω gqt = ω qt . So the point ω qt is not fixed by ˆ γ g . Since qt ∈ qP \ S q ( q − R ∩ P ), it follows that ω qt is in V ( qP, C ) and hence in V ( S ; C ). This shows that theset of the fixed points of ˆ γ g has empty interior and completes the proof. (cid:3) Corollary 5.10.
The equivalent conditions of Theorem imply that every nontrivial ideal of T λ ( P ) has nontrivial intersection with D r . If λ + : T u ( P ) → T λ ( P ) is an isomorphism, then the converseholds. Proof.
Suppose J is a nontrivial ideal in T λ ( P ). By Theorem 5.1, J ∩ ( D r (cid:111) γ,r P ∗ ) is a nontrivialideal in D r (cid:111) γ,r P ∗ . Since the action of P ∗ is topologically free, [2, Theorem 2] implies that theideal J ∩ D r = ( J ∩ ( D r (cid:111) γ,r P ∗ )) ∩ D r is nontrivial as wanted. When the kernel of the left regularrepresentation λ + : T u ( P ) → T λ ( P ) is trivial, the converse simply becomes the implication (4) = ⇒ (1)in Theorem 5.9, which is from [1, Theorem 4.5]. (cid:3) When we combine the results from Section 3 with topological freeness of the action of P ∗ , we obtainthe following uniqueness theorem for C ∗ -algebras generated by jointly proper representations of P . Theorem 5.11.
Let P be a submonoid of a group G . Suppose that any of the equivalent conditionsof Theorem hold and that the conditional expectation E u : T u ( P ) → D u is faithful. Let W p be afamily of elements of a C ∗ -algebra satisfying the presentation (T1)–(T4) given in Definition . Thenthere is a homomorphism π W : T λ ( P ) → C ∗ ( W ) such that π W ( L p ) = W p for all p ∈ P , and π W is anisomorphism if and only if W is jointly proper.Proof. By the universal property of T u ( P ), there is a representation ρ W : T u ( P ) → C ∗ ( W ) suchthat ρ ( t p ) = W p for all p ∈ P . Since E u : T u ( P ) → D u is faithful, the left regular representation λ + : T u ( P ) → T λ ( P ) is an isomorphism by Corollary 3.20. Thus π W := ρ W ◦ ( λ + ) − is the requiredhomomorphism.It follows that W is jointly proper if π W is faithful because the identity representation of T λ ( P )is obviously jointly proper. Assume now W is jointly proper. Then π W is faithful on the diagonalsubalgebra D r by Corollary 3.19, and so it is faithful on T λ ( P ) as well by Corollary 5.10. (cid:3) Remark . Corollary 5.10 generalizes [15, Corollary 5.7.3] to monoids with nontrivial units, providedtheir action is topologically free, for which we give a criterion in terms of the semigroup itself inTheorem 5.9(3). We would also like to note that there is a relation between Theorem 5.11 and theuniqueness result [3, Theorem 4.3] and postpone its discussion until Section 10.6.
Strong covariance and a full boundary quotient
In this section we analyze a universal boundary quotient for T u ( P ). We will see that this differsfrom the boundary quotient ∂ T λ ( P ) in the sense of Li [15, Definition 5.7.9] only by the failure ofan amenability condition. Our approach involves the covariance algebra C (cid:111) C P P associated tothe canonical product system C P over P , see [38, Theorem 3.10]. As we study representations of C (cid:111) C P P , a notion of foundation sets emerges naturally, generalizing the foundation sets from [39]and [10, Definition 3.4]. Thus, our findings extend [39, Proposition 5.6] to submonoids of groups thatare not quasi-lattice ordered. The partial crossed product structure of T u ( P ) is not needed to definethe covariance algebra, but it will be used to give sufficient (and in some cases necessary) conditionsfor simplicity of ∂ T λ ( P ).6.1. Strongly covariant representations.
Let P be a submonoid of a group G . Following [39], welet C P = ( C p ) p ∈ P denote the canonical product system over P with one-dimensional fibers. Thatis, C p = C carries the natural structure of a Hilbert space over C and the left action of C given bymultiplication, and the multiplication map C p ⊗ C C q ∼ = C pq is also just multiplication in C . Thus everyisometric representation of P in a unital C ∗ -algebra B extends uniquely by linearity to a nondegeneraterepresentation of C P in B , and every (nondegenerate) representation of C P in B arises this way. Sinceeach fiber of C P is a copy of C , the Fock space of C P can be naturally identified with ‘ ( P ). Underthis identification, the Fock representation of C P corresponds to the left regular representation of P on ‘ ( P ). We refer to [22] for further details on product systems.We aim to find necessary and sufficient conditions for a representation ρ : T u ( P ) → B to factorthrough the covariance algebra C (cid:111) C P P of C P [38]. First of all, we need to ensure that C (cid:111) C P P isindeed a quotient of T u ( P ), via a ∗ -homomorphism that identifies the canonical generating isometries.Let us give a description of strongly covariant isometric representations of P that is equivalent to[38, Definition 3.2] in the special case of C P , and also recall how C (cid:111) C P P is constructed. For eachfinite set F ⊆ G , we define a subset ∆ F of P as follows: an element p ∈ P lies in ∆ F if and only if forall g ∈ F , either pP ∩ gP = ∅ or pP ∩ gP = pP . This gives rise to a closed subspace ‘ (∆ F ) ⊂ ‘ ( P ).Observe that ‘ (∆ F ) corresponds to the Hilbert C -module C PF from [38, Equation (3.1)]. If F ⊂ F are finite subsets of G , then ∆ F ⊂ ∆ F and so ‘ (∆ F ) is a closed subspace of ‘ (∆ F ). OEPLITZ ALGEBRAS OF SEMIGROUPS 27
Definition 6.1.
We let F range in the directed set formed by all finite subsets of G ordered byinclusion and define an ideal in D r by D o r := { b ∈ D r | lim F k b (cid:22) ‘ (∆ F ) k = 0 } . We will say that an isometric representation w : P → B of P in a C ∗ -algebra B is strongly covariant if the map that sends ˙ L α ∈ D r to ˙ w α ∈ B extends to a well-defined ∗ -homomorphism D r → B thatfactors through the quotient D r /D o r . The covariance algebra C (cid:111) C P P is the universal C ∗ -algebra forstrongly covariant isometric representations of P .Let j : P → C (cid:111) C P P be the universal representation of P in C (cid:111) C P P . We will simply write j p instead of j p (1) for the range of 1 ∈ C p under the inclusion j p : C p → C (cid:111) C P P . So p j p is anisometric representation of P in C (cid:111) C P P . Notation 6.2.
We will denote by D o u the ideal of D u ⊂ T u ( P ) isomorphic to D o r via the left regularrepresentation.We will gradually work towards a more concrete description of the ideal D o u , and thus also ofstrongly covariant representations. We begin by showing that C (cid:111) C P P is a quotient of T u ( P ) and bygiving a characterization of strongly covariant isometric representations in terms of a dense subalgebraof D u . Lemma 6.3.
Let j : P → C (cid:111) C P P be the universal representation of P in C (cid:111) C P P . Then thereis a surjective ∗ -homomorphism q u : T u ( P ) → C (cid:111) C P P that sends a generating element t p ∈ T u ( P ) to j p ∈ C (cid:111) C P P . A representation ρ : T u ( P ) → B factors through C (cid:111) C P P if and only if ρ (cid:0) X α ∈ A λ α ˙ t α (cid:1) = 0 for every finite collection of neutral words A ⊂ W and scalars { λ α | α ∈ A } ⊂ C such that P α λ α ˙ t α ∈ D o u .Proof. We will first prove that j : P → C (cid:111) C P P satisfies relations (T1)–(T4) of Definition 3.6. Clearly j e = 1 since C (cid:111) C P P is generated by j ( C P ) as a C ∗ -algebra. Also, notice that ˙ j α = 0 for everyneutral word α ∈ W such that K ( α ) = ∅ , because then ˙ L α = 0 in D r , and ˙ j α = ˙ j β for every pairof neutral words α, β ∈ W such that K ( α ) = K ( β ), because then ˙ L α − ˙ L β = 0. To see that j alsosatisfies (T4), suppose K ( α ) = S β ∈ A K ( β ), where ˙ α = e and A is a finite collection of neutral words.Then Q β ∈ A ( ˙ L α − ˙ L β ) = 0 in D r and thus Q β ∈ A (˙ j α − ˙ j β ) = 0 in the covariance algebra C (cid:111) C P P by Definition 6.1. This shows that j also satisfies (T4) and hence t p j p gives rise to a surjective ∗ -homomorphism q u : T u ( P ) → C (cid:111) C P P as wished.It remains to establish the last assertion in the lemma. Clearly ρ (cid:0) P α ∈ A λ α ˙ t α (cid:1) = 0 if ρ : T u ( P ) → B factors through C (cid:111) C P P and P α λ α ˙ t α ∈ D o u , where A ⊂ W is a finite collection of neutral words andthe λ α ’s are scalars. To prove the converse implication, let C ⊂ J be an ∩ -closed finite collection ofconstructible right ideals of P . For each S ∈ C , choose a neutral word α S ∈ W such that K ( α S ) = S .Then A ( C ) = span { ˙ t α | K ( α ) ∈ C , ˙ α = e } = span { ˙ t α S | S ∈ C} is a finite dimensional ∗ -subalgebra of D u . By assumption, ρ (cid:0) X S λ S ˙ t α S (cid:1) = 0in B whenever P S λ S ˙ t α S ∈ D o u . We deduce that ρ vanishes on A ( C ) ∩ D o u . Hence it will factor through C (cid:111) C P P because C (cid:111) C P P is precisely the quotient of T u ( P ) by the ideal generated by D o u and D o u = lim C A ( C ) ∩ D o u . (cid:3) Foundation sets and generating projections of D o r . In order to find a more concretedescription of the kernel of the quotient map q u : T u ( P ) → C (cid:111) C P P , we give in the next lemma aclass of projections in D o u . Lemma 6.4.
Let α ∈ W be a neutral word. Suppose that A ⊂ W is a finite collection of neutralwords with S β ∈ A K ( β ) ⊂ K ( α ) . Then Q β ∈ A ( ˙ t α − ˙ t β ) belongs to D o u if and only if for all p ∈ K ( α ) ,one has pP ∩ (cid:16) [ β ∈ A K ( β ) (cid:17) = ∅ . Proof.
We begin by observing that for each finite set F ⊂ G such that Q β ∈ A ( K ( α ) − K ( β ) ) doesnot vanish on ‘ (∆ F ), one has (cid:13)(cid:13)(cid:13) λ + (cid:16) Y β ∈ A ( ˙ t α − ˙ t β ) (cid:17) (cid:22) ‘ (∆ F ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) Y β ∈ A ( K ( α ) − K ( β ) ) ∆ F (cid:13)(cid:13)(cid:13) = 1because Q β ∈ A ( K ( α ) − K ( β ) ) and ∆ F are commuting projections. So Q β ∈ A ( ˙ t α − ˙ t β ) lies in D o u if and only if there exists a finite set F ⊂ G such that Q β ∈ A ( K ( α ) − K ( β ) ) vanishes on ‘ (∆ F ) . Since Q β ∈ A ( K ( α ) − K ( β ) ) already vanishes on the closed subspace of ‘ ( P ) determined by the union S β ∈ A K ( β ), the proof of the lemma reduces to showing that there is a finite set F ⊂ G such that∆ F ∩ (cid:0) K ( α ) \ S β ∈ A K ( β ) (cid:1) = ∅ if and only if pP ∩ (cid:0) S β ∈ A K ( β ) (cid:1) = ∅ for all p ∈ K ( α ).Assume that ∆ F ∩ (cid:0) K ( α ) \ S β ∈ A K ( β ) (cid:1) = ∅ for some finite set F ⊂ G . We will prove that pP ∩ (cid:0) S β ∈ A K ( β ) (cid:1) = ∅ for all p ∈ K ( α ). In case K ( α ) = S β ∈ A K ( β ) we are done. Suppose that K ( α ) \ S β ∈ A K ( β ) = ∅ and take p ∈ K ( α ) \ S β ∈ A K ( β ) . It suffices to show that pP ∩ ∆ F = ∅ because∆ F ∩ (cid:0) K ( α ) \ S β ∈ A K ( β ) (cid:1) = ∅ . Since p ∆ F , we can find g ∈ F with pP ∩ g P = ∅ but p g P . Let r ∈ pP ∩ g P . If r ∈ ∆ F , we are done. Otherwise, the set F := F \ { g } must be nonempty and wehave r ∆ F because r ∆ F and r ∈ g P . Then we can find g ∈ F such that r P ∩ g P = ∅ but r g P . Take r ∈ r P ∩ g P . Again, if r ∈ ∆ F , we are done. Otherwise, the set F := F \ { g , g } is nonempty. As above, we deduce that r ∆ F since r ∆ F and r ∈ g P ∩ g P . Thus we can find g ∈ F such that r P ∩ g P = ∅ but r g P . Take r ∈ r P ∩ g P . We are done in case r ∈ ∆ F .Otherwise, notice that F (cid:40) F (cid:40) F . Since F is a finite set, this process must stop after finitely manysteps, and so we can find r l ∈ ∆ F ∩ pP ⊂ K ( α ). We conclude that r l ∈ S β ∈ A K ( β ).For the converse, assume that pP ∩ (cid:0) S β ∈ A K ( β ) (cid:1) = ∅ for all p ∈ K ( α ). We have to find a finiteset F ⊂ G with ∆ F ∩ (cid:0) K ( α ) \ S β ∈ A K ( β ) (cid:1) = ∅ . Of course this holds whenever K ( α ) = S β ∈ A K ( β ).Otherwise, set F := [ β ∈ A Q ( β ) . We claim that ∆ F ∩ (cid:16) K ( α ) \ [ β ∈ A K ( β ) (cid:17) = ∅ . To see this, let p ∈ K ( α ) \ S β ∈ A K ( β ). Let β ∈ A be such that pP ∩ K ( β ) = ∅ . Since p K ( β ), theremust be g ∈ Q ( β ) such that p gP . For such a g , we have pP ∩ gP = ∅ because ∅ 6 = pP ∩ K ( β ) ⊂ pP ∩ gP . Hence p ∆ F . This completes the proof of the lemma. (cid:3) Lemma 6.4 motivates the following generalization of the concept of a foundation set, originallydefined in the context of quasi-lattice orders [39], see also [10, Definition 3.4].
