Amenability and paradoxicality in semigroups and C*-algebras
aa r X i v : . [ m a t h . OA ] F e b AMENABILITY AND PARADOXICALITY IN SEMIGROUPS ANDC ∗ -ALGEBRAS PERE ARA , FERNANDO LLED ´O , AND DIEGO MART´INEZ Abstract.
We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable,unital) semigroups and corresponding semigroup rings. We consider also Følner type characteri-zations of amenability and give an example of a semigroup whose semigroup ring is algebraicallyamenable but has no Følner sequence.In the context of inverse semigroups S we give a characterization of invariant measures on S (inthe sense of Day) in terms of two notions: domain measurability and localization . Given a unitalrepresentation of S in terms of partial bijections on some set X we define a natural generalization ofthe uniform Roe algebra of a group, which we denote by R X . We show that the following notionsare then equivalent: (1) X is domain measurable; (2) X is not paradoxical; (3) X satisfies thedomain Følner condition; (4) there is an algebraically amenable dense *-subalgebra of R X ; (5) R X has an amenable trace; (6) R X is not properly infinite and (7) [0] = [1] in the K -group of R X . Wealso show that any tracial state on R X is amenable. Moreover, taking into account the localizationcondition, we give several C*-algebraic characterizations of the amenability of X . Finally, we showthat for a certain class of inverse semigroups, the quasidiagonality of C ∗ r ( X ) implies the amenabilityof X . The reverse implication (which is a natural generalization of Rosenberg’s conjecture to thiscontext) is false. Contents
1. Introduction 22. Groups and Uniform Roe Algebras 43. Semigroups 64. Inverse semigroups 94.1. A characterization of invariant measures 104.2. Domain measurable inverse semigroups 114.3. Representations of inverse semigroups 124.4. Amenable inverse semigroups 185. Inverse semigroups, C ∗ -algebras and traces 205.1. Domain measures as amenable traces 215.2. Traces and amenable traces 245.3. Traces in amenable inverse semigroups 25References 27 Date : February 26, 2020.2010
Mathematics Subject Classification.
Key words and phrases. amenability, paradoxical decompositions, Følner condition, semigroups, semigroup rings,inverse semigroup C*-algebra, proper infiniteness, amenable traces. Partially supported by MINECO and European Regional Development Fund, jointly, through the grantMTM2017-83487-P, by MINECO through the Mar´ıa de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445) and by the Generalitat de Catalunya through the grant 2017-SGR-1725. Supported by research projects MTM2017-84098-P and Severo Ochoa SEV-2015-0554 of the Spanish Ministryof Economy and Competition (MINECO), Spain. Supported by research projects MTM2017-84098-P, Severo Ochoa SEV-2015-0554 and BES-2016-077968 of theSpanish Ministry of Economy and Competition (MINECO), Spain. Introduction
The notion of an amenable group was first introduced by von Neumann in [42] to explain whythe paradoxical decomposition of the unit ball in R n (the so-called Banach-Tarski paradox) occursonly for dimensions greater than two (see [55, 43, 50]). Later, Følner provided in [28] a very usefulcombinatorial characterization of amenability in terms of nets of finite subsets of the group thatare almost invariant under left multiplication. This alternative approach was then used to studyamenability in the context of algebras over arbitrary fields by Gromov [31, § ∗ -algebras (see, e.g., the pioneering works [18, 47, 36] as well as the recentsurvey by Lawson in [38] and references therein). In this article we address von Neumann’s originaldichotomy - amenable versus paradoxical - in the context of semigroups and semigroup rings, andconnect this analysis to C ∗ -algebraic structures associated to inverse semigroups. In particular, wedefine a C ∗ -algebra for an inverse semigroup which generalizes the uniform Roe-algebra of a group,and then study its trace space in relation to the amenability of the original inverse semigroup.Amenability of semigroups has been studied since Day’s seminal article (see [18] as well as otherclassical references [6, 34, 21, 41]). However, the category of semigroups is too broad to obtainclassical equivalences like that between amenability, existence of Følner sequences, absence of para-doxical decompositions or algebraic amenability of the corresponding semigroup ring. This has ledto a variety of approaches that modify classical definitions and introduce new notions, such as strongand weak Følner conditions or fair amenability to mention only a few [20, 29, 59]. Some other recentresults exploring (geo)metrical aspects of discrete semigroups are presented in [27, 30]. Furthermore,following the dynamical point of view mentioned above, semigroups are closer to algebras than theyare to groups, since the action of an element s ∈ S on subsets of S can be singular. In the case that S has a zero element, for instance, its action drastically shrinks the size of any subset of S undermultiplication. As an illustration of the singular dynamics involved we show in Theorem 3.8 that if S has a Følner sequence but does not have a Følner sequence exhausting the semigroup (which we call proper Følner sequence in Definition 2.1 (3)) then S has a finite principal left ideal. This behavioris characteristic of the dynamics given by multiplication in an algebra (cf., [4, Theorem 3.9]) and isnot present in the context of groups, where one can easily modify a Følner sequence of a group toturn it exhausting.An alternative approach to understand the dichotomy on a given category is to use operatoralgebra techniques for a canonical C*-algebra associated to the initial structure. In the special caseof groups two important C*-algebras are the reduced group C*-algebra, denoted by C ∗ r ( G ), and theuniform Roe algebra of a group, which we denote by R G = ℓ ∞ ( G ) ⋊ r G , where G acts on ℓ ∞ ( G ) byleft translation. Among other things, Rørdam and Sierakowski establish in [49] a relation betweenparadoxical decompositions of G and properly infinite projections in R G . Nevertheless, it is notobvious how to associate a C*-algebra to a general semigroup, since the naive approach would beto define the generators of a possible C*-algebra via V s δ t := δ st on the Hilbert space ℓ ( S ). This,in general, gives unbounded operators due to the singular dynamics involved. Therefore, when weconnect our analysis to C*-algebras we will restrict to the class of inverse semigroups, where thedynamics induced by left multiplication are only locally injective, i.e., injective on the correspondingdomains. Some general references for inverse semigroups and, also, in relation to C*-algebras are[7, 27, 32, 37, 40, 43, 44, 52, 56]. In Theorem 3.19 of [35], Kudryavtseva, Lawson, Lenz and Resende MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 3 prove a Tarski’s type alternative where the invariant measure and the paradoxical decompositionrestricts to the space of projections E ( S ) of the inverse semigroup. In the context of groupoids,B¨onicke and Li (see [11]) and Rainone and Sims (see [46]) establish a sufficient condition on an ´etalegroupoid that ensures pure infiniteness of the reduced groupoid C*-algebra in terms of paradoxicalityof compact open subsets of the unit space. See also [1, 26, 24] for additional references on the relationbetween inverse semigroups, C*-algebras and groupoids.Recall that an inverse semigroup S is a semigroup such that for every s ∈ S there is a unique s ∗ ∈ S satisfying ss ∗ s = s and s ∗ ss ∗ = s ∗ . We will assume that our semigroups are unital, discreteand countable. In Proposition 4.6 we will characterize invariant measures in the sense of Day, i.e.,finitely additive probability measures satisfying µ ( s − A ) = µ ( A ), s ∈ S , A ⊂ S (where s − A denotesthe preimage of A by s ), by means of the two following conditions:(a) Localization: µ ( A ) = µ ( A ∩ s ∗ sA ), for any s ∈ S , A ⊂ S .(b) Domain-measurability: µ ( s ∗ sA ) = µ ( sA ), for any s ∈ S , A ⊂ S .Fixing a representation α : S → I ( X ) of S in terms of partial bijections on some discrete set X onecan consider a natural *-representation V : S → B ( ℓ ( X )). Define R X,alg as the *-algebra generatedby the family of partial isometries { V s | s ∈ S } and ℓ ∞ ( X ). The C*-algebra R X is the norm closureof R X,alg . In particular, taking the left regular representation ι : S → I ( S ) we obtain a Roe algebra R S , which is a natural generalization of the uniform Roe algebra R G of a discrete group. Recallthat uniform Roe algebras associated to general discrete metric spaces are an important class ofC*-algebras that naturally encode properties of the metric space, such as amenability, property(A) or lower dimensional aspects (see, e.g., [5, Theorem 4.9] or [39, Theorem 2.2]). We will usethis strategy to characterize in different ways amenability aspects of the inverse semigroup. In thiscontext one can define notions like S -domain Følner condition and S -paradoxical decompositionwhich correspond, in essence, to the usual notions but restricted to the corresponding domainsgiven by the representation α . In this way, one of the main results in this article is Theorem 1 (cf., Theorem 5.4) . Let S be a countable and discrete inverse semigroup with identity ∈ S , and let α : S → I ( X ) be a representation of S on X . Then the following conditions areequivalent: (1) X is S -domain measurable. (2) X is not S -paradoxical. (3) X is S -domain Følner. (4) R X,alg is algebraically amenable. (5) R X has an amenable trace. (6) R X is not properly infinite. (7) [0] = [1] in the K -group of R X . Note that this characterization involves the notions corresponding to domain-measurability (see(b) above). We also characterize the full force of amenability of the action in Theorem 5.11, obtainingin particular that it is equivalent to the fact that no projection associated to an idempotent of S isproperly infinite in R X (compare with Theorem 1(6)).An important step in the proof of the previous theorem is the construction and analysis of atype semigroup Typ( α ) (see Definition 4.12) associated to the representation α . Recall that typesemigroups have been considered recently in many interesting situations (see, e.g., [2, 45]).Moreover, we also show in this section that every tracial state on R X is amenable. Theorem 2 (cf., Theorem 5.9) . Let S be a countable and discrete inverse semigroup with identity ∈ S , and let α : S → I ( X ) be a representation. A positive linear functional on R X is a trace ifand only if it is an amenable trace. Given the representation V : S → B (cid:0) ℓ ( X ) (cid:1) introduced above, one can also consider the reducedsemigroup C ∗ -algebra , that is, the C ∗ -algebra C ∗ r ( X ) generated by { V s } s ∈ S . In particular we havethe following inclusions: C ∗ r ( X ) := C ∗ ( { V s | s ∈ S } ) ⊂ R X := C ∗ ( { V s | s ∈ S } ∪ ℓ ∞ ( X )) ⊂ B (cid:0) ℓ ( X ) (cid:1) . PERE ARA, FERNANDO LLED ´O, AND DIEGO MART´INEZ
Lastly, using the theorems above we also prove a generalization to a result by Rosenberg (cf., [19])in the setting of inverse semigroups.
Theorem 3 (cf., Theorem 5.13) . Let S be a countable and discrete inverse semigroup with identity ∈ S and with a minimal projection. Let α : S → I ( X ) be a representation on some set X andsuppose C ∗ r ( X ) is quasidiagonal. Then X is S -amenable. The structure of the article is as follows. In Section 2 we recall different results around the notionof amenability in the context of groups and algebras that partly motivate our analysis. In particular,we introduce the notion of uniform Roe algebra R G of a group G and mention in Theorem 2.4 avariety of ways in which one may characterize the amenability of G via R G . In Section 3 we focusfirst on amenability and Følner sequences for general semigroups and semigroup rings. We givean example of a semigroup S which has an algebraically amenable group ring C S , but no Følnersequence (see Example 3.7).In the final two sections we restrict our analysis to the case of inverse semigroups. In Section 4 wefocus on the algebraic (read as non-C*) aspects of amenability in inverse semigroups. In particularwe split Day’s invariance condition for measures over amenable inverse semigroups S into the twonotions (a) and (b) above, and introduce the type semigroup construction. In Section 5 we presentthe C*-aspects of our analysis. For example, we introduce the algebra R X and prove that all itstraces factor through ℓ ∞ ( X ) via a canonical conditional expectation. We finish the article studyingthe relation between the quasidiagonality of C ∗ r ( X ) and the S -amenability of X . We also mentionsome questions in relation to this problem. Conventions:
We denote by A ⊔ B the disjoint union of two sets A and B . Unless otherwisespecified, any measure µ on a set X will be a finitely additive probability measure , i.e., µ : P ( X ) → [0 , P ( X ) is the power set of X , satisfies µ ( X ) = 1 and µ ( A ⊔ B ) = µ ( A ) + µ ( B ), forevery A, B ⊂ X . All semigroups S considered will be countable, discrete and with unit 1 ∈ S . Arepresentation of an inverse semigroup S on a set X is a unital semigroup homomorphism α : S →I ( X ), where I ( X ) denotes the inverse semigroup of partial bijections of X . We will denote by B ( H )the algebra of bounded linear operators on a complex separable Hilbert space H . Acknowledgements:
We thank an anonymous referee for his helpful remarks on a previousversion of the manuscript. 2.
Groups and Uniform Roe Algebras
This section aims to give a brief summary to some aspects of amenability needed later. We beginwith classical notions in the context of groups and relate these with C*-algebraic concepts using theuniform Roe algebra of a group.
Definition 2.1.
Let G be a countable group and A a unital C -algebra of countable dimension.(1) G is (left) amenable if there exists a (left) invariant measure on G , i.e., a finitely additiveprobability measure µ : P ( G ) → [0 ,
1] such that µ (cid:0) g − A (cid:1) = µ ( A ) for all g ∈ G , A ⊂ G .(2) G satisfies the Følner condition if for every ε >
F ⊂ G , there is a finite non-empty F ⊂ G such that | gF ∪ F | ≤ (1 + ε ) | F | , for every g ∈ F .(3) G satisfies the proper Følner condition if, in addition, the finite set F can be taken to containany other set A ⊂ G , i.e., for every ε > , F ⊂ G and finite A ⊂ G there is a finite non-empty F ⊂ G as above such that A ⊂ F .(4) G is paradoxical if there are sets A i , B j ⊂ G and elements a i , b j ∈ G such that G = a A ⊔ · · · ⊔ a n A n = b B ⊔ · · · ⊔ b m B m ⊃ A ⊔ · · · ⊔ A n ⊔ B ⊔ · · · ⊔ B m . (5) A is algebraically amenable if for every ε > F ⊂ A there is a non-zero finitedimensional subspace W ≤ A such that dim ( AW + W ) ≤ (1 + ε ) dim ( W ) for every A ∈ F .Along the lines of these notions, but in the C ∗ -scenario, we can define when a C ∗ -algebra A cap-tures some aspects of amenability , the Følner condition or paradoxicality . For additional motivationsand results see, e.g., [16, 17, 9, 10, 3] and references therein. MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 5 Definition 2.2.
Let
A ⊂ B ( H ) be a unital C ∗ -algebra of bounded linear operators on a complexseparable Hilbert space H . A state on A is a positive and linear functional on A with norm one.(1) A state τ on A is called an amenable trace if there is a state φ on B ( H ) extending τ , i.e., φ | A = τ , and satisfying φ ( AT ) = φ ( T A ) , T ∈ B ( H ) , A ∈ A . The state φ is called a hypertrace for A . The concrete C*-algebra A is a called a FølnerC*-algebra if it has an amenable trace.(2) A satisfies the Følner condition if for every ε >
F ⊂ A there is a non-zerofinite rank orthogonal projection P ∈ B ( H ) such that || P A − AP || ≤ ε || P || for every A ∈ F , where k · k denotes the Hilbert-Schmidt norm.(3) A projection P ∈ A is properly infinite if there are V, W ∈ A such that P = V ∗ V = W ∗ W ≥ V V ∗ + W W ∗ . Note that, in this case, the range projections V V ∗ and W W ∗ are orthogonal.The algebra A is called properly infinite when 1 ∈ A is properly infinite. Remark 2.3.