Definition 6.5.
Let P be a submonoid of a group G and let S ∈ J be a constructible right idealof P . We shall say that a finite collection C ⊂ J of constructible ideals is a foundation set for S , or an S -foundation set , if R ⊂ S for all R ∈ C and for all p ∈ S , one has pP ∩ (cid:16) [ R ∈C R (cid:17) = ∅ . We will say that an S -foundation set C is proper if S \ (cid:0) S R ∈C R (cid:1) = ∅ . In general, we will say that afinite collection of constructible ideals C ∈ J is a relative foundation set for S if { S ∩ R | R ∈ C} isan S -foundation set.We can now describe a set of generating projections for D o u . OEPLITZ ALGEBRAS OF SEMIGROUPS 29
Corollary 6.6.
The ideal D o u of D u ⊂ T u ( P ) is the closed linear span of projections of the form Q β ∈ A ( ˙ t α − ˙ t β ) , where α is a neutral word and A ⊂ W is a finite collection of neutral words such that { K ( β ) | β ∈ A } is a proper foundation set for K ( α ) . Moreover, a map w : P → B into a C ∗ -algebra B is a strongly covariant isometric representation of P if and only if it satisfies the relations (T1)–(T4) of Definition and the boundary relations , that is, Y β ∈ A ( ˙ w α − ˙ w β ) = 0 for every neutral word α in W and proper foundation set { K ( β ) | β ∈ A } for K ( α ) , where A is afinite collection of neutral words in W .Proof. Let a ∈ D o u . Using that D o u = lim C ( A ( C ) ∩ D o u ), where A ( C ) is the (finite-dimensional)C ∗ -subalgebra of D u spanned by the projections { ˙ t α | ˙ α = e, K ( α ) ∈ C} , we may assume that a is afinite linear combination P α ∈ F λ α ˙ t α , where F ⊂ W is a finite collection of neutral words and λ α ∈ C for all α ∈ F . Thus by Lemma 3.13, a can be decomposed as a finite linear combination of orthogonalprojections a = X ∅6 = A ⊂ F λ A Q A , where Q A = Q α ∈ A ˙ t α Q β ∈ F \ A ( t e − ˙ t β ) and λ A = P α ∈ A λ α . We rewrite Q A = Y α ∈ A ˙ t α Y β ∈ F \ A ( t e − ˙ t β ) = Y β ∈ F \ A (cid:0) ˙ t Q A α − ˙ t Q A αβ (cid:1) and use Lemma 6.4 to conclude that a ∈ D o u if and only if { K ( Q A αβ ) | β ∈ F \ A } is a foundation setfor K ( Q A α ) whenever λ A = 0. Now observe that if K ( α ) = S β ∈ A K ( β ) so that { K ( β ) | β ∈ A } isobviously a foundation set for K ( α ), then Q β ∈ A ( ˙ t α − ˙ t β ) = 0 already holds in T u ( P ) by (T4). Hencethe nonzero generators of D o u arise from proper foundation sets. This proves the first assertion ofthe corollary. The second assertion follows because C (cid:111) C P P is the quotient of T u ( P ) by the idealgenerated by D o u . (cid:3) Remark . Take an element a = P α ∈ F λ α ˙ t α ∈ D u and suppose that a D o u . Using the samedecomposition as in the first part of the proof of Corollary 6.6, we see that the image of a in thequotient D u /D o u is the linear combination of the images of mutually orthogonal basic projections thatare not given by foundation sets. Remark . In the case that (
G, P ) is a quasi-lattice ordered group, the last statement of Corollary 6.6is proved in [39, Proposition 5.6]. Indeed, in this case the nonempty constructible right ideals of P arethe principal ideals. So let p ∈ P and let A ⊂ P be a finite set. Then { qP | q ∈ A } is a foundationset for pP in the sense of Definition 6.5 if and only if { p − qP | q ∈ A } is a foundation set for P asdefined in [39]. A foundation set { qP | q ∈ A } for pP is proper if and only if p A or, equivalently, e p − A := { p − q | q ∈ A } .We could deduce from [38, Lemma 3.3] that the ideal of T u ( P ) generated by D o u is invariant underthe canonical gauge coaction of G on T u ( P ), and so an element a ∈ D u lies in the kernel of the quotientmap q u : T u ( P ) → C (cid:111) C P P if and only if a ∈ D o u (see [38, Lemma 3.4]). If we then apply Lemma 6.4,we see that the kernel of the quotient map q u : T u ( P ) → C (cid:111) C P P does not contain any projection ofthe form Q β ∈ A ( ˙ t α − ˙ t β ) when { K ( β ) | β ∈ A } is not a K ( α )-foundation set.We would like to give next a direct proof of a stronger result that makes it clear why theseprojections cannot vanish under any nontrivial representation of T u ( P ) when { K ( β ) | β ∈ A } is not afoundation set for K ( α ). Proposition 6.9.
Let ρ be a representation of T u ( P ) in a C ∗ -algebra B . Let α ∈ W be a neutralword and let A ⊂ W be a finite (possibly empty) collection of neutral words such that K ( β ) ⊂ K ( α ) for all β ∈ A . If { K ( β ) | β ∈ A } is not a foundation set for K ( α ) and ρ (cid:16) Y β ∈ A ( ˙ t α − ˙ t β ) (cid:17) = 0 , then ρ ≡ . As a consequence, the restriction of q u to D u induces an embedding of D u /D o u in C (cid:111) C P P ,and a ∗ -homomorphism ˆ ρ : C (cid:111) C P P → B is faithful on D u /D o u if and only if ˆ ρ = 0 . Proof.
The proof is based on an observation that goes back to [24, Lemma 5.1]. Regarding the firstpart of the proposition, it suffices to show that if { K ( β ) | β ∈ A } is not a foundation set for K ( α ),then the projection ρ (cid:0) Q β ∈ A ( ˙ t α − ˙ t β ) (cid:1) dominates a range projection of the form t p t ∗ p for some p ∈ P .Indeed, if { K ( β ) | β ∈ A } is not a K ( α )-foundation set, there must be p ∈ K ( α ) satisfying pP ∩ (cid:16) [ β ∈ A K ( β ) (cid:17) = ∅ . It follows from the defining relation (T2) of T u ( P ) that t p t ∗ p ˙ t β = 0 for all β ∈ A . Also, from relations(T1) and (T3) of Definition 3.6 we have t p t ∗ p ˙ t α = t p t ∗ p . Hence t p t ∗ p Y β ∈ A ( ˙ t α − ˙ t β ) = t p t ∗ p ˙ t α = t p t ∗ p as wished. Thus ρ (cid:0) Q β ∈ A ( ˙ t α − ˙ t β ) (cid:1) = 0 forces ρ ( t p t ∗ p ) = 0, and since t p is an isometry, this meansthat ρ has to be the zero representation.Suppose now that ρ : T u ( P ) → B is a nontrivial strongly covariant representation. We will showthat D u ∩ ker ρ = D o u . By definition, D o u ⊂ D u ∩ ker ρ . For the reverse inclusion take a ∈ D u ∩ ker ρ .As before, it suffices to consider elements of the form a = P α ∈ F λ α ˙ t α ∈ span { ˙ t α | α ∈ W , ˙ α = e } .Arguing as in the proof of Corollary 6.6, we may write a as a finite linear combination of orthogonalprojections a = X ∅6 = A ⊂ F λ A Q A , where Q A = Q α ∈ A ˙ t α Q β ∈ F \ A ( t e − ˙ t β ) . If we had a D o u , Corollary 6.6 would produce an A ⊂ F such that λ A = 0 and { K ( Q A αβ ) | β ∈ F \ A } is not a foundation set for K ( Q A α ). In this case ρ ( a ) = 0 would imply ρ ( Q A ) = 0 and thus ρ ≡ D u ∩ ker ρ ⊂ D o u .Taking now ρ = q u : T u ( P ) → C (cid:111) C P P , which is a nontrivial strongly covariant representation by[38, Theorem 3.10](C3), by the preceding argument we conclude that D u ∩ ker q u = D o u . This showsthat q u induces an embedding of D u /D o u in C (cid:111) C P P .Finally, if ˆ ρ : C (cid:111) C P P → B is a nonzero representation, then the representation ρ := ˆ ρ ◦ q u of T u ( P ) is strongly covariant and nontrivial, so D u ∩ ker ρ = D o u . Since the embedded copy of D u /D o u in C (cid:111) C P P is q u ( D u ), to finish the proof we only need to notice that if ˆ ρ ( q u ( a )) = 0, then ρ ( a ) = 0and hence a ∈ D o u . (cid:3) The covariance algebra as a full boundary quotient.
We can now provide a few charac-terizations of
C (cid:111) C P P . Among them, we show that C (cid:111) C P G coincides with the full partial crossedproduct C( ∂ Ω P ) (cid:111) G , of the partial action of G restricted to ∂ Ω P , see [15, Definition 5.7.8] and[15, Definition 5.7.9]. In view of [15, Corollary 5.7.6], it is then natural to regard C (cid:111) C P P as a full boundary quotient for T u ( P ). In order to establish the desired crossed product picture for C (cid:111) C P P , weneed to show that the partial action γ = ( { A g } g ∈ G , { γ g } g ∈ G ) of G on D r from Theorem 4.7 induces apartial action of G on the quotient D r /D o r .Let γ = ( { A g } g ∈ G , { γ g } g ∈ G ) be a partial action of G on a C ∗ -algebra A . Recall from [21, Defini-tion 2.7] that a C ∗ -subalgebra D of A is invariant under γ if γ g ( D ∩ A g − ) ⊂ D for all g ∈ G . If I / A is an invariant ideal, then γ restricts to a partial action on I in a natural manner. The underlyingcollection of ideals is simply { I ∩ A g } g ∈ G and the isomorphism I ∩ A g − → I ∩ A g is simply the re-striction of γ g . The quotient A/I also carries a partial action of G given by ˙ γ = ( { A g /I } g ∈ G , { ˙ γ } g ∈ G ),where ˙ γ g : A g − /I → A g /I is the isomorphism a + I γ g ( a ) + I . By [21, Proposition 3.1], there is ashort exact sequence 0 −→ I (cid:111) γ (cid:22) I G −→ A (cid:111) γ G −→ ( A/I ) (cid:111) ˙ γ G −→ , where the inclusion I (cid:111) γ (cid:22) I G , → A (cid:111) γ G extends the embedding ( I ∩ A g ) δ g , → A g δ g for g ∈ G whilethe second ∗ -homomorphism sends aδ g ∈ A g δ g to ( a + I ) δ g ∈ ( A g /I ) δ g . Lemma 6.10.
Let P be a submonoid of a group G . Then the ideal D o r of D r is invariant under thepartial action γ = ( { A g } g ∈ G , { γ g } g ∈ G ) from Proposition . OEPLITZ ALGEBRAS OF SEMIGROUPS 31
Proof.