The class of Følner C ∗ -algebras is also known in the literature as weakly hypertracial C ∗ -algebras (cf., [9]). In [3, Section 4] the first and second authors gave an abstract (i.e., represen-tation independent) characterization of this class of algebras in terms of a net of unital completelypositive (u.c.p.) maps into matrices which are asymptotically multiplicative in a weaker norm thanthe operator norm (see also [5, Theorem 3.8]). It can be shown that an abstract C ∗ -algebra is a Følner C ∗ -algebra if there exists a non-zero representation π : A → B ( H ) such that π ( A ) has anamenable trace (cf., [3, Theorem 4.3]). In general, quasidiagonality is a stronger notion than Følner(see, e.g., the examples given in the context of general uniform Roe algebras over metric spaces in[5, Remark 4.14]). However, if A is a unital nuclear C ∗ -algebra, then it is a Følner C ∗ -algebra ifand only if A admits a tracial state (see [12, Proposition 6.3.4]). Note this fact implies that everystably finite unital nuclear C ∗ -algebra is in the Følner class.A classical construction relating C ∗ -algebras and groups is given via the so-called left regularrepresentation : the unitary representation λ : G → B (cid:0) ℓ ( G ) (cid:1) defined by ( λ g ( f )) ( h ) := f (cid:0) g − h (cid:1) .The uniform Roe algebra R G of G is the C ∗ -algebra generated by { λ g } g ∈ G and ℓ ∞ ( G ) viewed asmultiplication operators in ℓ ( G ), that is, R G := C ∗ (cid:16) { λ g | g ∈ G } ∪ ℓ ∞ ( G ) (cid:17) ⊂ B (cid:0) ℓ ( G ) (cid:1) . The following result shows how one can characterize amenability and paradoxicality of the group interms of C ∗ -properties of the algebra R G . Theorem 2.4.
Let G be a countable and discrete group. The following are equivalent: (1) G is amenable. (2) G is not paradoxical. (3) G has a Følner sequence. (4) C G is algebraically amenable. (5) R G has an amenable trace (and hence is a Følner C ∗ -algebra). (6) R G is not properly infinite. (7) [0] = [1] in the K -group of R G .Proof. The equivalences (1) ⇔ (2) ⇔ (3) are classical (see, e.g., [44, 33]). Their equivalence to (4)is due to Bartholdi [8]. To show the equivalences (1) ⇔ (5) ⇔ (6) recall that the uniform Roe algebraof the group G can be also seen as a reduced crossed product, i.e., R G = ℓ ∞ ( G ) ⋊ r G , where theaction of G on ℓ ∞ ( G ) is given by left translation of the argument (see, e.g., [12, Proposition 5.1.3]).Rørdam and Sierakowski show in [49, Proposition 5.5] a direct equivalence between paradoxicalityof G and proper infiniteness for this class of crossed products. In fact, they show that E ⊂ G is paradoxical if and only if the characteristic function P E is properly infinite in ℓ ∞ ( G ) ⋊ r G (seealso [5, Theorem 4.9]). Finally, using the reasoning in [5, Theorem 4.6] one can also prove (1) ⇔ (7). (cid:3) PERE ARA, FERNANDO LLED ´O, AND DIEGO MART´INEZ
Remark 2.5.
For a general study of the relation between Følner C ∗ -algebras and crossed productssee also [9, 10]. Moreover, note that the C ∗ -algebra R G contains the reduced group C ∗ -algebraC ∗ r ( G ) = C ∗ ( { λ g | g ∈ G } ) . However the characterization in terms of proper infiniteness in the preceding theorem would not betrue if we replaced the former by the latter. In fact, it is well known (see [15]) that the reducedC ∗ -algebra of the free group on two generators F has no non-trivial projection and hence is vac-uously not properly infinite. However F is indeed paradoxical. Thus observe that a C ∗ -algebraiccharacterization of amenability via proper infiniteness requires the existence of non-trivial projec-tions in the C ∗ -algebra, and that in R G the existence of nontrivial projections is guaranteed by thecharacteristic functions in ℓ ∞ ( G ). 3. Semigroups
We begin next our analysis of amenability in the context of semigroups. We will see that, from adynamical point of view, semigroups are closer to algebras than to groups. At this level of generalityit is not possible to define a natural C*-algebra which provides the variety of characterizations givenin Theorem 2.4.Recall that a semigroup is a non-empty set S equipped with an associative binary operation( s, t ) st . The notions treated in Section 2 do have an analogue in the semigroup scenario, whichrelies on the preimage of a set. Given s ∈ S and A ⊂ S , the preimage of A by s is defined by s − A := { t ∈ S | st ∈ A } . The following definition for semigroups is due to Day [18] (see also [28, 41]). Recall that by probabilitymeasure we mean finitely additive probability measure . Definition 3.1.
Let S be a semigroup.(1) S is (left) amenable if there exists an (left) invariant measure on S , i.e., a probability measure µ : P ( S ) → [0 ,
1] such that µ (cid:0) s − A (cid:1) = µ ( A ) for every s ∈ S , A ⊂ S .(2) S satisfies the (left) Følner condition if for all ε > F ⊂ S there is a finitenon-empty F ⊂ S such that | sF ∪ F | ≤ (1 + ε ) | F | for every s ∈ F .(3) S satisfies the proper Følner condition if, in addition, the Følner set F can be taken tocontain any other set A ⊂ S , i.e., for every ε >
0, finite
F ⊂ S and finite A ⊂ S there is afinite non-empty F ⊂ S as in (2) that, in addition, satisfies A ⊂ F . Remark 3.2. (1) For the rest of the text we will just consider left amenability as defined inDefinition 3.1 and we will omit the prefix left .(2) We mention the Følner condition given in (2) is equivalent to the existence of a net (a se-quence if S is countable) { F i } i ∈ I of finite non-empty subsets of S such that | sF i \ F i | / | F i | → s ∈ S . These conditions will be used indistinctly throughout the text.We introduce next a stronger notion than amenability in the context of semigroups. Definition 3.3.
A semigroup S is called measurable if there is a probability measure µ on S suchthat µ ( sA ) = µ ( A ), s ∈ S , A ⊂ S .It is a standard result that any measurable semigroup is amenable as well. The reverse implicationis false in general, although it holds in some classes of semigroups, e.g., for left cancellative ones (seeSorenson’s Ph.D. thesis [51] as well as Klawe [34]).The following proposition justifies why we can assume a semigroup S to be countable and unital,as we will normally do in the following sections. In general, given a possibly non-unital semigroup S we can always consider its unitization S ′ := S ⊔ { } and define a multiplication in S ′ extendingthat of S so that 1 behaves as a unit. Moreover, as in the case of groups and algebras, the propertyof amenability is in essence a countable one, at least for a large class of semigroups (including theinverse). Recall from [29] that a semigroup S satisfies the Klawe condition whenever sx = sy for s, x, y ∈ S implies there is some t ∈ S such that xt = yt . As mentioned in [29], the Klawe conditionis very general and, in particular, left cancellative as well as inverse semigroups satisfy it. MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 7 Proposition 3.4.