Let a ∈ D o r ∩ A g − . As usual we may assume that a = X α ∈ F λ α K ( α ) = X α ∈ F λ α K (˜ αα ) , where F ⊂ W is a finite collection of words and λ α ∈ C for all α ∈ F . What is not immediately clear isthat we may choose such a linear combination so that ˙ α = g . To see that we may, suppose first ˙ α = e for α ∈ F and consider the decomposition of the identity associated to F as in Lemma 3.13. Thus a isa finite linear combination a = P ∅6 = A ⊂ F λ A Q A , where the Q A ’s are mutually orthogonal projections.Since D o r ∩ A g − is an ideal of D r , it follows that Q A = Y α ∈ A K ( α ) Y α ∈ F \ A ( − K ( α ) ) = λ α Q A a ∈ D o r ∩ A g − whenever λ A = 0. Suppose ∅ 6 = A ⊂ F satisfies λ A = 0 and let 0 < ε < . Let b = P β ∈ E λ β K ( β ) ∈ A g − be such that k Q A − b k < ε , where E ⊂ W is a finite collection of words with ˙ β = g for all β ∈ E . Using Proposition 2.6(5) write Q α ∈ A K ( α ) = K ( α A ) with α A := Q A α , (the order of thisconcatenation is irrelevant because each α is neutral). Since we have chosen ε < , the support of Q A is contained in S β ∈ E K ( β ), and we see that K (cid:0) α A (cid:1) = K (cid:0) α A (cid:1) \ (cid:18)(cid:18) [ α ∈ F \ A K ( α ) (cid:19) [ (cid:18) [ β ∈ E K ( β ) (cid:19)(cid:19) = (cid:18) [ α ∈ F \ A K (cid:0) αα A (cid:1)(cid:19) [ (cid:18) [ β ∈ E K (cid:0) βα A (cid:1)(cid:19) , (by (5) of Proposition 2.6).Then K ( α A ) = X ∅6 = B ⊂ (( F \ A ) ∪ E ) α A ( − | B | +1 Y β ∈ B K ( β ) , (6.11)where (( F \ A ) ∪ E ) α A stands for the collection of words { βα A | β ∈ ( F \ A ) ∪ E } . If B ∩ ( F \ A ) α A = ∅ ,then one of the factors in the product Q β ∈ B K ( β ) equals K ( α A ) K ( α ) because α A is neutral, hence Y β ∈ B K ( β ) Y α ∈ F \ A ( − K ( α ) ) = 0 . Thus only the factors of (6.11) with B ⊂ Eα A contribute to Q A , so that Q A = K ( α A ) Y α ∈ F \ A ( − K ( α ) ) = X ∅6 = B ⊂ Eα A ( − | B | +1 Y β ∈ B K ( β ) Y α ∈ F \ A ( − K ( α ) ) . Considering now each term in the sum on its own, suppose ∅ 6 = B ⊂ Eα A and take β B ∈ B . Let β B ∈ E be such that β B = β B α A . Define the concatenation σ B := β B Y β ∈ B \{ β B } ˜ β β . Then ˙ σ B = ˙ β B = ˙ β B ˙ α A = ˙ β B = g and Q β ∈ B K ( β ) = K ( σ B ) , by properties (3) and (5) ofProposition 2.6. Hence Y β ∈ B K ( β ) Y α ∈ F \ A ( − K ( α ) ) = K ( σ B ) Y α ∈ F \ A ( − K ( α ) ) = X A ⊂ F \ A ( − | A | K ( σ B ) Y α ∈ A K ( α ) is a finite linear combination of projections of the form K ( σ ) , where σ ∈ W satisfies ˙ σ = g . So thesame will be true for Q A and for a = P ∅6 = A ⊂ F λ A Q A .For the remainder of the proof, assume a = X α ∈ F λ α K ( α ) , where F ⊂ W is a finite collection of words with ˙ α = g and λ α ∈ C for all α ∈ F . We can seein the proof of Corollary 6.6 using the identification D u ∼ = D r that each projection Q A appearingin the decomposition of a will have the form Q n A i =1 ( K ( α A ) − K ( β i,A ) ), where ˙ α A = ˙ β i,A = g and K ( β i,A ) ⊂ K ( α A ) for all i = 1 , . . . , n A , by another application of (3) and (5) of Proposition 2.6. Since a also lies in D o r , Lemma 6.4 implies that { K ( β i,A ) | i = 1 , . . . , n } has to be a foundation set for K ( α A ) whenever λ A = 0.Thus in order to prove that γ g ( a ) ∈ D o r , all we have to show is that if λ A = 0, then γ g ( Q A ) = n A Y i =1 ( K ( ˜ α A ) − K ( ˜ β i,A ) ) ∈ D o r . But this happens if and only if { K ( ˜ β i,A ) | i = 1 , . . . , n } is a foundation set for K (˜ α A ). So let p ∈ K (˜ α A ) \ S n A i =1 K ( ˜ β i,A ). Notice that g − p ∈ K ( α A ). Using that { K ( β i,A ) | i = 1 , . . . , n } is afoundation set for K ( α A ), we can find r ∈ g − pP ∩ (cid:0)S n A i =1 K ( β i,A ) (cid:1) . It follows that gr ∈ pP ∩ g n A [ i =1 K ( β i,A ) ! = pP ∩ n A [ i =1 K ( ˜ β i,A ) ! . So γ g ( a ) ∈ D o r as wished. (cid:3) Li introduced a notion of a boundary quotient ∂ T λ ( P ) of T λ ( P ) [15, Definition 5.7.9], cf. [29,Definition 7.14], based on a more general construction in the context of inverse semigroups due to Exel[18, 19]. Before we can relate the covariance algebra C (cid:111) C P P to ∂ T λ ( P ), we recall the descriptionof the boundary ∂ Ω P of Ω P . Let ˆ J max be the subset of ˆ J formed by all characters χ ∈ ˆ J such that χ − (1) = { S ∈ J | χ ( S ) = 1 } is maximal among all characters. That is, if χ ∈ ˆ J , χ = χ , then χ − (1) χ (1) . By [15, Lemma 5.7.7], the closure ˆ J max is contained in Ω P , and it is invariant underthe partial action ˆ γ = ( { U g } g ∈ G , { ˆ γ g } g ∈ G ) of G on Ω P by [15, Lemma 5.7.5]. Set ∂ Ω P := ˆ J max . ThenC( ∂ Ω P ) is invariant under the partial action ˆ γ ∗ on C(Ω P ), see (5.8). By [15, Corollary 5.7.6], the boundary quotient ∂ T λ ( P ) of T λ ( P ) as defined in [15, Definition 5.7.9] is canonically isomorphic tothe reduced partial crossed product C( ∂ Ω P ) (cid:111) r G of C( ∂ Ω P ) by the partial action of G obtained byrestricting ˆ γ ∗ to C( ∂ Ω P ). We will now see that under the canonical identification of Ω P with thespectrum ˆ D r of D r , we have Ω P \ ∂ Ω P ∼ = ˆ D o r , and consequently ∂ T λ ( P ) ∼ = ( D r /D o r ) (cid:111) ˙ γ,r G . Noticethat ∂ T λ ( P ) is indeed a quotient of T λ ( P ) by [20, Proposition 21.3]. Lemma 6.12.
Let P be a submonoid of a group G . Let V ( S ; C ) = { τ ∈ Ω P | τ ( S ) = 1; τ ( R ) =0 for R ∈ C} be a basic open set for the topology of Ω P as in Lemma . Then V ( S ; C ) ∩ ∂ Ω P = ∅ ifand only if C is not a relative S -foundation set. In particular, the collection of subsets of ∂ Ω P given by { V ( S ; C ) ∩ ∂ Ω P | C is not a relative S -foundation set } is a basis of nonempty open sets for the topology of ∂ Ω P . Moreover, ∂ Ω P = { τ ∈ Ω P | τ (cid:22) D o r = 0 } and D r /D o r ∼ = C( ∂ Ω P ) .Proof. In case ∅ 6∈ J , that is, when P is left reversible, then ∂ Ω P consists of a single point, given bythe character τ : J → { , } , τ ( S ) = 1 for all S ∈ J . So clearly V ( S ; C ) ∩ ∂ Ω P = ∅ if and only if C = ∅ .Since any nonempty collection C of constructible ideals contained in R ⊂ S produces a foundation setin this case, this proves the result for left reversible P .Assume ∅ ∈ J . Let us argue by contradiction and suppose that C is a relative foundation set for S and there is some τ ∈ ∂ Ω P ∩ V ( S ; C ). We may assume τ ∈ ˆ J max because V ( S ; C ) is open. Let R ∈ C .Since τ ( R ) = 0, there is ∅ 6 = S R ∈ J with τ ( S R ) = 1 and S R ∩ R = ∅ by [15, Lemma 5.7.4]. Put R := T R ∈C S R ∈ J . Notice that R ∩ (cid:16) S R ∈C R (cid:17) = ∅ and S ∩ R = ∅ because τ ( S ∩ R ) = τ ( S ) τ ( R ) = 1.Take p ∈ S ∩ R . Using that { S ∩ R | R ∈ C} is an S -foundation set, we can find r ∈ pP ∩ (cid:16) S R ∈C R (cid:17) .But r ∈ R because R is a right ideal. This gives a contradiction. Hence V ( S ; C ) ∩ ∂ Ω P = ∅ only if C is not a relative foundation set for S .To see that V ( S ; C ) ∩ ∂ Ω P = ∅ if C is not a relative foundation set for S , notice that the image ofthe projection S Y R ∈C ( − R ) = Y R ∈C ( S − S ∩ R ) OEPLITZ ALGEBRAS OF SEMIGROUPS 33 in C( ∂ Ω P ) has to be nonzero by Proposition 6.9. This is so because ∂ T λ ( P ) is nonzero. Hence anycharacter τ ∈ ∂ Ω P with τ (cid:18) Y R ∈C ( S − S ∩ R ) (cid:19) = 0will belong to V ( S ; C ) and so V ( S ; C ) ∩ ∂ Ω P is nonempty.It remains to establish the identification ∂ Ω P = { τ ∈ Ω P | τ (cid:22) D o r = 0 } . We identify the spectrum ˆ D o r of D o r with the open subspace U o of Ω P consisting of the characters τ ∈ Ω P satisfying the followingproperty: there exist a nonempty constructible ideal S , and a finite collection of nonempty constructibleideals C ⊂ J such that C is a proper foundation set for S , with τ ( S ) = 1 and τ ( R ) = 0 for R ∈ C . Fromthis identification and from the above we immediately see the inclusion ∂ Ω P ⊂ { τ ∈ Ω P | τ (cid:22) D o r = 0 } .To prove the reverse inclusion, take τ ∈ Ω P with τ ≡ D o r . Let V ( S ; C ) ⊂ Ω P be a basic open setcontaining τ . Then C is not a relative foundation set for S because τ vanishes on D o r . By the firstpart of the proof, we have V ( S ; C ) ∩ ∂ Ω P = ∅ . So τ ∈ ∂ Ω P since ∂ Ω P is closed and V ( S ; C ) is anarbitrary basic open set around τ . This implies the identification Ω P \ ∂ Ω P ∼ = ˆ D o r and the isomorphism D r /D o r ∼ = C( ∂ Ω P ), and completes the proof of the lemma. (cid:3) Theorem 6.13.
Let P be a submonoid of a group G . The following C ∗ -algebras are isomorphic, viaisomorphisms that identify the canonical generating elements: (1) the covariance algebra C (cid:111) C P P of the canonical product system C P over P ; (2) the universal C ∗ -algebra with generating set { v p | p ∈ P } subject to the relations (T1)–(T3) ofDefinition together with the relations Y β ∈ A ( ˙ v α − ˙ v β ) = 0 (6.14) for every neutral word α in W and foundation set { K ( β ) | β ∈ A } for K ( α ) , where A is afinite collection of neutral words in W ; (3) the universal C ∗ -algebra with generating set { v p | p ∈ P } subject to the relations (T1)–(T4) ofDefinition together with the boundary relations from Corollary ; (4) the full partial crossed product ( D r /D o r ) (cid:111) ˙ γ G ; (5) the full partial crossed product C( ∂ Ω P ) (cid:111) G of C( ∂ Ω P ) by ˆ γ ∗ (cid:22) C( ∂ Ω P ) .Proof. The relation (6.14) corresponds to relation (T4) of Definition 3.6 when K ( α ) = S β ∈ A K ( β ).Hence the universal C ∗ -algebra from item (2) is canonically isomorphic to the one with presentation(T1)–(T4) of Definition 3.6 and the boundary relations. These in turn are isomorphic to C (cid:111) C P P byCorollary 6.6.In order to establish the isomorphism C (cid:111) C P P ∼ = ( D r /D o r ) (cid:111) ˙ γ G , observe that C (cid:111) C P P is thequotient of T u ( P ) by the ideal generated by D o u ∼ = D o r . Such an ideal is simply L g ∈ G ( D o u ∩ A g ) δ g by γ -invariance of D o r obtained in Lemma 6.10. Hence it is canonically isomorphic to D o r (cid:111) γ (cid:22) D o r G by theobservation preceding the statement of Lemma 6.10 and so C (cid:111) C P P ∼ = T u ( P ) / h D o u i ∼ = ( D r (cid:111) γ G ) / ( D o r (cid:111) γ (cid:22) D o r G ) ∼ = ( D r /D o r ) (cid:111) ˙ γ G as asserted.Finally, to see that C (cid:111) C P P is also canonically isomorphic to C( ∂ Ω P ) (cid:111) G , we apply Lemma 6.12.The isomorphism T u ( P ) ∼ = D r (cid:111) γ G ∼ = C(Ω P ) (cid:111) G identifies D o r (cid:111) γ (cid:22) D o r G with C(Ω P \ ∂ Ω P ) (cid:111) G sinceΩ P \ ∂ Ω P ∼ = ˆ D o r . Hence C (cid:111) C P P ∼ = ( D r /D o r ) (cid:111) ˙ γ G ∼ = C( ∂ Ω P ) (cid:111) G canonically. (cid:3) Let us recount the type of relations we have encountered thus far. First we had the original relations,corresponding to (T3), saying that for neutral α the expression ˙ v α depends only on the ideal K ( α );these give rise to Li’s C ∗ s ( P ). We then introduced the additional relations (T4) saying that a productof defect projections must vanish if it does in the reduced diagonal; these give rise to our T u ( P ). Whenwe add the boundary relations, saying that products corresponding to proper S -foundation sets mustvanish, Theorem 6.13(3), we obtain C (cid:111) C P P . No more relations of this type can be added without causing a total collapse, by Proposition 6.9. In view of Theorem 6.13(5), we will say that C (cid:111) C P P isthe full boundary quotient of T u ( P ). Notation 6.15.