Let S be a semigroup and denote by S ′ its unitization. Then (i) S is amenable if and only if S ′ is amenable. (ii) If any countable subset in S is contained in an amenable countable subsemigroup of S , then S is amenable. If, in addition, S satisfies the Klawe condition, then the reverse implicationis also true.Proof. (i) The proof directly follows from the definition. Indeed, an invariant measure on S canbe extended to an invariant measure on S ′ defining µ ( { } ) = 0. Conversely, { } is null for anyinvariant measure on S ′ , so any invariant measure on S ′ is also an invariant measure on S .(ii) For the first part, let A ( S ) denote the set of countable and amenable subsemigroups of S .Furthermore, for T ∈ A ( S ) denote by µ T an invariant measure on T ⊂ S . We may, without loss ofgenerality, extend it to S by defining µ T ( S \ T ) := 0. Observe that then µ T (cid:0) t − A (cid:1) = µ T ( A ) forevery t ∈ T , A ⊂ T . Consider the measure: µ : P ( S ) → [0 , , A µ ( A ) := lim T ∈A ( S ) µ T ( A ) = lim T ∈A ( S ) µ T ( A ∩ T ) , where the limit is taken along a free ultrafilter of A ( S ). It follows from a straightforward compu-tation that µ is an invariant measure on S .For the second part we follow a similar route to that of [4, Proposition 3.4]. Recall from [29]that a semigroup satisfying the Klawe condition is amenable if and only if for every ε > F ⊂ S there is a ( ε, F )-Følner set F ⊂ S such that | F | = | sF | for every s ∈ F (see Theorem 2.6 in[29] in relation with the notion of strong Følner condition).Let C = { c n } n ∈ N ⊂ S be a countable subset. In order to construct an amenable semigroup T ⊃ C we define an increasing sequence { T n } n ∈ N of countable subsemigroups of S by: • T is the subsemigroup generated by C . • Suppose T i = { t j } j ∈ N has been defined. By [29] and the previous paragraph, for every k ∈ N we may find an (1 /k, { t , . . . , t k } )-Følner set F k ⊂ S such that | t j F k | = | F k | forevery j = 1 , . . . , k . We thus define the semigroup T i +1 to be the semigroup generated by T i ∪ F ∪ F ∪ . . . .Finally, consider the semigroup T = ∪ i ∈ N T i . It is straightforward to prove that then T is amenable,countable and contains C . (cid:3) Remark 3.5.
As in the case of metric spaces or algebras (see, e.g., [5, Section 2.1] and [3, Section 4]),an amenable semigroup can have non-amenable sub-semigroups. For instance, take S := F ⊔ { } ,where 0 ω = ω ω ∈ F . This semigroup S is amenable, since it has a 0 element, buthas a non-amenable sub-semigroup. A more striking fact is that amenable groups may contain non-amenable semigroups. For instance, the group G of isometries of R is a solvable group containinga non-commutative free semigroup (see [55, Theorem 1.8, Theorem 14.30]).For the purpose of this article the main difference between a semigroup and a group is the lackof injectivity under left multiplication. This fact, among other things, makes it impossible to definea canonical regular representation in the general semigroup case. We recall next some well-knownfacts. Theorem 3.6.
Let S be a countable discrete semigroup. Consider the assertions: (1) S is amenable. (2) S has a Følner sequence. (3) C S is algebraically amenable.Then (1) ⇒ (2) ⇒ (3).Proof. Følner proved the implication (1) ⇒ (2) in the case of groups and, later, Frey and Namiokaextended the proof for semigroups (cf., [28, 41]). To show (2) ⇒ (3) choose a Følner sequence { F n } n ∈ N for S . Then the linear span of these subsets W n := span { f | f ∈ F n } defines a Følnersequence for C S . In fact, note that dim ( W n ) = | F n | and for any s ∈ S we havedim ( sW n + W n )dim ( W n ) ≤ | sF n ∪ F n || F n | n →∞ −−−→ , PERE ARA, FERNANDO LLED ´O, AND DIEGO MART´INEZ which concludes the proof. (cid:3)
We remark that none of the reverse implications in Theorem 3.6 hold in general. It is well knownthat a finite semigroup may be non-amenable and any such semigroup is a counterexample to theimplication (2) ⇒ (1), because if S is finite it has a trivial (constant) Følner sequence F n = S . Aconcrete example was first given by Day in [18]: let S = { a, b } , where ab = aa = a and ba = bb = b .Note that in this case any invariant measure µ must satisfy µ (cid:0) b − { a } (cid:1) = µ (cid:0) a − { b } (cid:1) = µ ( ∅ ) = 0.Therefore, no probability measure on S can be invariant and, hence, S not amenable.The following example is a counterexample to the implication (3) ⇒ (2) in Theorem 3.6. Example 3.7.
Consider the additive semigroup of natural numbers N = { , , , . . . } and the freesemigroup on two generators F +2 = { a, b, ab, . . . } , where we assume that the semigroup F +2 has noidentity. Denote by α the action of F +2 y N given by α : F +2 → End ( N ), a, b α a ( n ) = α b ( n ) = n − n ≥ α a (0) = α b (0) = 0.We claim the semigroup S := N ⋊ α F +2 does not satisfy the Følner condition, while its complexgroup algebra is algebraically amenable. Note that the element s = (0 , a ) − (1 , a ) ∈ C S clearly satisfies ( n, ω ) s = 0 for every ( n, ω ) ∈ S . Therefore W := C s is trivially a Følner subspacefor C S , since it is a one-dimensional left ideal. This proves that C S is algebraically amenable.In order to prove that S does not satisfy the Følner condition, we shall prove that for any non-empty finite subset F ⊂ S either | (0 , a ) F \ F | ≥ | F | /
50 or | (0 , b ) F \ F | ≥ | F | /
50. First observethat | (0 , a ) F | ≥ | F | /
2, and that equality holds if and only if(3.1) F = { (0 , w ) , (1 , w ) , . . . , (0 , w k ) , (1 , w k ) } for some w i ∈ F +2 , i = 1 , . . . , k. Indeed, if F is of this form then clearly | (0 , a ) F | = | F | /
2. And, conversely, given ( n, u ) = ( m, v )one has that (0 , a ) ( n, u ) = (0 , a ) ( m, v ) only when u = v and n = 0 , m = 1 or n = 1 , m = 0.Now suppose F is of the form given in Eq. (3.1) and satisfies | (0 , a ) F \ F | ≤ | F | /
5. Note that,by the observation in the previous paragraph, | (0 , a ) F \ F | ≥ | F | / − N a , where N a is the numberof words w i of F that begin with a . Since a word cannot begin with a and with b , it follows thatthe number N b of words that begin with b satisfies N b ≤ | F | /
5. Therefore, again, we conclude that | (0 , b ) F \ F | ≥ | F | / − | F | / ≥ | F | /
5, as desired. This proves that no set of the form (3.1) can beFølner.Finally, given an arbitrary F ⊂ S we may decompose it into F = F ∗ ⊔ F ′ , where F ∗ is of the form(3.1) and F ′ does not contain pairs of elements of the form (0 , ω ) , (1 , ω ), with ω ∈ F +2 . We have | ((0 , a ) F ∪ (0 , b ) F ) \ F | ≥ (cid:12)(cid:12) (0 , a ) F ′ (cid:12)(cid:12) + | (0 , a ) F ∗ | + (cid:12)(cid:12) (0 , b ) F ′ (cid:12)(cid:12) + | (0 , b ) F ∗ | − | F | = (cid:12)(cid:12) F ′ (cid:12)(cid:12) . Note that the last equality follows from the fact that | (0 , a ) F ′ | = | F ′ | = | (0 , b ) F ′ | . Therefore, if F ′ is relatively large when compared to F , then F itself will not be Følner. Suppose hence that | F ′ | ≤ | F | /
25 and | (0 , a ) F \ F | ≤ | F | /
25. Then we have | F ∗ | ≥ (24 / | F | . Now observe that(0 , a ) F ∗ \ F ∗ = [(0 , a ) F ∗ \ F ] ⊔ [(0 , a ) F ∗ ∩ F ′ ] ⊆ ((0 , a ) F \ F ) ∪ F ′ , and so | (0 , a ) F ∗ \ F ∗ | ≤ | F |
25 + | F | ≤ · · | F ∗ | · < | F ∗ | . Since F ∗ is of the form (3.1), it follows that | (0 , b ) F ∗ \ F ∗ | ≥ | F ∗ | /
5. Hence | (0 , b ) F ∗ \ F ∗ | ≥ | F ∗ | ≥ | F | · . Finally, | (0 , b ) F \ F | ≥ | (0 , b ) F ∗ \ F | = | (0 , b ) F ∗ \ F ∗ | − | (0 , b ) F ∗ ∩ F ′ |≥ | F | · − | F | ≥ | F | . MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 9 It remains to consider what happens when | F ′ | ≥ | F | /
25. In this case, by the above computation,we get 2 max {| (0 , a ) F \ F | , | (0 , b ) F \ F |} ≥ | ((0 , a ) F ∪ (0 , b ) F ) \ F | ≥ | F ′ | ≥ | F | , and we deduce that either | (0 , a ) F \ F | or | (0 , b ) F \ F | is greater or equal than | F | / F ⊂ S can be ( ε, { a, b } )-invariant for ε < / S itself does not satisfy the Følner condition.In the following result we establish the difference between the Følner condition and the properFølner condition. This result is analogous to [5, Proposition 2.15] (see also [5, Theorem 3.9]). Wewill use this statement in Proposition 4.31. Its proof is inspired by the corresponding result in thealgebra setting [5, Theorem 3.9]. Theorem 3.8.
Let S be a semigroup. Suppose that S satisfies the Følner condition but not theproper Følner condition. Then there is an element a ∈ S such that | Sa | < ∞ .Proof. Given ε >
F ⊂ S defineFøl ( ε, F ) := (cid:26) F ⊂ S | < | F | < ∞ and max s ∈F | sF \ F || F | ≤ ε (cid:27) ,M ε, F := sup F ∈ Føl( ε, F ) | F | ∈ N ∪ {∞} . Since S is not properly Følner there is a pair ( ε , F ) with finite M ε , F . Note that the pairs ( ε, F )are partially ordered by ( ε , F ) ≤ ( ε , F ) if and only if F ⊂ F and ε ≤ ε . This partial orderinduces a partial order on M ε, F and thus we may suppose that ε M ε , F <
1. Indeed, simplysubstitute ε with some ε ′ < min { ε , /M ε , F } .We first claim that for any ε ∈ (0 , ε ] and F ⊃ F we have Føl (0 , F ) = Føl ( ε, F ). Indeed, theinclusion ⊂ is obvious. Moreover, for F ∈ Føl ( ε, F ) and s ∈ F we have | sF \ F | ≤ ε | F | ≤ εM ε, F ≤ ε M ε , F < | sF \ F | = 0. Therefore F ∈ Føl (0 , F ). Thus it makes sense to consider the largestFølner sets with ε = 0:Føl max (0 , F ) := (cid:8) F ∈ Føl (0 , F ) | | F | ≥ (cid:12)(cid:12) F ′ (cid:12)(cid:12) for all F ′ ∈ Føl (0 , F ) (cid:9) . Next we claim that if
F ⊂ F ′ and F m ∈ Føl max (0 , F ) , F ′ m ∈ Føl max (0 , F ′ ), then F ′ m ⊂ F m .Indeed, suppose the contrary. Then b F := F m ∪ F ′ m would be in Føl max (0 , F ) and strictly largerthan F m , contradicting the maximality condition in the definition of Føl max (0 , F ). In particular,this means that Føl max (0 , F ) has only one element, for if F , F ∈ Føl max (0 , F ) then F ⊂ F ⊂ F .Finally, denote by F F the element of Føl max (0 , F ) and consider the net {| F F |} F∈J , where J := {F ⊂ S | |F | < ∞ and F ⊂ F } . This net is decreasing and contained in [1 , | F F | ] ∩ N and, thus,has a limit, which is attained by some F . This means that sF F ⊂ F F for all s ∈ S . Thereforeany a ∈ F F will meet the requirements of the theorem. (cid:3) Inverse semigroups
In the rest of the article we will incorporate into the analysis notions of paradoxical decompositionsand the relation to C*-algebras in the category of inverse semigroups, i.e., where one only has alocally injective action. While the rest of the text will be devoted to inverse semigroups, this sectionfocuses only on the algebraic (meaning non-C ∗ ) properties and Section 5 will focus on how theseproperties of S translate into properties of a C*-algebra defined as a generalization of the uniformRoe algebra of a group.First we recall the definition of inverse semigroup as well as some important structures andexamples. Definition 4.1. An inverse semigroup is a semigroup S such that for every s ∈ S there is a unique s ∗ ∈ S satisfying ss ∗ s = s and s ∗ ss ∗ = s ∗ . Example 4.2.
The most important example of an inverse semigroup is that of the set of partialbijections on a given set X , denoted by I ( X ). Elements ( s, A, B ) ∈ I ( X ) are bijections s : A → B ,where A, B ⊂ X . The operation of the semigroup is just the composition of maps where it canbe defined. This semigroup contains both a zero element, namely (0 , ∅ , ∅ ), and a unit, namely(id , X, X ). Just as the elements of a group G can be thought of as bijections of G on itself by leftmultiplication, every inverse semigroup S can be thought as contained in I ( S ) via the Wagner-Preston representation (see, e.g., [56, 44, 7]). Remark 4.3.
Given an inverse semigroup S , the set E ( S ) = { s ∗ s | s ∈ S } is the set of all idempo-tents (or projections ) of S , i.e., elements satisfying e = e ∈ S . Observe that in an inverse semigroupall idempotents commute and satisfy e ∗ = e (see [56] or [37, Theorem 3]). Moreover, E ( S ) has thestructure of a meet semi-lattice with respect to the order e ≤ f ⇐⇒ ef = e , and S is a group if andonly if E ( S ) only has a single element (the identity in the group). If one considers S as containedin I ( S ), then an idempotent e ∈ S will be identified with the identity function id eS : eS → eS .Note also that S is unital if and only if E ( S ) has a greatest element. We shall assume that allour inverse semigroups are unital with unit denoted by 1.We will show in Theorems 4.27 and 5.4 that all the different amenability notions are againrelated in the inverse semigroup case, but not quite as elegantly intertwined as in groups (seeSection 2). Such a conclusion might seem surprising, since it is known that the amenability of aninverse semigroup is closely related to the amenability of its group homomorphic image G ( S ), asthe following result of Duncan and Namioka in [21] shows. In the literature, this fact has led to theopinion that amenability in the inverse semigroup case can be traced back to the group case. Ourresults later will refine this line of thought. Theorem 4.4.
A countable discrete inverse semigroup S is amenable if and only if the group G ( S ) is amenable, where G ( S ) = S/ ∼ and s ∼ t if and only if es = et for some projection e ∈ E ( S ) . A characterization of invariant measures.
Recall from Definition 3.1 that a semigroup S is called amenable if there is an invariant probability measure µ : P ( S ) → [0 , s i , t j whose regular representation induce properly infinite projectionsin the C ∗ -algebra R G , as in the group case. Avoiding this drawback will be a critical step in theproof of Theorem 5.4. We will present an alternative approach taking into account the domain ofthe action of the semigroup. In this way we can directly relate the non-amenability of S with theproper infiniteness of the identity of the associated C*-algebra. However, before developing the newapproach, we present some basic results for inverse semigroups. In particular, the following lemma,whose proof is elementary, will be very useful in the rest of the text. Lemma 4.5.
Let S be an inverse semigroup. For any s ∈ S and A, B ⊂ S the following relationshold: (i) s (cid:0) s − A ∩ s ∗ ss − A (cid:1) = A ∩ ss ∗ A = ss − A ⊂ A ⊂ s − sA . (ii) ss ∗ ( A ∩ ss ∗ B ) = A ∩ ss ∗ B . (iii) s − ( A \ ss ∗ A ) = ∅ .Proof. The inclusions ss − A ⊂ A ⊂ s − sA follow directly from the definition and (ii) is straight-forward to check. To show s (cid:0) s − A ∩ s ∗ ss − A (cid:1) = A ∩ ss ∗ A choose t ∈ s − A ∩ s ∗ ss − A . Then st ∈ A and t = s ∗ sq for some q ∈ S with sq ∈ A , hence st = ss ∗ sq ∈ ss ∗ A . To show the re-verse inclusion consider t ∈ A ∩ ss ∗ A , i.e., A ∋ t = ss ∗ a for some a ∈ A . Then s ∗ a ∈ s − A and t = ss ∗ s ( s ∗ a ) ∈ s (cid:0) s − A ∩ s ∗ ss − A (cid:1) . The remaining equalities are proved in a similar vein. (cid:3) The following result gives a useful characterization of invariant measures that avoids the use ofpreimages.
Proposition 4.6.