We will denote by Λ ∂ the canonical ∗ -homomorphism C (cid:111) C P P → ∂ T λ ( P ) obtainedby combining the isomorphisms C (cid:111) C P P ∼ = C( ∂ Ω P ) (cid:111) G and C( ∂ Ω P ) (cid:111) r G ∼ = ∂ T λ ( P ) with the leftregular representation Λ : C( ∂ Ω P ) (cid:111) G → C( ∂ Ω P ) (cid:111) r G. Corollary 6.16.
Let P be a submonoid of a group G . Suppose that G is exact. Then a ∗ -homomorphism ρ : T λ ( P ) → B factors through the boundary quotient ∂ T λ ( P ) if and only if ρ (cid:16) Y R ∈C ( S − R ) (cid:17) = 0 whenever C is a foundation set for S ∈ J .Proof. Recall that ( D r /D o r ) (cid:111) ˙ γ,r G is canonically isomorphic to ∂ T λ ( P ), see Lemma 6.12 and thepreceding comment. View T λ ( P ) as the reduced partial crossed product D r (cid:111) γ,r G via the isomorphismgiven in Theorem 4.7. Since T λ ( P ) carries a faithful conditional expectation onto D r , the inclusion D o r , → D r induces an embedding D o r (cid:111) γ (cid:22) D o r ,r G , → T λ ( P ) by [20, Proposition 21.3]. Now supposethat G is exact. By [20, Theorem 21.18], there is a short exact sequence0 −→ D o r (cid:111) γ (cid:22) D o r ,r G −→ T λ ( P ) −→ ( D r /D o r ) (cid:111) ˙ γ,r G −→ . Since ( D r /D o r ) (cid:111) ˙ γ,r G ∼ = C( ∂ Ω P ) (cid:111) r G ∼ = ∂ T λ ( P ), the result follows from an application of Corollary 6.6because D o r (cid:111) γ (cid:22) D o r ,r G is precisely the ideal of T λ ( P ) ∼ = D r (cid:111) γ,r G generated by D o r . (cid:3) As an application of Theorem 6.13, we can characterize the situation in which the universal Toeplitzalgebra coincides with the full boundary quotient.
Corollary 6.17.
Let P be a submonoid of a group G . The following are equivalent: (1) the quotient map q u : T u ( P ) → C (cid:111) C P P is an isomorphism; (2) no finite collection of proper constructible right ideals is a foundation set for P ; (3) for every nonempty constructible right ideal S in P , no finite collection of constructible rightideals is a proper foundation set for S .Proof. That (1) is equivalent to (3) is a consequence of the presentation given in Theorem 6.13(3) for
C (cid:111) C P P , and that (3) implies (2) is obvious.Next we prove that (2) implies (3) by contrapositive. Suppose there is a nonempty constructibleideal S ∈ J and a finite collection of ideals C ⊂ J such that C is a proper foundation set for S . Take p ∈ S \ (cid:16) [ R ∈C R (cid:17) . We claim that { p − R ∩ P | R ∈ C} is a proper foundation set for P . Indeed, let q ∈ P . Then pq ∈ S and so pqP ∩ (cid:0) S R ∈C R (cid:1) = ∅ since C is a foundation set for S . Thus qP ∩ (cid:0) [ R ∈C p − R ∩ P (cid:1) = ∅ , and hence { p − R ∩ P | R ∈ C} is a proper foundation set for P because e ∈ P \ (cid:0) S R ∈C p − R ∩ P (cid:1) . (cid:3) Characterization of purely infinite simple boundary quotients.
Let P be a submonoidof G and let G be the subgroup of G given by G = { g ∈ G | gP ∩ S = ∅ , g − P ∩ S = ∅ , for all ∅ 6 = S ∈ J } . By [15, Proposition 5.7.13], the partial action of G on ∂ Ω P is topologically free if and only if itsrestriction to G is so. We give next equivalent characterizations of topological freeness of this actionin terms of constructible ideals of P . Proposition 6.18.
Let P be a submonoid of a group G . The following statements are equivalent: (1) The partial action of G on ∂ Ω P is topologically free; (2) The partial action of G on ∂ Ω P is topologically free; OEPLITZ ALGEBRAS OF SEMIGROUPS 35 (3)
For every g ∈ G , g = e , and for every p ∈ P , there is q ∈ pP with qP ∩ gqP = ∅ ;(4) For all p, t ∈ P with p = t , there is s ∈ P such that psP ∩ tsP = ∅ ; (5) Every proper ideal of
C (cid:111) C P P is contained in the kernel of the canonical map Λ ∂ : C (cid:111) C P P → ∂ T λ ( P ) . Proof.
The equivalence (1) ⇔ (2) is [15, Proposition 5.7.13] and (1) ⇔ (5) follows from [1, Theorem 4.5].So we prove (2) ⇔ (3) ⇔ (4). Let g ∈ G , g = e , and p ∈ P . Consider the open subset V ( pP ) := V ( pP ; ∅ ) ∩ ∂ Ω P = { τ ∈ ∂ Ω P | τ ( pP ) = 1 } . We claim that V ( pP ) ∩ U g − = ∅ . Indeed, considering that pP is a nonempty constructible ideal and g ∈ G , take s ∈ g − P ∩ pP and let t ∈ P be such that s = g − t . Put α := ( e, t, s, e ). Then ˙ α = g and K ( α ) = P ∩ sP ∩ st − P = sP , so that sP ∈ A g − . We deduce that any character τ ∈ ∂ Ω P satisfying τ ( sP ) = 1 lies in V ( pP ) ∩ U g − . Hence V ( pP ) ∩ U g − is nonempty.Now using that the action of G on ∂ Ω P is topologically free and g = e , we can find τ ∈ V ( pP ) ∩ U g − such that τ = ˆ γ g ( τ ) = τ ◦ γ g − . Thus for some β ∈ W with ˙ β = g − we have τ ( K ( β )) = ( τ ◦ γ g − )( K ( β )) = τ ( g − K ( β )) = τ ( K ( ˜ β )) . Since τ is a character and τ ( pP ) = 1, this implies that τ ( pP ∩ K ( β )) = τ ( pP ∩ K ( ˜ β )). If τ ( K ( β )) = 1,we see that pP ∩ K ( ˜ β ) ∩ K ( β ) is not a foundation set for pP ∩ K ( β ) since τ ∈ ∂ Ω P . So there mustbe q ∈ pP ∩ K ( β ) with qP ∩ K ( ˜ β ) = ∅ . It follows that qP ∩ g − qP = ∅ and thus qP ∩ gqP = ∅ aswished, because g − qP ⊂ g − K ( β ) = K ( ˜ β ). In the case τ ( K ( ˜ β )) = 1 and τ ( K ( β )) = 0, we exchangethe roles of β and ˜ β and reason as above to obtain an element q ∈ pP ∩ K ( ˜ β ) with qP ∩ K ( β ) = ∅ .This implies qP ∩ gqP = ∅ , completing the proof of (2) ⇒ (3).Assume now that condition (3) holds. Let e = g ∈ G and let V ( S ; C ) ∩ ∂ Ω P be a basic open set ofthe topology of ∂ Ω P as in Lemma 6.12. Because { R | R ∈ C} is not a relative S -foundation set, thereis p ∈ S satisfying pP ∩ (cid:18) [ R ∈C R (cid:19) = ∅ . Take s ∈ pP ∩ g − P and let t ∈ P be such that s = g − t . Then sP ∈ A g − because sP = K ( α ),where α = ( e, t, s, e ). Using our hypothesis, we can find q ∈ sP ⊂ pP such that qP ∩ gqP = ∅ . Wehave qP ∈ A g − since q ∈ sP . Let τ ∈ ∂ Ω P be a character with τ ( qP ) = 1. Then τ ∈ V ( S ; C ) ∩ U g − because q ∈ sP ⊂ pP ⊂ S . Also, qP ∩ gqP = ∅ gives 1 = ˆ γ g ( τ )( gqP ) = τ ( qP ) = τ ( gqP ) = 0. Weconclude that for every e = g ∈ G , the set { τ ∈ U g − ∩ ∂ Ω P | ˆ γ g ( τ ) = τ } has empty interior. Thus the action of G on ∂ Ω P is topologically free. We have established (2) ⇔ (3).In order to prove the implication (3) ⇒ (4), take p, t ∈ P with p = t . Suppose first that there exists q ∈ pP such that qP ∩ tP = ∅ . Setting s := p − q , we obtain psP ∩ tsP ⊂ qP ∩ tP = ∅ . In case there is q ∈ tP satisfying qP ∩ pP = ∅ , we also have psP ∩ tsP = ∅ with s := t − q . Otherwise,if those two previous cases do not occur, then pP ∩ tP is a foundation set for both pP and tP . Weclaim that p − t ∈ G . Indeed, let q ∈ P . Using that pP ∩ tP is a tP -foundation set, take r, s ∈ P such that ps = tqr . Then qr = t − ps ∈ t − pP and so qP ∩ t − pP = ∅ . Because pP ∩ tP is also afoundation set for pP , one can similarly show that qP ∩ p − tP = ∅ . Since q ∈ P is arbitrary, we obtain p − t ∈ G , proving the claim. Applying (3) with g = p − t and the identity e playing the role of p , wecan find q ∈ P such that p − tqP ∩ qP = ∅ . Thus pqP ∩ tqP = ∅ as wished. This gives (3) ⇒ (4).Next we prove (4) ⇒ (3). Let g ∈ G , g = e , and p ∈ P . Using that g ∈ G , we can find r ∈ pP ∩ g − P .Let t ∈ P be such that r = g − t . Thus g = tr − and notice that r = t because g = e . By (4),there is s ∈ P such that tsP ∩ rsP = ∅ . Put q := rs . Then q ∈ pP and gqP = tr − rsP = tsP . So gqP ∩ qP = ∅ . This establishes the implication (4) ⇒ (3) and completes the proof of the proposition. (cid:3) It follows from [15, Corollary 5.7.17] that ∂ T λ ( P ) is purely infinite simple provided the partialaction of G on ∂ Ω P is topologically free and P = { e } . By recent work of Abadie–Abadie [1], there isa converse to [15, Corollary 5.7.17] whenever the full and reduced partial crossed products associatedwith the partial action of G on ∂ Ω P are the same. Combining this with the criteria for topological freeness given in Proposition 6.18, we then get a characterization of purely infinite simple boundaryquotients in terms of properties of the semigroup. We specify this in the next corollary. Corollary 6.19.
Let P be a submonoid of a group G with P = { e } . Any of the conditions fromProposition implies that ∂ T λ ( P ) is purely infinite simple. The converse implication also holds ifwe further have C (cid:111) C P P ∼ = ∂ T λ ( P ) via the canonical map Λ ∂ .Proof. If the partial action of G on ∂ Ω P is topologically free, then ∂ T λ ( P ) is purely infinite simpleby [15, Corollary 5.7.17]. This gives the first assertion in the statement. For the last assertion, supposethat Λ ∂ in an isomorphism and that ∂ T λ ( P ) is purely infinite simple. It follows that the left regularrepresentation Λ : C( ∂ Ω P ) (cid:111) G → C( ∂ Ω P ) (cid:111) r G is an isomorphism and C( ∂ Ω P ) (cid:111) G has no nontrivialproper ideal. Thus the partial action of G on ∂ Ω P is topologically free by [1, Theorem 4.5]. Socondition (1) of Proposition 6.18 is satisfied. This completes the proof of the corollary. (cid:3) Remark . In the case that P is a right LCM monoid that embeds in a group, the characterizationof purely infinite simple boundary quotients in terms of constructible ideals from Corollary 6.19 followsfrom Theorem 4.12 and Theorem 4.15 of [40]. Indeed, by [40, Theorem 4.12] it suffices to verifycondition (4) of Proposition 6.18 for elements p and t in the core submonoid P := { p ∈ P | pP ∩ qP = ∅ for all q ∈ P } ⊂ P to deduce that the action of G on ∂ Ω P is topologically free. Thus if C (cid:111) C P P ∼ = ∂ T λ ( P ), then ∂ T λ ( P )is simple if and only if for all p, t ∈ P , p = t , there exists s ∈ P with psP ∩ tsP = ∅ . This is so becauseif p, t ∈ P are such that e = p − t ∈ G and r ∈ P satisfies pP ∩ tP = rP , then p − r, t − r ∈ P . So p − rsP ∩ t − rsP = ∅ yields pt − rs ∩ rsP = ∅ . Hence condition (4) of Proposition 6.18 restricted topairs of elements in P implies Proposition 6.18(3). In general, if P is not a right LCM monoid we donot know whether Proposition 6.18(4) restricted to elements in the core submonoid implies topologicalfreeness of the partial action of G on ∂ Ω P .7. A numerical semigroup
An additive submonoid P of N such that N \ P is finite is called a numerical semigroup . Aspointed out by Li [15, Section 5.6.5] numerical semigroups do not satisfy independence. We wish todemonstrate the concrete application of condition (T4) to the specific example Σ := N \ { } studiedin [36]. We begin with an explicit description of the set J (Σ) of constructible ideals of Σ. Lemma 7.1.