Let S be a countable and discrete inverse semigroup with identity ∈ S and µ be a probability measure on it. Then the following conditions are equivalent: MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 11 (1) µ is invariant, i.e., µ ( A ) = µ (cid:0) s − A (cid:1) for all s ∈ S , A ⊂ S . (2) µ satisfies the following conditions for all s ∈ S , A ⊂ S : (2.a) µ ( A ) = µ ( A ∩ s ∗ sA ) . (2.b) µ ( s ∗ sA ) = µ ( sA ) .Proof. For notational simplicity we show conditions (2.a) and (2.b) interchanging the roles of s and s ∗ . The implication (1) ⇒ (2) follows from two simple observations. First, note that µ ( A \ ss ∗ A ) = µ (cid:0) s − ( A \ ss ∗ A ) (cid:1) = µ ( ∅ ) = 0 . Therefore µ ( A ) = µ ( A ∩ ss ∗ A ) + µ ( A \ ss ∗ A ) = µ ( A ∩ ss ∗ A ), as required. Secondly, observe that s ∗ A ⊂ s − ss ∗ A . Thus µ ( ss ∗ A ) = µ (cid:0) s − ss ∗ A (cid:1) = µ (cid:0) s − ss ∗ A \ s ∗ A (cid:1) + µ ( s ∗ A )= µ (cid:16) ( s ∗ ) − (cid:0) s − ss ∗ A \ s ∗ A (cid:1)(cid:17) + µ ( s ∗ A ) = µ ( s ∗ A ) . The reverse implication (2) ⇒ (1) follows from (2.a), (2.b) and Lemma 4.5(i). In fact, µ (cid:0) s − A (cid:1) = µ (cid:0) s − A ∩ s ∗ ss − A (cid:1) = µ (cid:0) s (cid:0) s − A ∩ s ∗ ss − A (cid:1)(cid:1) = µ ( A ∩ ss ∗ A ) = µ ( A ) , which proves (1). (cid:3) Remark 4.7.
Observe that this characterization indeed restricts to the usual one in the case ofgroups since then s ∗ = s − and A ∩ ss − A = A . Therefore condition (2.a) is empty in the groupcase.In the following corollary we combine conditions (2.a) and (2.b) into a single one. Corollary 4.8.
Let S be a countable and discrete inverse semigroup and µ be a probability measureon it. Then µ is invariant if and only if µ ( A ) = µ ( s ( A ∩ s ∗ sA )) for all s ∈ S , A ⊂ S .Proof. Assume that µ is invariant, hence satisfies conditions (2.a) and (2.b). Since A ∩ s ∗ sA ⊂ s ∗ sA we have µ ( A ) = µ ( A ∩ s ∗ sA ) = µ ( s ∗ s ( A ∩ s ∗ sA )) = µ ( s ( A ∩ s ∗ sA )) . To show the reverse implication we prove first condition (2.a) which follows from µ ( A ) = µ ( s ( A ∩ s ∗ sA )) = µ (cid:16) s ∗ (cid:0) s ( A ∩ s ∗ sA ) ∩ ss ∗ s ( A ∩ s ∗ sA ) (cid:1)(cid:17) = µ ( s ∗ s ( A ∩ s ∗ sA )) = µ ( A ∩ s ∗ sA ) , where for the last equation we used Lemma 4.5(ii). The condition (2.b) follows directly from µ ( A ) = µ ( s ( A ∩ s ∗ sA )) just by replacing the set A by s ∗ sA . (cid:3) Domain measurable inverse semigroups.
The characterization of an invariant measure µ given in Proposition 4.6 means that µ is measurable (see Definition 3.3) via s but only when theaction of s is restricted to its domain (namely µ ( s ∗ sA ) = µ ( sA )). In addition, the measure of anyset A is localized within the domain of every s ∈ S (namely µ ( A ) = µ ( A ∩ s ∗ sA )).We will show in Theorem 5.4 that a necessary and sufficient condition for the C ∗ -algebra R S tohave an amenable trace is the measurability condition on domains given by µ ( s ∗ sA ) = µ ( sA ). Thisfact justifies the next definition. Definition 4.9.
Let S be a countable and discrete inverse semigroup and A ⊂ S a subset. Then A is domain measurable if there is a measure µ : P ( S ) → [0 , ∞ ] such that the following conditionshold:(1) µ ( A ) = 1.(2) µ ( s ∗ sB ) = µ ( sB ) for all s ∈ S and B ⊂ S .We say that S is domain measurable when the latter holds for A = S and call the correspondingmeasures domain measures .Domain measurable semigroups can be understood as a possible generalization of the amenablegroups. To further specify this idea see Theorem 5.4 and compare it to Theorem 2.4. Remark 4.10.
Recall from [20] that a semigroup is called fairly amenable if it has a probabilitymeasure µ such that(4.1) µ ( A ) = µ ( sA ) if s acts injectively on A ⊂ S .
Observe that if µ satisfies Eq. (4.1), then it satisfies condition (2.b) in Proposition 4.6 as well, since s acts injectively on s ∗ sA . Therefore, if S is fairly amenable then it is also domain measurable.However, a domain measure satisfying the condition of domain measurability need not satisfy (4.1).In fact, consider an inverse semigroup S with a 0 element and some other element s ∈ S , s = 0.Then S is domain measurable since it is amenable with an invariant measure µ satisfying µ ( { } ) = 1.This measure, however, cannot implement fair amenability. Example 4.11.
We build a class of non-amenable, domain measurable semigroups. Let
A, N bedisjoint inverse semigroups, with A amenable and N non-amenable. Consider then the semigroup S = A ⊔ N , where an := n =: na for every a ∈ A , n ∈ N . It is routine to show S is an inversesemigroup. Furthermore, we claim that it is non-amenable and domain measurable. Indeed, supposeit is amenable and let µ be an invariant measure on it. For any n ∈ N , µ ( A ) = µ (cid:0) n − A (cid:1) = µ ( ∅ ) = 0and hence µ ( N ) = 1. Therefore µ would restrict to an invariant mean on N , contradicting thehypothesis.To prove now that S is domain measurable, just choose an invariant measure ν on A (that existssince A is amenable) and extend it to S as b ν ( A ′ ⊔ N ′ ) = ν ( A ′ ), for any A ′ ⊂ A and N ′ ⊂ N . Thismeasure will satisfy b ν ( s ∗ sB ) = b ν ( sB ) for every s ∈ S and B ⊂ S .4.3. Representations of inverse semigroups.
Following [37], we define a representation of aunital inverse semigroup S on a (discrete) set X as a unital semigroup homomorphism α : S → I ( X ).One can check that any action θ of S on X gives a representation α of S on X by the rule α s = ( θ s | θ s ∗ s ( X ) : θ s ∗ s ( X ) → θ ss ∗ ( X )) . Indeed the domain for θ s | θ s ∗ s ( X ) ◦ θ t | θ t ∗ t ( X ) is θ t ∗ ( θ tt ∗ ( X ) ∩ θ s ∗ s ( X )) = θ t ∗ ( X ) ∩ θ t ∗ s ∗ s ( X ) = θ t ∗ s ∗ s ( X ) = θ ( st ) ∗ ( st ) ( X ) . If α is a representation, we denote by D s ∗ s the domain of α s . Note that α s is a bijection from D s ∗ s onto D ss ∗ , with inverse α s ∗ . Definition 4.12.
Let α : S → I ( X ) be a representation of the inverse semigroup S on a set X .We define the type semigroup Typ( α ) as the commutative monoid generated by symbols [ A ] with A ∈ P ( X ) and relations(1) [ ∅ ] = 0.(2) [ A ] = [ α s ( A )] if A ⊆ D s ∗ s .(3) [ A ∪ B ] = [ A ] + [ B ] if A ∩ B = ∅ .This definition is very natural and allows to easily check if a map from Typ( α ) to another semi-group is a homomorphism. We show next that Typ( α ) is indeed isomorphic to a type semigroupwhich is constructed based on Tarski’s original ideas. Definition 4.13.
Let α : S → I ( X ) be a representation of the inverse semigroup S on a set X . Wesay A, B ⊂ X are equidecomposable , and write A ∼ B , if there are sets A i ⊂ X and elements s i ∈ S , i = 1 , . . . , n , such that A i ⊆ D s ∗ i s i for i = 1 , . . . , n , and A = A ⊔ · · · ⊔ A n and α s ( A ) ⊔ · · · ⊔ α s n ( A n ) = B. It is routine to show that ∼ is an equivalence relation. Indeed, note that since 1 ∈ S we have A ∼ A . Furthermore the relation ∼ is clearly symmetric by choosing B i := α s i ( A i ) and the dynamics t i := s ∗ i . Finally, if A ∼ B ∼ C then there are A i , B j ⊂ X and s i , t j ∈ S such that A i ⊆ D s ∗ i s i , B j ⊆ D t ∗ j t j , and A = A ⊔ · · · ⊔ A n and α s ( A ) ⊔ · · · ⊔ α s n ( A n ) = B,B = B ⊔ · · · ⊔ B m and α t ( B ) ⊔ · · · ⊔ α t m ( B m ) = C. MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 13 In this case the sets A ij = α s ∗ i ( α s i ( A i ) ∩ B j ) and the elements r ij = t j s i implement the relation A ∼ C .Given a representation α : S → I ( X ), consider the following extensions: • The semigroup S × Perm ( N ), where Perm ( N ) is the finite permutation group of N , that isthe group of permutations moving only a finite number of elements. • A set A ⊂ X × N is called bounded if A ⊂ X × F , with F ⊂ N finite. These sets aresometimes expressed as A × { i } ⊔ · · · ⊔ A k × { i k } and, if so, each A j is called a level .Then there is an obvious representation of S × Perm on X × N given coordinate-wise, which willbe also denoted by α . Hence, it makes sense to ask when two bounded sets A, B ⊂ X × N areequidecomposable. Define b X := { A ⊂ X × N | A is bounded } / ∼ . This set has the natural structure of a commutative monoid with 0 = ∅ and sum defined asfollows. Given two bounded sets A, B ⊂ X × N , let k ∈ N be such that A ∩ B ′ = ∅ , where B ′ := { ( b, n + k ) | ( b, n ) ∈ B } . Then define [ A ] + [ B ] := [ A ⊔ B ′ ]. One can verify that + is well-defined, associative and commutative. This construction was first done by Tarski in [53], and hasbeen used since then extensively (see, e.g., [55, 48, 45, 43]). We only need the notation b X temporarilyand after the proof of the following result we will only use the symbol Typ( α ). Proposition 4.14.
Let α be a representation of the unital inverse semigroup S on X . Then themap γ : Typ( α ) −→ b X, γ ([ A ]) = [ A × { } ] is a monoid isomorphism.Proof. Since [ A × { } ] = [ A × { i } ] in b X , we easily see that this map is well-defined and surjective.To show it is injective, assume that γ ( P ni =1 [ A i ]) = γ ( P mj =1 [ B j ]) for subsets A i , B j of X . Then A := n G i =1 A i × { i } ∼ m G j =1 B j × { j } =: B, and so by definition there are subsets W , . . . , W l of X , and numbers n , . . . , n l , m , . . . , m l ∈ N andelements s , . . . , s l ∈ S such that W k ⊆ D s ∗ k s k for k = 1 , . . . , l and A = l G k =1 W k × { n k } , B = l G k =1 α s k ( W k ) × { m k } . It follows that there is a partition { , . . . , l } = F ni =1 I i such that for each i ∈ { . . . , n } we have A i = F j ∈ I i W j . We thus get in Typ( α ) n X i =1 [ A i ] = n X i =1 X j ∈ I i [ W j ] = n X i =1 X j ∈ I i [ α s j ( W j )] = l X j =1 [ α s j ( W j )] = m X j =1 [ B j ] , showing injectivity. (cid:3) For simplicity we will often denote α s ( x ) ∈ X by sx and sA will stand for α s ( A ) for any s ∈ S , x ∈ X and A ⊂ X . Recall that sx is defined only if x ∈ D s ∗ s . We extend next Definition 4.9 aboveto representations. Definition 4.15.
Let α : S → I ( X ) be a representation of S and let A ⊂ X be a subset.(1) The set A is S -domain measurable if there is a measure µ : P ( X ) → [0 , ∞ ] satisfying thefollowing conditions:(a) µ ( A ) = 1.(b) µ ( B ) = µ ( sB ) for all s ∈ S and B ⊂ D s ∗ s .We say that X is S -domain measurable when the latter holds for A = X . (2) The set A is S -domain Følner if there is a sequence { F n } n ∈ N of finite, non-empty subsets of A such that | s ( F n ∩ D s ∗ s ) \ F n || F n | n →∞ −−−→ s ∈ S .(3) The set A is S -paradoxical if there are A i , B j ⊂ X and s i , t j ∈ S , i = 1 , . . . , n , j = 1 , . . . , m ,such that A i ⊆ D s ∗ i s i , B j ⊆ D t ∗ j t j and A = s A ⊔ . . . ⊔ s n A n = t B ⊔ . . . ⊔ t m B m ⊃ A ⊔ . . . ⊔ A n ⊔ B ⊔ . . . ⊔ B m . Remark 4.16. (i) Note that S is domain measurable in the sense of Definition 4.9 preciselywhen S is S -domain measurable with respect to the canonical representation α : S → I ( S ).(ii) Note also that A ⊂ X being paradoxical is the same as saying that 2[ A ] ≤ [ A ] in Typ( α ).Recall that, in a commutative semigroup S , we denote by n · β the sum β + · · · + β of n terms.Also, the only (pre-)order that we use on S is the so-called algebraic pre-order , defined by x ≤ y ifand only if x + z = y for some z ∈ S . Lemma 4.17.