Every nonempty ideal of
Σ = { , , , , , . . . } is constructible, and J (Σ) = { p + Σ | p ∈ Σ } t { p + (2 + N ) | p ∈ Σ } t { N } . Proof.
We first use Proposition 2.6 to compute two key constructible nonprincipal ideals: K (3 , , ,
3) = Σ ∩ ( − ∩ ( − N ; K (2 , , ,
2) = Σ ∩ ( − ∩ (1 + Σ) = 3 + N .If I ⊂ Σ is a nonempty ideal and m is its smallest element, then m + Σ ⊂ I . If m + 1 / ∈ I , then I = m + Σ, which is in the first set. If m + 1 ∈ I then I = m + N = ( m −
2) + (2 + N ), which is in thesecond set unless m = 3, in which case I = 3 + N . Since p + (2 + N ) = K (0 , p, , , , , p, ∅ / ∈ J (Σ) because Σ is abelian. (cid:3) It is clear that independence can only fail at nonprincipal ideals. We choose the equality2 + N = { , , , , . . . } = { , , , , . . . } ∪ { , , , , . . . } = (2 + Σ) ∪ (3 + Σ) , (7.2)as the basic failure of independence, and then show that all other failures follow from this one. Lemma 7.3.
Suppose that independence fails at the constructible ideal S of Σ , in the sense that S = S R ∈C R for a finite family C of constructible ideals not containing S . Let m = min S . (1) If m = 3 , then S = p + (2 + N ) for p := m − ∈ Σ , and there exist R , R ∈ C such that R = p + (2 + Σ) and R ⊃ p + (3 + Σ) . In this case p + (2 + N ) = S = [ R ∈C R = R ∪ R = ( p + (2 + Σ)) ∪ ( p + (3 + Σ)) , which is simply the p -translate of (7.2) . OEPLITZ ALGEBRAS OF SEMIGROUPS 37 (2) If m = 3 , then S = 3 + N and there exist R , R ∈ C such that R = (3 + Σ) and R ⊃ .In this case N = S = [ R ∈C R = R ∪ R = (3 + Σ) ∪ (4 + Σ) , which is the ( − -translate of (7.2) .Proof. The ideal m + N is the only nonprincipal ideal containing m as its smallest element, so it mustbe equal to S . Let R be an ideal in C that contains m , necessarily as its smallest element; since R = S we must have R = m + Σ. Since m + 1 is in S but not in R , there must be another ideal R ∈ C that contains m + 1, again, necessarily as its smallest element; hence R is either m + 1 + Σ or m + 1 + N . The rest consists of rewriting this in terms of p = m − m = 3. (cid:3) Proposition 7.4.
Suppose { V p } p ∈ Σ is a family of elements of a C ∗ -algebra satisfying the relations (T1)–(T3) together with the extra relation (T4) N : V ∗ V V ∗ V = V V ∗ + V V ∗ − V V ∗ V V ∗ .Then L p V p extends to a ∗ -homomorphism π V : T λ (Σ) → C ∗ ( V p | p ∈ Σ) . Moreover, π V is anisomorphism if and only it V ∗ V V ∗ V = 1 .Proof. Since Σ is a submonoid of the abelian group Z , we know that T λ (Σ) has the universal propertyof Definition 3.6, by [15, Corollary 5.6.45] (see also Corollary 4.10 above). In order to conclude that L p V p extends, it suffices to show that V satisfies relation (T4) in that definition. By Lemma 3.8(2)we only need to consider the cases in which independence fails. So let S = S R ∈C R and m = min S asin Lemma 7.3.If m = 3 we put p = m −
2. If I is a constructible ideal, let E I := ˙ V α for any neutral word α ∈ W (Σ) such that I = K ( α ). Then, using the earlier computation of 2 + N and 3 + N andProposition 2.6(6), we can write E S = E p +(2+ N ) = V p E N V ∗ p = V p ( V ∗ V V ∗ V ) V ∗ p , and similarly E p +(2+Σ) = V p E (2+Σ) V ∗ p = V p ( V V ∗ ) V ∗ p and E p +(3+Σ) = V p E (3+Σ) V ∗ p = V p ( V V ∗ ) V ∗ p . By Lemma 7.3we have 0 ≤ Y R ∈C ( E S − E R ) ≤ ( E S − E R )( E S − E R ) ≤ V p ( E N − E ) V ∗ p V p ( E N − E ) V ∗ p = V p ( V ∗ V V ∗ V − V V ∗ )( V ∗ V V ∗ V − V V ∗ ) V ∗ p , which vanishes because of (T4) N . If m = 3 then0 ≤ Y R ∈C ( E S − E R ) ≤ ( E S − E R )( E S − E R ) ≤ ( E N − E )( E N − E )= ( V ∗ V V ∗ V − V V ∗ )( V ∗ V V ∗ V − V V ∗ )= (cid:0) V ∗ V ( V ∗ V V ∗ V − V V ∗ ) V ∗ V (cid:1)(cid:0) V ∗ V ( V ∗ V V ∗ V − V V ∗ ) V ∗ V (cid:1) = V ∗ V ( V ∗ V V ∗ V − V V ∗ )( V ∗ V V ∗ V − V V ∗ ) V ∗ V , which, again, vanishes because of (T4) N . This completes the proof that V satisfies (T4), giving acanonical ∗ -homomorphism π V : T λ (Σ) → C ∗ ( V ).If π V is injective, then 1 − V ∗ V V ∗ V = 0 because 1 − L ∗ L L ∗ L is the projection onto thesubspace generated by δ ∈ ‘ (Σ). For the converse, assume 1 − V ∗ V V ∗ V = 0 and let K ( α ) bea proper ideal for each α in a finite collection A ⊂ W (Σ) of neutral words. It is easy to see that2 + N is the largest proper ideal of Σ, so V ∗ V V ∗ V = E K (3 , , , ≥ E K ( α ) = ˙ V α for every α ∈ A .Hence Q α ∈ A (1 − ˙ V α ) ≥ − V ∗ V V ∗ V = 0, proving that V is jointly proper. By Theorem 5.11, π V : T λ (Σ) → C ∗ ( V ) is faithful. (cid:3) Since Σ is generated by the elements 2 and 3, we would like to characterize representations satisfying(T1)–(T3) and (T4) N in terms of generating isometries W and W . Notice that (T1) is obviousand (T2) never applies because Σ is abelian, so the issue is to characterize pairs that generate afamily satisfying (T3). Fortunately, in the present case we can rely on the spatial results from [36] to characterize the pairs that yield a presentation of T λ (Σ) and thus obtain the following simplificationof our uniqueness result. Corollary 7.5.
Suppose W and W are two commuting isometries in a C ∗ -algebra having commutingrange projections and satisfying (T3) Σ : W = W and (T4) N : W ∗ W W ∗ W = W W ∗ + W W ∗ − W W ∗ W W ∗ . Then there exists a unique ∗ -homomorphism π W : T λ (Σ) → C ∗ ( W , W ) such that π W ( L ) = W and π W ( L ) = W , and π W is an isomorphism if and only if W ∗ W W ∗ W = 1 . Therefore, all the C ∗ -algebras generated by pairs W and W of isometries as above such that W ∗ W W ∗ W = 1 arecanonically isomorphic.Proof. By [36, Proposition 1.5] W and W generate a representation V of Σ by isometries withcommuting range projections. By [36, Theorem 2.1] V has a decomposition into three subrepre-sentations. Clearly the extra condition W ∗ W W ∗ W = W W ∗ + W W ∗ − W W ∗ W W ∗ has to besatisfied separately in the three subrepresentations. But if we compute with the representation S from[36, Example 1.2] we see that S ∗ S S ∗ S = 1 = S S ∗ = S S ∗ + S S ∗ − S S ∗ S S ∗ . The only alternativeis that the S -component of V is trivial, so V is unitarily equivalent to a unitary representation and amultiple of the left regular representation of Σ on ‘ (Σ), the latter multiple being nontrivial if andonly if W ∗ W W ∗ W = 1, in which case π W is injective. (cid:3) ax + b -monoids of integral domains Basics on ax + b -monoids of integral domains. Let R be an integral domain, which we willalways assume to have a unit 1 = 0, and let R × := R \ { } be its multiplicative semigroup. The ax + b -monoid associated to R is R (cid:111) R × , where the action of R × by endomorphisms of R is given bymultiplication. Hence the operation in R (cid:111) R × is given by( b, a )( d, c ) = ( b + ad, ac ) , b, d ∈ R, a, c ∈ R × . The monoid R (cid:111) R × embeds in the group G := Q (cid:111) Q ∗ , where Q denotes the field of fractions of R . Notation 8.1.
Following [30], we will denote by I ( R ) the set of constructible ring-theoretic ideals of R , I ( R ) = (cid:8) \ g ∈ F gR | F ⊂ Q × , F is finite and 1 ∈ F (cid:9) . Thus, a nonzero ideal I of R lies in I ( R ) if and only if it is the intersection of finitely many principalfractional ideals, which is the same as saying that I × is a constructible ideal in R × .By [30, Lemma 2.11], the nonempty constructible right ideals in R (cid:111) R × are indexed by pairs ( r, I )in which I ∈ I ( R ) and r ∈ R represents a class in R/I . Specifically, when R is not a field, J ( R (cid:111) R × ) = { ( r + I ) × I × | r ∈ R and I ∈ I ( R ) } ∪ {∅} . Topological freeness for ax + b -monoids of integral domains. The following propositiongives algebraic conditions on R that are equivalent to Theorem 5.9(3) for the associated ax + b -monoid R (cid:111) R × , enabling the application of Theorem 5.11. Proposition 8.2.
Let R be an integral domain. Suppose that R/I is finite for every ideal I ∈ I ( R ) .Then R (cid:111) R × satisfies condition (3) of Theorem if and only if for every x ∈ R × and everyfinite (possibly empty) collection C ⊂ I ( R ) of proper constructible ring-theoretic ideals there exists a ∈ R \ S I ∈C I such that x / ∈ aR .Proof. Condition (3) of Theorem 5.9 says that for every unit ( x, u ) ∈ R (cid:111) R ∗ \ { (0 , } and every finitecollection C (cid:111) = { ( r + I ) × I × | r ∈ R, I ∈ C} of proper constructible ideals in R (cid:111) R × , there exists( s, a ) ∈ R (cid:111) R × \ S ( r + I ) × I × ∈C (cid:111) ( r + I ) (cid:111) I × such that ( x, u )( s, a ) P = ( s, a ) P .We claim that the above condition is equivalent to the existence, for every nontrivial unit ( x, u )and every collection C (cid:111) as above, of a ∈ R \ S I ∈C I such that( u − R + x aR. (8.3)First notice that since the condition is to be satisfied by every family C (cid:111) , and since each | R/I | < ∞ ,we may assume without loss of generality that C (cid:111) has been augmented so that all the classes modulo OEPLITZ ALGEBRAS OF SEMIGROUPS 39 each of the I are covered. Thus ( s, a ) ∈ R (cid:111) R × \ S I ∈C S r ∈ R/I ( r + I ) (cid:111) I × for some s if and only if a ∈ R \ S I ∈C I . Recall from Lemma 5.3 that ( x, u )( s, a ) P = ( s, a ) P is equivalent to( x + us, ua ) = ( x, u )( s, a ) = ( s, a )( y, v ) = ( s + ay, av ) ∀ ( y, v ) ∈ P ∗ . The multiplicative parts can always be matched by taking v = u so this is really a condition on theadditive parts, which says that a and s satisfy x + ( u − s = ay for every y ∈ R . This completes theproof of the claim.Suppose now that R (cid:111) R × satisfies condition (3) of Theorem 5.9 and, given x ∈ R × , consider thenontrivial unit ( x,
1) of R (cid:111) R × . By the equivalence proved above, there exists a ∈ R \ S I ∈C I suchthat (8.3) holds, which in this case says that { x } = (1 − R + x aR , so x / ∈ aR .To prove the converse, suppose that C (cid:111) = { ( r + I ) × I × | r ∈ R, I ∈ C} is a finite collection ofproper constructible ideals in R (cid:111) R × , and let ( x, u ) be a nontrivial unit. Assume first x = 0. Thenthere exists a ∈ R \ S I ∈C I such that x / ∈ aR . Since x = ( u − x ∈ ( u − R + x , we conclude that(8.3) holds. Assume now x = 0. Then u − = 0, and there exists a ∈ R \ S I ∈C I such that u − / ∈ aR .Since u − u − ∈ ( u − R + x , we conclude that (8.3) holds in this case too. (cid:3) Boundaries for ax + b -monoids of integral domains. A characterization of pure infinitenessand simplicity of C ∗ -algebras constructed from rings that may have zero-divisors is given in [27,Theorem 2]. As a consequence, for R an integral domain that is not a field, the boundary quotient ∂ T λ ( R (cid:111) R ) is purely infinite simple [27, Corollary 8]. We can give a direct proof of this result, byverifying Proposition 6.18(4) and then applying Corollary 6.19. Corollary 8.4.
Let R be an integral domain that is not a field and let R × R × be the associated ax + b -monoid. Then condition (4) of Proposition holds in R (cid:111) R × , and hence the boundaryquotient ∂ T λ ( R (cid:111) R × ) is purely infinite simple.Proof. In order to lighten the notation, we write P := R (cid:111) R × . Let p, t ∈ P with p = t . We need tofind s ∈ P satisfying psP ∩ tsP = ∅ . Let b, d ∈ R , a, c ∈ R × be such that p = ( b, a ) and t = ( d, c ). Itsuffices to find such an element s when b = d because the case b = d entails a = c and we may take anonzero element r ∈ R and substitute p and t by p ( r,
1) = ( b + ar, a ) , t ( r,
1) = ( d + cr, c ) , which satisfy b + ar = d + cr . Finding s for the pair of elements p ( r,
1) and t ( r, s = ( r, s for the pair of elements p and t . Assuming thus b − d = 0, we separate the proof of existence of s intotwo cases. Case 1: b − d acR . Set s := (0 , ac ). Then ps = ( b, a c ) and ts = ( d, c a ) . We claim that psP ∩ tsP = ∅ . Indeed, looking for a contradiction, assume that there are q = ( f , e )and q = ( f , e ) in P such that psq = tsq . We would then have psq = ( b + a cf , a ce ) and tsq = ( d + c af , c ae ) . Thus the equality psq = tsq would imply b + a cf = d + c af and hence b − d = ac ( cf − af ) . This contradicts our assumption that b − d acR . Case 2: b − d ∈ acR × . Let ¯ x ∈ R × be the unique element satisfying b − d = ac ¯ x . Let r ∈ R be anoninvertible element. We set s := (0 , ac ¯ xr ). Thus ps = ( b, a c ¯ xr ) and ts = ( d, c a ¯ xr ) . We claim that psP ∩ tsP = ∅ . Looking for a contradiction, assume that there are q = ( f , e ) and q = ( f , e ) in P such that psq = tsq . In this case we would have that b + a c ¯ xrf = d + c a ¯ xrf , and hence b − d = ac ¯ x ( crf − arf ) = ( b − d )( crf − arf ) . Since R has no zero-divisors we would have crf − arf = 1 and thus r ( cf − af ) = 1. We havearrived at a contradiction because r is not invertible in R × . So we must have psP ∩ tsP = ∅ . We have shown that Proposition 6.18(4) holds, so Corollary 6.19 gives the rest. (cid:3) ax + b -monoids of orders in number fields Recall that an algebraic number field K is a finite degree extension of the field Q of rational numbers.The ring of algebraic integers O K of K is the integral closure of Z in K ; it is a free Z -module of rank d := [ K : Q ]. The rings of integers in algebraic number fields form an important class of integraldomains, in fact of Dedekind domains. The Toeplitz C ∗ -algebras of the corresponding ax + b -monoidsand their equilibrium states were studied in [14], and their ideal structure and K -theory in [17] and[30], respectively.Here we will be interested in the more general class of orders in K . These are subrings of O K thatare free of full rank d as Z -modules. We will use the letter O to denote an order in a number field; O ∗ will denote the set of invertible elements of O and O × the multiplicative monoid O \ { } . Thefraction field of O equals that of O K , so that ( O × ) − O = K . We refer to [33] for the basic definitionsand results about rings of integers in number fields and to [41] for orders; we also found K. Conrad’sset of notes [8] very helpful.9.1. A uniqueness result for T λ ( O (cid:111) O × ) . As an application of Proposition 8.2 and Theorem 5.11,we prove next a uniqueness result for the Toeplitz C ∗ -algebra T λ ( O (cid:111) O × ) of the ax + b -monoid of anorder O in a number field K . Corollary 9.1.
Let O be an order in an algebraic number field K , and let W : O (cid:111) O × → B bea map into a C ∗ -algebra B satisfying relations (T1)–(T4) from Definition . Then there is a ∗ -homomorphism π W : T λ ( O (cid:111) O × ) → C ∗ ( W ) such that π W ( L p ) = W p for all p ∈ O (cid:111) O × , and π W is an isomorphism if and only if W is jointly proper.Proof. We will establish that the assumptions of Theorem 5.11 hold for O (cid:111) O × . First, since thegroup K (cid:111) K ∗ is amenable, Corollary 4.10 and Theorem 4.9 show that the left regular representation λ + is faithful. Then so is the conditional expectation E u : T u ( O (cid:111) O × ) → D u by Corollary 3.20.Next we show that O (cid:111) O × satisfies condition (3) of Theorem 5.9. Since O satisfies the assumptionsof Proposition 8.2, it suffices to show that for every x ∈ O × and every finite collection C ⊂ I ( O )of proper ideals there exists a ∈ O \ S S ∈C S such that x / ∈ a O . Let x ∈ O × and C be such acollection. Suppose first that x is not invertible, that is, x O is a proper ideal in O . The collection C x := { S | S is a proper ideal in O and x ∈ S } is finite because O /x O is finite. Consider now thefinite collection C ∪ C x . Since each proper ideal in O can contain at most one rational prime, we maychoose a prime number p ∈ N such that p / ∈ (cid:16) [ S ∈C S (cid:17) ∪ (cid:16) [ S ∈C x S (cid:17) . Then p ∈ O \ S S ∈C S , and x / ∈ p O because, otherwise, p O would be a proper ideal between x O and O ,hence in C x and this would contradict the choice of p . Suppose now that x is invertible, that is, x O = O and just take a rational prime p / ∈ (cid:16) [ S ∈C S (cid:17) . Then p ∈ O \ S S ∈C S , and x / ∈ p O because 1 /p is not an algebraic integer.We have verified that the conditional expectation E u : T u ( O (cid:111) O × ) → D u is faithful, and that theequivalent conditions of Theorem 5.9 hold, so Theorem 5.11 completes the proof. (cid:3) Remark . The key feature in the proof of Theorem 9.1 is the availability of an infinite set of rationalprimes. So a slight modification of the proof also yields a uniqueness theorem for representationsof the ax + b -monoid of a congruence monoid. See [5, Theorem 6.1] for a sharper result that onlyrequires the jointly proper condition at a restricted class of ideals. OEPLITZ ALGEBRAS OF SEMIGROUPS 41
Independence fails for every nonmaximal order.
The multiplicative monoid O × K and the ax + b -monoid O K (cid:111) O × K of the ring O K of algebraic integers in a number field K both satisfyindependence [15, Lemma 5.6.36], see also [28, Lemma 2.30]. We will show that, in contrast, when O is a nonmaximal order in K , then O × does not satisfy independence, and neither does O (cid:111) O × ,because [30, Lemma 2.12] shows that independence of R × and of R (cid:111) R × are equivalent for all integraldomains. It follows that the behaviour exhibited by the example Z [ √−
3] given in [15, Section 5.6.5] istypical for nonmaximal orders. We will discuss this example later in detail following a recipe thatcould be used with every nonmaximal order, see Example 9.6.Let O be an order in an algebraic number field K . The conductor of O is given by c := ( O : O K ) = { x ∈ K | x O K ⊂ O} , where we use ( R : I ) := { x ∈ Q | xR ⊂ I } to indicate the ideal quotient of an integral domain R withquotient field Q over a fractional ideal I . Notice that c is an ideal in both O and O K , and is actuallythe largest ideal of O K contained in O . Since O has full rank in O K , there is a positive integer m ∈ Z such that m O K ⊂ O . For example, one such integer is the index [ O K : O ]. In particular, wemay indeed regard O K as a fractional O -ideal. Whenever m O K ⊂ O , then m O K is an ideal of O K contained in the conductor c of O .A key concept for us to analyze the failure of independence in O is that of a divisorial fractionalideal. Recall that a nonzero fractional ideal I of an integral domain R is said to be divisorial whenever( R : ( R : I )) = I ; equivalently, whenever I is the intersection of an arbitrary collection of principalfractional ideals. Notice that a nonzero ideal I of a noetherian integral domain R is divisorial ifand only if it is a constructible ring-theoretic ideal, e.g. if and only if I × is a constructible ideal in R × , because the arbitrary intersection can be replaced by a finite one. Interested in the question ofwhether m O K is a constructible ring-theoretic ideal of O , we prove next a number-theoretic resultthat is possibly well-known to the experts, but for which we have been unable to find a reference. Lemma 9.3.
Let K be an algebraic number field and O K the ring of integers in K . Let O be anorder in K . Then O K is divisorial as a fractional O -ideal.Proof. Of course we may assume that O is a nonmaximal order in K . We have to show that O K = ( O : ( O : O K )) = ( O : c ) = { x ∈ K | x c ⊂ O} . Let g ∈ K be such that g c ⊂ O . If h ∈ c , then h O K ⊂ c and so gh O K ⊂ g c ⊂ O . In particular, gh ∈ c and hence g c is contained in c , not just in O .Now since O K is a Dedekind domain and c ⊂ O K is a nonzero ideal, it follows that c is invertible(as a fractional O K -ideal) and so there exists a fractional O K -ideal c − with c − c = O K . Thus g O K = c − g c ⊂ c − c = O K . This shows that g O K ⊂ O K and therefore g ∈ O K as wished. (cid:3) In order to prove that O × and O (cid:111) O × do not satisfy independence whenever O is a nonmaximalorder in K , we introduce first some notation. Given h ∈ O K , we denote by n h the positive generatorof the ideal h O K ∩ Z of Z ; equivalently, n h is the least positive integer such that n h O K ⊂ h O K . Then h divides n h in O K and we let h := n h h ∈ O K . Proposition 9.4.
Let O be a nonmaximal order in a number field K . Let m ∈ Z be the least positiveinteger such that m O K ⊂ O and let H ⊂ O K be a complete set of representatives for the nontrivialcosets of m O K in O K . For each h ∈ H define a fractional O -ideal I h by I h := ( h m O ∩ O K if h ∈ c , h O ∩ O K if h c . Then h + m O K ⊂ I h (cid:40) O K and O K = [ h ∈ H I h . As a consequence, the monoids O × and O (cid:111) O × do not satisfy independence. Proof.
We begin by proving that O K = S h ∈ H I h . Since H is a complete set of representatives for thenontrivial cosets of m O K in O K , we have O K = m O K t (cid:16) G h ∈ H (cid:16) h + m O K (cid:17)(cid:17) . Next we show that each nontrivial class h + m O K can be replaced by the corresponding I h . We let h ∈ H and compute h + m O K = n h hn h + n h n h m O K = n h hn h + hh n h m O K = hn h (cid:16) n h + mh O K (cid:17) = 1 h (cid:16) n h + mh O K (cid:17) ⊂ h ( Z + O ) = 1 h O . Therefore h + m O K ⊂ h O ∩ O K . When h belongs to the conductor c of O , we further have that n h = hh ∈ c ∩ Z = m Z and h O K ⊂ c ⊂ O . Thus the inclusion in the last line of the abovecomputation can be strengthened to h (cid:0) n h + mh O K (cid:1) ⊂ h (cid:0) m Z + m O (cid:1) ⊂ h m O . This yields h + m O K ⊂ h m O ∩ O K in the case h ∈ c and completes the proof that O K = m O K ∪ (cid:0) S h ∈ H I h (cid:1) . In order to see that thetrivial class m O K is superfluous in this union, observe that m O K = h + m O K − h ⊂ I h for every h ∈ H . Hence O K = S h ∈ H I h .It remains to show that I h is properly contained in O K for every h ∈ H . Let h ∈ H and supposefirst that h c . Looking for a contradiction, assume that O K = h O ∩ O K . Multiplying this equality by h ∈ O K , we deduce that h O K ⊂ O and so h is in the conductor c . Thiscontradicts our assumption that h c and thus we must have I h = O K whenever h ∈ H satisfies h c .Suppose next that h ∈ c . Again looking for a contradiction, assume that O K = h m O ∩ O K . In this case we deduce that h m O K ⊂ O , and hence h m is in the conductor c of O . In particular h m isin O K , so that n h m = h h m ∈ Q + ∩ O K is a positive integer such that n h m O K = h h m O K ⊂ h O K . Butsince 0 = h h m = n h m < n h because m = 1, this contradicts our choice of n h as the smallest positive integer satisfying n h O K ⊂ h O K .Therefore I h = O K also for h ∈ H with h ∈ c .Finally, by Lemma 9.3 O K is divisorial as a fractional O -ideal, and thus it is a finite intersectionof principal fractional O -ideals because O is a noetherian domain. Hence m O K is a constructiblering-theoretic ideal of O and so is mI h for each h ∈ H ; moreover, mI h = m O K . Thus we haverealized m O × K as a union m O × K = [ h ∈ H mI × h of constructible ideals in O × none of which is equal to m O × K . We conclude that O × and thecorresponding ax + b -semigroup O (cid:111) O × do not satisfy independence. (cid:3) Remark . In an algebraic number field K with maximal order O K , every nonzero fractional O K -idealis invertible, hence divisorial. By [30, Corollary 7.1], among the orders in K for which all fractionalideals are divisorial, O K is the only one that satisfies independence. We have shown in Lemma 9.3that for an arbitrary order O in K the fractional O -ideal O K is divisorial, and this was enough toconclude that nonmaximal orders do not satisfy independence. We do not know whether all fractional OEPLITZ ALGEBRAS OF SEMIGROUPS 43 O -ideals are divisorial, but many are. For example, since the map a a ∩ O is a bijection of the setof (integral) ideals in O K that are relatively prime to the conductor c onto the ideals in O relativelyprime to c , then the latter inherit from the former the property of being divisorial. Since the fractionfield of O is the same as that of O K , it also follows from Lemma 9.3 that the integral ideals of O K that are contained in O are divisorial, in particular, the conductor is divisorial. Example 9.6.
Let K = Q ( √−
3) and consider the order O = Z [ √−
3] in K . So O is a nonmaximalorder as the ring of algebraic integers O K is given by O K = Z [ √− ] = Z + i √ Z = (cid:8) ( x + yi √ | x, y ∈ Z and x − y = 0 (mod 2) (cid:9) because − ≡ O K = (cid:8) a + bi √ | a, b ∈ Z and a − b = 0 (mod 2) (cid:9) ⊂ O , and notice that in this case 2 O K is the conductor c of O . Let ω := √− . Then H := { , ω, ω } ⊂ O K is a complete set of representatives for the nontrivial cosets of 2 O K in O K . Since each h ∈ H isinvertible in O K , it follows that n h = 1, h = h and so h c . Hence Proposition 9.4 gives O K = O ∪ ω O ∪ ω O . Multiplying this equality by 2 and removing the zero element from the ideals involved, we obtain2 O × K = 2 O × ∪ ω O × ∪ ω O × . (9.7)Notice that 2 O × K corresponds to the constructible ideal K (2 , ω ) = K (2 , i √
3) = K (2 ω, , , ω ) of O × because 2 O × K = 2( O × ∩ ω O × ) = O × ∩ ω O × , while the ideals appearing in the union above are simply principal ideals of O × . We have arrivedexactly at the instance of failure of independence for O × = Z [ √− × provided in [15, Section 5.6.5]. Example 9.8.
Let K = Q ( √−
1) and O = Z [2 √−
1] = Z + 2 Z i . In this case O K is the ring ofGaussian integers Z [ √−
1] = Z + Z i . The conductor of O is c = 2 O K = 2 Z + 2 Z i . One can show that H = { , i, i } is a complete set of representatives for the nontrivial cosets of 2 O K in O K . We have2 O K = (1 + i )(1 − i ) O K ⊂ (1 + i ) O K , so that n i = 2. Thus Proposition 9.4 gives O K = O ∪ i O ∪ (cid:0) i O ∩ O K (cid:1) . We then obtain a concrete failure of independence in the monoid O × given by2 O × K = O × ∩ i O × = 2 O × ∪ i O × ∪ (cid:0) (1 + i ) O × ∩ O × ∩ i O × (cid:1) . Example 9.9.
Consider the cubic field K = Q ( √
19) = Q + Q √
19 + Q √ , and let O be the order Z [ √
19] = Z + Z √
19 + Z √ in K , see [8, Example 2.3]. The ring of algebraic integers O K is O K = Z + Z √
19 + Z √
19 + √ . Clearly 3 is the smallest positive integer m with the property that m O K ⊂ O . The conductor c of O is c = { a + b √
19 + c √ | a + b + c ≡ } , while 3 O K consists of the ideal in O given by all elements of the form a + b √
19 + c √
19 with a, b, c integers that are equal modulo 3. Hence 3 O K (cid:40) c .We set ω := √ √ . So { , √ , ω } is a Z -basis for O K . One can show that the set H := { q + q √
19 + q ω | q i ∈ { , , } for i = 1 , , q + q + q = 0 } is a complete set of representatives for the nontrivial cosets of 3 O K in O K . Let h = q + q √ q ω ∈ H and let n ∈ Z . Then h divides n in O K if and only if there are integers r , r and r such that h ( r + r √
19 + r ω ) = n. Thus, given h , we seek the smallest positive integer n h , for which the system q x + (6 q − q ) y + (6 q + 4 q ) z = n h q x + ( q − q ) y + 2 q z = 0 q x + (3 q + q ) y + ( q + q + q ) z = 0 (9.10)has a solution ( r , r , r ) ∈ Z , which gives h = r + r √
19 + r ω .All the possible fractional O -ideals obtained from the elements in H as in Proposition 9.4 aredisplayed below in Table 1. We have eliminated the obvious repetitions resulting from pairs h , h ∈ H with h = 2 h , which yield h = h and hence I h = I h . Notice that the table has three elements h that lie in the conductor, namely − √ − − √
19 + 3 ω and − − √
19 + 18 ω ; their respectiverows are indexed by (0 , , , , , , Table 1. ( q , q , q ) n h h I h (1 , ,
0) 1 1 O (0 , ,
0) 19 − − √
19 + 3 ω √ O ∩ O K (1 , ,
0) 20 − √
19 + 3 ω √ O ∩ O K (2 , ,
0) 9 1 − √
19 + ω √ O ∩ O K (1 , ,
0) 51 − − √
19 + 4 ω √ O ∩ O K (0 , ,
1) 6 − √ ω O ∩ O K (1 , ,
1) 8 2 √ − ω ω O ∩ O K (2 , ,
1) 6 2 + √ − ω ω O ∩ O K (0 , ,
1) 10 − ω √ ω O ∩ O K (1 , ,
1) 27 − − √
19 + 4 ω √ ω O ∩ O K (2 , ,
1) 12 − − √
19 + 3 ω √ ω O ∩ O K (0 , ,
1) 60 − − √
19 + 4 ω √ ω O ∩ O K (1 , ,
1) 66 − − √
19 + 5 ω √ ω O ∩ O K (2 , ,
1) 82 − − √
19 + 7 ω √ ω O ∩ O K (1 , ,
2) 75 − √ − ω ω O ∩ O K (0 , ,
2) 153 −
23 + 5 √
19 + 7 ω √ ω O ∩ O K (1 , ,
2) 94 −
20 + 4 √
19 + 5 ω √ ω O ∩ O K (2 , ,
2) 15 − √
19 + ω √ ω O ∩ O K (1 , ,
2) 303 − − √
19 + 18 ω √ ω O ∩ O K We already know from Lemma 9.3 that O K is a divisorial fractional O -ideal, and hence 3 O K is aconstructible ring-theoretic ideal of O . But it is not difficult to describe them explicitly as O K = − √ O ∩ O and 3 O K = − √ O ∩ O . Indeed, take an arbitrary element x = a + b √
19 + c √ ∈ O and compute(1 − √ x = (1 − √ a + b √
19 + c √ ) = a − c + ( b − a ) √
19 + ( c − b ) √ . Observe that a − c ≡ a − c (mod 3), and so − √ x ∈ O if and only if x ∈ O K . Consequently,3 O K = − √ O ∩ O
OEPLITZ ALGEBRAS OF SEMIGROUPS 45 as claimed and thus O K = − √ O∩ O . This also shows that 3 O × K = K (3 , − √
19) as a constructibleideal of O × . By Proposition 9.4, after multiplying by 3 and removing the zero element, each fractional O -ideal in the last column of Table 1 yields a constructible ideal of O × that is properly containedin 3 O × K . The union of all these ideals is 3 O × K . This illustrates a concrete failure of independence in O × .9.3. Reduction of relation (T4) for Z [ √− (cid:111) Z [ √− × . In analogy to what we saw in Section 7,it sometimes suffices to verify relation (T4) from Definition 3.6 on a particular instance of failure ofindependence also for ax + b -monoids of orders. We would like to illustrate this by examining in moredetail the example Z [ √− F is a finite collection of fractional O -ideals such that S (cid:40) O K for each S ∈ F and O K = [ S ∈F S, then we must have {O , ω O , ω O} ⊂ F . The constructible ideals of O × are given in [31], see also[41, Example 4.2]; specifically, J ( O × ) = { x O × | x ∈ O × } ∪ { y O × K | y ∈ O × K } . Yet another description of J ( O × ) will be more convenient for our purposes: because O K = S j =0 ω j O ,an arbitrary element y ∈ O × K has the form y = ω j x for some j ∈ { , , } and x ∈ O × . Using that ω ∈ O ∗ K and thus ω O K = O K , we may rewrite J ( O × ) as J ( O × ) = { x O × | x ∈ O × } ∪ { x O × K | x ∈ O × } . It will also be convenient to represent J ( O × ) in terms of neutral words in W ( O × ) by J ( O × ) = { K (1 , x, x, | x ∈ O × } ∪ { K (1 , x, ω, , , ω, x, | x ∈ O × } . We observe that [31, Lemma 6.3] also implies that if x ∈ O × and C ⊂ J ( O × ) is a finite collectionof constructible ideals such that 2 x O × K = ∪ S ∈C S and S (cid:40) x O × K for all S ∈ C , then C must contain { x O × , xω O × , xω O × } . Therefore, all instances of failure of independence for O × can be obtainedfrom (9.7) via translations by elements in O × . This leads to the following proposition at the level ofrepresentations of O × . Proposition 9.11.
Let K = Q ( √− and O = Z [ √− . Let w : O × → B be a map into a C ∗ -algebra B satisfying relations (T1)–(T3) from Definition . Then w satisfies relation (T4) if and only if w satisfies (T4) at (9.7) , that is, Y j =0 ( w ∗ ω w w ∗ w ω − w ω j w ∗ ω j ) = 0 . Proof.
The "only if" direction is clear. In order to prove the converse, suppose Y j =0 ( w ∗ ω w w ∗ w ω − w ω j w ∗ ω j ) = 0 . By Lemma 3.8, all we need to show is that w satisfies (T4) at special cases in which the independencecondition fails. Let α ∈ W ( O × ) be a neutral word and let F ⊂ W ( O × ) be a finite set of neutral wordssuch that K ( β ) (cid:40) K ( α ) for all β ∈ F and K ( α ) = [ β ∈ F K ( β ) . We may assume that α = (1 , x, ω, , , ω, x,
1) for some x ∈ O × because w satisfies relations (T1)–(T3) from Definition 3.6 and the independence condition can only fail at the nonprincipal ideals of O × .By the discussion preceding the statement of the proposition, we deduce that { x O × , xω O × , xω O × } ⊂ { K ( β ) | β ∈ F } . Thus, for each j ∈ { , , } , we can find a word β j ∈ F with K ( β j ) = 2 xω j O × . By relations (T1) and(T3), we have ˙ w β j = w xω j w ∗ xω j = w x w ω j w ∗ ω j w ∗ x , ( j = 0 , , . Also, ˙ w α = w x w ∗ ω w w ∗ w ω w ∗ x . Then Y j =0 ( ˙ w α − ˙ w β j ) = w x (cid:16) Y j =0 ( w ∗ ω w w ∗ w ω − w ω j w ∗ ω j ) (cid:17) w ∗ x = 0 . Hence Y β ∈ F ( ˙ w α − ˙ w β ) = Y j =0 ( ˙ w α − ˙ w β j ) Y β ∈ F ( ˙ w α − ˙ w β ) = 0as wished. So w satisfies relation (T4) from Definition 3.6. (cid:3) We now turn our attention to the ax + b -monoid O (cid:111) O × . It follows from [30, Lemma 2.11] thatthe family of constructible right ideals of O (cid:111) O × is given by J ( O (cid:111) O × ) = { ( r + x O ) × x O × | x ∈ O × , r ∈ O} ∪ { ( r + 2 x O K ) × x O × K | x ∈ O × , r ∈ O} ∪ {∅} = { ( r, x )( O (cid:111) O × ) | r ∈ O , x ∈ O × } ∪ { ( r, x )(2 O K (cid:111) O × K ) | r ∈ O , x ∈ O × } ∪ {∅} . The nonprincipal ideal ( r, x )(2 O K (cid:111) O × K ) equals K ( α ), where α ∈ W ( O (cid:111) O × ) is the neutral (andsymmetric) word α = ((0 , , ( r, x ) , (0 , ω ) , (0 , , (0 , , (0 , ω ) , ( r, x ) , (0 , . In O (cid:111) O × , we find a failure of independence at the ideal 2 O K (cid:111) O × K because2 O K × O × K = (cid:16) [ j =0 ω j O × ω j O × (cid:17) ∪ (cid:16) [ j =0 ( r j + 2 ω j O ) × ω j O × (cid:17) , where r j ∈ O K is any element in the unique nontrivial class of the quotient ring 2 O K / ω j O foreach j = 0 , ,
2. Because 2 O K = 2 ω j O t ( r j + 2 ω j O ) for each j ∈ { , , } , another application of[31, Lemma 6.3] shows that if F ⊂ J ( O (cid:111) O × ) is a finite collection of constructible right ideals suchthat 2 O K × O × K = S S ∈F S , then F contains { ω j O × ω j O × , ( r j + 2 ω j O ) × ω j O × } for j = { , , } . An analogue of this fact also holds with an arbritary constructible ideal ( r + 2 x O K ) × x O × K ∈ J ( O (cid:111) O × ) in place of 2 O K × O × K and { xω j O × xω j O × , ( r + xr j + 2 xω j O ) × xω j O × } in place of { ω j O × ω j O × , ( r j + 2 ω j O ) × ω j O × } for j = 0 , , ax + b -monoid O (cid:111) O × , let us first introducesome notation, following [14]. Given an isometric representation w : O (cid:111) O × → B in a C ∗ -algebra B ,we regard O as a group with its additive operation and let u : O → B be the unitary representationgiven by r w ( r, . Similarly, w gives rise to an isometric representation s : O × → B via x w (0 ,x ) .The next result is at the same time a simplification of relation (T4) for O (cid:111) O × and an application ofCorollary 9.1. Corollary 9.12.
Let K = Q ( √− and O = Z [ √− . Let W : O (cid:111) O × → B be a map into a C ∗ -algebra satisfying relations (T1)–(T3) from Definition . Let s and u be the restrictions of W tothe multiplicative and additive parts of O (cid:111) O × , respectively, and suppose that Y j =0 ( s ∗ ω s s ∗ s ω − s ω j s ∗ ω j ) Y j =0 ( s ∗ ω s s ∗ s ω − u r j s ω j s ∗ ω j u ∗ r j ) = 0 , where r j is any element in the nontrivial class of O K / ω j O for j = 0 , , . Then there is a ∗ -homomorphism π W : T λ ( O (cid:111) O × ) → B that sends L ( r,x ) to W ( r,x ) . Moreover, π W is faithful if andonly if for every finite set F of primes in O \ O K , one has Q F := (cid:16) − ( X r ∈O / O K u r s ∗ ω s s ∗ s ω u ∗ r ) (cid:17) Y p ∈ F (cid:16) − ( X r ∈O /p O u r s p s ∗ p u ∗ r ) (cid:17) = 0 . (9.13) OEPLITZ ALGEBRAS OF SEMIGROUPS 47
Proof.
In order to see that w satisfies relation (T4) from Definition 3.6, observe that if α is a neutralword in W ( O (cid:111) O × ) with K ( α ) = ( r + 2 x O K ) × x O × K , where r ∈ O and x ∈ O × , then˙ W α = u r s x s ∗ ω s s ∗ s ω s ∗ x u ∗ r since w satisfies relations (T1)–(T3) from Definition 3.6. Similarly, if β ∈ W ( O (cid:111) O × ) is neutral and K ( β ) = ( r + xr j + 2 xω j O ) × xω j , then˙ W β = u r s x u r j s ω j s ∗ ω j u ∗ r j s ∗ x u ∗ r . Hence, the same reasoning used to prove Proposition 9.11 shows that w satisfies relation (T4). ByCorollary 9.1, there exists a ∗ -homomorphism π W : T λ ( O (cid:111) O × ) → B mapping a canonical generator L ( r,x ) to W ( r,x ) = u r s x , and π W is faithful if and only if W is jointly proper. Thus all we need toprove is that W is jointly proper if and only if (9.13) holds for every finite set F of primes in O \ O K .The ‘only if’ direction is clear. For the converse, suppose that (9.13) holds for every finite set F of primes in O \ O K . Since 2 O K is a maximal ideal in O , it follows that an ideal in O is eithercontained in 2 O K or it is relatively prime to 2 O K . If I is a proper ideal in O that is relatively primeto 2 O K , then I is contained in a prime ideal p of O that is itself relatively prime to 2 O K because O ⊃ p + 2 O K ⊃ I + 2 O K = O . Since the ideals of O that are relatively prime to 2 O K are principal ideals, we have p = p O for aprime element p ∈ O .Now take a constructible ideal S = ( r + I ) × I × in J ( O (cid:111) O × ) with S (cid:40) O (cid:111) O × . By the aboveparagraph, we have either S ⊂ ( r + 2 O K ) × O × K or S ⊂ ( r + p O ) × p O × for some prime element p ∈ O that is relatively prime to 2 O K . Then if α ∈ W ( O (cid:111) O × ) is a neutral word with K ( α ) = S , itfollows that1 − ˙ W α ≥ (cid:16) − (cid:0) X r ∈O / O K u r s ∗ ω s s ∗ s ω u ∗ r (cid:1)(cid:17)(cid:16) − (cid:0) X r ∈O /p O u r s p s ∗ p u ∗ r (cid:1)(cid:17) = Q { p } . for some prime p ∈ O \ O K . So if we take a finite set A of neutral words in O (cid:111) O × such that K ( α ) (cid:40) O (cid:111) O × for all α ∈ A , we can find a finite set F ⊂ O \ O K of primes such that Y α ∈ A (1 − ˙ W α ) ≥ Y α ∈ A (1 − ˙ W α ) Q F = Q F = 0 . This shows that W is jointly proper. (cid:3) Right LCM semigroups
Our results give new insight also in the particular – and important – case of right LCM semigroups.In order to demonstrate this point we present in this section two applications of our uniqueness resultsto the universal Toeplitz algebras of right LCM submonoids of groups.10.1.
Topological freeness and uniqueness.
The first application stems from the observation thatthere is an obvious parallel between Theorem 5.11 and [3, Theorem 4.3] because the condition givenin Equation (4.1) in [3] corresponds to our joint properness condition, Definition 3.14, when appliedto right LCM semigroups. We would like to elaborate on this parallel here in order to provide asimplification of the hypothesis and a strengthening of [3, Theorem 4.3]. Along the way, we also shedconceptual light on the technical condition (D2) from [3, Definition 4.1]:(D2) if s ∈ P , s ∈ s P and F ⊂ P is a finite subset with s P ∩ (cid:16) P \ S q ∈ F qP (cid:17) = ∅ , then forevery x ∈ P ∗ \ { e } , there is s ∈ s P satisfying s P ∩ (cid:16) P \ [ q ∈ F qP (cid:17) = ∅ and s − s P ∩ xs − s P = ∅ . Recall that when P is a right LCM semigroup, the nonempty constructible right ideals of P areall of the form qP , q ∈ P , hence an ideal qP ∈ J is proper if and only if q ∈ P \ P ∗ . Thus, it seemsworth recasting Theorem 5.9 specifically for right LCM monoids, giving algebraic conditions on P that are equivalent to topological freeness of the partial action of the underlying group. Theorem 10.1.
Let P be a right LCM submonoid of a group G . The following are equivalent: (1) the partial action of G on Ω P is topologically free; (2) the action of P ∗ on Ω P is topologically free; (3) if u ∈ P ∗ \ { e } and F ⊂ P \ P ∗ is a finite set, then there exists t ∈ P \ S q ∈ F qP such that utP = tP (or, equivalently, such that ut / ∈ tP ∗ ); (4) every ideal of T u ( P ) that has trivial intersection with D u is contained in the kernel of the leftregular representation. When we apply this characterization, we see that (D2) implies topological freeness.
Corollary 10.2.
Let P be a right LCM submonoid of a group. If P satisfies (D2) , then the action of P ∗ on Ω P is topologically free.Proof. It suffices to show that condition (3) of Theorem 10.1 holds. Suppose u ∈ P ∗ \ { e } and let F ⊂ P \ P ∗ be a finite set. When we apply (D2) to the set F and to s = e , s = e and x = u weobtain an element s ∈ s P = P such that s P ∩ ( P \ S q ∈ F qP ) = ∅ and us P ∩ s P = ∅ . Thatis, us P and s P are disjoint subsets of P . This implies condition (3) of Theorem 10.1, which onlyrequires those subsets to be different. (cid:3) If P is a right LCM monoid, an isometric representation w : P → B induces a representation of T u ( P ) if and only if it satisfies the relation w p w ∗ p w q w ∗ q = ( w r w ∗ r if pP ∩ qP = rP, pP ∩ qP = ∅ . (10.3)The diagonal subalgebra D u ⊂ T u ( P ) is the closed linear span of the range projections { t p t ∗ p | p ∈ P } .We simplify Theorem 5.11 when P is a right LCM monoid, and obtain, in particular, the group-embeddable case of [3, Theorem 4.3] as a consequence of Corollary 10.2. Theorem 10.4 (cf. [3, Theorem 4.3]) . Let P be a right LCM submonoid a group G . Supposethat any of the equivalent conditions of Theorem holds and that the conditional expectation E u : T u ( P ) → D u is faithful. Let W : P → B be an isometric representation of P satisfying (10.3) .Then the canonical map ρ W : T u ( P ) → C ∗ ( W ) is an isomorphism if and only if Y p ∈ F (1 − W p W ∗ p ) = 0 for every finite subset F ⊂ P \ P ∗ . It is important to keep in mind that [3, Theorem 4.3] also applies to C ∗ -algebras of right LCMsemigroups that do not embed in groups, which are not covered by our results. It is nonethelessplausible that replacing (D2) by Theorem 10.1(3) would still produce a version of Theorem 10.4 alsofor semigroups that do not embed in groups.10.2. Pure infiniteness and simplicity.
We now turn our attention to the main findings in [3] con-cerning pure infiniteness and simplicity of the Toeplitz algebra of a right LCM semigroup. Specifically,we aim to show that for P a right LCM submonoid of a group, the conclusion of [3, Theorem 5.3]follows from a combination of Theorem 5.9, Corollary 6.17 and Corollary 6.19. We start by interpretingCorollary 6.17 in the special case of a right LCM monoid. Corollary 10.5.
Let P be a right LCM submonoid of a group. The following are equivalent: (1) the quotient map q u : T u ( P ) → C (cid:111) C P P is an isomorphism; (2) for every finite set F ⊂ P \ P ∗ , there exists s ∈ P such that s P ∩ qP = ∅ for all q ∈ F ; (3) if s ∈ P and F is a finite subset of P with sP ∩ ( P \ S q ∈ F qP ) = ∅ , then there is s ∈ sP such that s P ∩ qP = ∅ for all q ∈ F . (This is condition (D3) of [3]) . Corollary 10.6.
Let P be a right LCM submonoid of a group G with P = { e } . Suppose that P satisfies any of the conditions of Theorem and any of the conditions of Corollary . Then T λ ( P ) is purely infinite simple.Proof. By Corollary 10.5(3), we have D o r = { } and so Ω P = ∂ Ω P by Lemma 6.12. Thus we have acanonical isomorphism T λ ( P ) ∼ = C(Ω P ) (cid:111) r G = C( ∂ Ω P ) (cid:111) r G ∼ = ∂ T λ ( P ) . By Theorem 10.1(1), the partialaction of G on Ω P is topologically free, and so T λ ( P ) is purely infinite simple by Corollary 6.19. (cid:3) OEPLITZ ALGEBRAS OF SEMIGROUPS 49
Remark . We emphasize that [3, Theorem 5.3] applies to not-necessarily group embeddablecancellative right LCM semigroups. Nevertheless, we would like to clarify the relationship betweenCorollary 10.6 and [3, Theorem 5.3] when applied to right LCM submonoids of a group.(i) Our assumption P = { e } in Corollary 10.6 is necessary to exclude the case T λ ( P ) = C , which isnot purely infinite. This assumption is not explicitly stated in [3, Theorem 5.3].(ii) Case (1) of [3, Theorem 5.3] assumes P ∗ = { e } , and thus (D2) holds vacuously. In addition, (D2)also holds in case (3) of [3, Theorem 5.3], because of [3, Lemma 5.1]. Since (D2) implies topologicalfreeness by Corollary 10.2, we see that our Corollary 10.6 recovers [3, Theorem 5.3] for right LCMsubmonoids of groups. References [1] Beatriz Abadie and Fernando Abadie,
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