Let α : S → I ( X ) be a representation of S , and consider the type semigroup Typ( α ) constructed above. Then the following hold: (1) For any bounded sets
A, B ⊂ X × N if A ∼ B , then there exists a bijection φ : A → B suchthat for any C ⊂ A and D ⊂ B one has C ∼ φ ( C ) and D ∼ φ − ( D ) . (2) For any [ A ] , [ B ] ∈ Typ( α ) , if [ A ] ≤ [ B ] and [ B ] ≤ [ A ] , then [ A ] = [ B ] . (3) A subset A of X is S -paradoxical if and only if [ A ] = 2 · [ A ] . (4) For any [ A ] , [ B ] ∈ Typ( α ) and n ∈ N , if n · [ A ] = n · [ B ] , then [ A ] = [ B ] . (5) If [ A ] ∈ Typ( α ) and ( n + 1) · [ A ] ≤ n · [ A ] for some n ∈ N , then [ A ] = 2 · [ A ] .Proof. The proof of this lemma is virtually the same as in the group case (see, e.g. [50, p. 10]). Forconvenience of the reader we include a sketch of the proofs.To construct the bijection φ : A → B in (1) just define it by multiplication by s i in each of thesubsets A i , where A = ⊔ i A i and B = ⊔ i s i A i .(2) There are [ A ] , [ B ] ∈ Typ( α ) such that [ A ] + [ A ] = [ B ] and [ B ] + [ B ] = [ A ]. In this casewithout loss of generality we can suppose that A ∩ A = ∅ = B ∩ B . Choose φ : A ⊔ A → B and ψ : B ⊔ B → A as in (1) and consider C := A , C n +1 := ψ ( φ ( C n )) and C := ∪ ∞ n =0 C n . It then follows that ( B ⊔ B ) \ φ ( C ) = ψ − ( A \ C ) = ψ − ( A ⊔ A \ C ) and hence A ⊔ A = ( A \ C ) ⊔ C ∼ ψ − ( A \ C ) ⊔ φ ( C ) = ( B ⊔ B \ φ ( C )) ⊔ φ ( C ) = B ⊔ B . Therefore B ∼ A ⊔ A ∼ B ⊔ B ∼ A .Now (3) follows from the definitions and (2).Claim (4) uses graph theory and follows from K¨onig’s Theorem (see [43, Theorem 0.2.4]). If n · [ A ] = n · [ B ] then there are sets A i , B j with the following properties:(a) A , . . . , A n are pairwise disjoint, just as B , . . . , B n .(b) n · [ A ] = [ A ] + · · · + [ A n ] = [ B ] + · · · + [ B n ] = n · [ B ].(c) For every i = 1 , . . . , n we have A i ∼ A and B i ∼ B .Consider then the bijections φ j : A → A j , ψ j : B → B j and χ : n · [ A ] → n · [ B ] induced by ∼ , asin (1). For a ∈ A denote by a the set { φ ( a ) , . . . , φ n ( a ) } (and analogously for b ∈ B ). Considernow the bipartite graph defined by: • Its sets of vertices are X = { a | a ∈ A } and Y = (cid:8) b | b ∈ B (cid:9) . • The vertices a and b are joined by an edge if χ ( φ j ( a )) ∈ b for some j = 1 , . . . , n . MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 15 Then this graph is n -regular and, by K¨onig’s Theorem, it has a perfect matching F . In this case itcan be checked that the sets C j,k := (cid:8) a ∈ A | ∃ b ∈ B such that (cid:0) a, b (cid:1) ∈ F and χ ( φ j ( a )) = ψ k ( b ) (cid:9) ,D j,k := (cid:8) b ∈ B | ∃ a ∈ A such that (cid:0) a, b (cid:1) ∈ F and χ ( φ j ( a )) = ψ k ( b ) (cid:9) , are respectively a partition of A and B . Furthermore ψ − k ◦ χ ◦ φ j is a bijection from C j,k to D j,k implementing the relations C j,k ∼ D j,k . These, in turn, implement A ∼ A ∼ B ∼ B .Finally, (5) follows from (2) and (4). Indeed, from the hypothesis2 · [ A ] + n · [ A ] = ( n + 1) · [ A ] + [ A ] ≤ n · [ A ] + [ A ] = ( n + 1) · [ A ] ≤ n · [ A ] . Iterating this argument we get 2 n · [ A ] ≤ n · [ A ] and, since the other inequality trivially holds, n · [ A ] = 2 n · [ A ]. Applying (4) we conclude that [ A ] = 2 · [ A ]. (cid:3) Finally, we next recall one of Tarski’s fundamental results [53] (see also [50, Theorem 0.2.10]).
Theorem 4.18.
Let ( S , +) be a commutative semigroup with neutral element and let ǫ ∈ S . Thefollowing are then equivalent: (i) ( n + 1) · ǫ n · ǫ for all n ∈ N . (ii) There is a semigroup homomorphism ν : ( S , +) → ([0 , ∞ ] , +) such that ν ( ǫ ) = 1 . In order to prove the main result of the section (see Theorem 4.23) we need to introduce severalactions of the inverse semigroup S on canonical spaces associated with X . In particular, we willconsider the behavior of domain measures as functionals on ℓ ∞ ( X ). Given a representation α : S →I ( X ) we define the action of S on ℓ ∞ ( X ) by(4.2) ( sf )( x ) := (cid:26) f ( s ∗ x ) if x ∈ D ss ∗ x / ∈ D ss ∗ . The next result establishes an invariance condition in the context of states on ℓ ∞ ( X ). Proposition 4.19.
Let α : S → I ( X ) be a representation of S . If X is domain measurable, withdomain measure µ (cf., Definition 4.15), then there is a state m : ℓ ∞ ( X ) → C such that (4.3) m ( sf ) = m ( f P s ∗ s ) , for f ∈ ℓ ∞ ( X ) , s ∈ S, where P s ∗ s denotes the characteristic function of D s ∗ s ⊂ X .Proof. For a set B ⊂ S define m ( P B ) := µ ( B ), where P B denotes the characteristic function on B ,and extend the definition by linearity to simple functions and by continuity to all ℓ ∞ ( X ). Then m satisfies Eq. (4.3) if and only if it does for any characteristic function P B , and this is a consequenceof the domain measurability of µ . Indeed, observe that sP B = P s ( B ∩ D s ∗ s ) and hence, by domain measurability, we obtain m ( sP B ) = m ( P s ( B ∩ D s ∗ s ) ) = µ ( s ( B ∩ D s ∗ s )) = µ ( B ∩ D s ∗ s ) = m ( P B ∩ D s ∗ s ) = m ( P B P s ∗ s ) , as claimed. (cid:3) We next observe that the functional m in the latter proposition can be approximated by func-tionals of finite support. Lemma 4.20.
Let α : S → I ( X ) be a representation of S . If X is domain measurable then forevery ε > and finite F ⊂ S there is a positive h ∈ ℓ ( X ) of norm one such that || s ∗ h − s ∗ sh || < ε for every s ∈ F . Furthermore, the support of h is finite.Proof. We denote by Ω the set of positive h ∈ ℓ ( X ) of norm one and finite support. By Propo-sition 4.19 there is a functional m : ℓ ∞ ( X ) → C such that m ( sf ) = m ( s ∗ sf ) for every s ∈ S, f ∈ ℓ ∞ ( X ). Since the normal states are weak-* dense in ( ℓ ∞ ( X )) ∗ there is a net { h λ } λ ∈ Λ in Ω such that(4.4) | φ h λ ( sf ) − φ h λ ( s ∗ sf ) | = | φ s ∗ h λ ( f ) − φ s ∗ sh λ ( f ) | → , where φ h ( f ) = P x ∈ X h ( x ) f ( x ) and h ∈ Ω , f ∈ ℓ ∞ ( X ). In order to transform the latter weakconvergence to norm convergence, we shall use a variation of a standard technique (see [18, 41, 16]).Consider the space E = (cid:0) ℓ ( X ) (cid:1) S , which, when equipped with the product topology, is a locallyconvex linear topological space. Consider the map T : ℓ ( X ) → E, h T ( h ) = ( s ∗ h − s ∗ sh ) s ∈ S . Since the weak topology coincides with the product of weak topologies on E , it follows from Eq. (4.4)that 0 belongs to the weak closure of T (Ω). Furthermore, since E is locally convex and T (Ω) isconvex, its closure in the weak topology and in the product of the norm topologies are the same.Thus we may suppose that the net { h λ } λ ∈ Λ actually satisfies that || s ∗ h λ − s ∗ sh λ || → s ∈ S ,which completes the proof. (cid:3) The following lemma is straightforward to check, but we mention it for convenience of the reader.
Lemma 4.21.
Let X be a set. Any h ∈ ℓ ( X ) + of norm and finite support can be written as h = ( β / | A | ) P A + · · · + ( β N / | A N | ) P A N for some finite A ⊃ A ⊃ · · · ⊃ A N , where β i ≥ and P Ni =1 β i = 1 .Proof. Let 0 =: a < a < · · · < a N be the distinct values of the function h . Then, defining A i := { x ∈ X | a i ≤ h ( x ) } we have that A ⊃ A ⊃ · · · ⊃ A N . Furthermore h = P Ni =1 γ i P A i , where γ i = a i − a i − for i ≥
1. To conclude the proof put β i := γ i | A i | , i = 1 , . . . , N , and note that k h k = 1implies P Ni =1 β i = 1. (cid:3) Lemma 4.22.
Let α : S → I ( X ) be a representation of S and consider s ∈ S , A ⊂ X . Then ( s ∗ P A − s ∗ sP A ) ( x ) < if and only if x ∈ A ∩ D s ∗ s \ s ∗ ( A ∩ D ss ∗ ) .Proof. By definition of the action given in Eq. (4.2) we compute( s ∗ P A − s ∗ sP A ) ( x ) = x ∈ D s ∗ s \ A and sx ∈ A − x ∈ A ∩ D s ∗ s and sx A . Thus, if ( s ∗ P A − s ∗ sP A ) ( x ) < x ∈ A ∩ D s ∗ s \ s ∗ ( A ∩ D ss ∗ ). The other implication is clear. (cid:3) We can finally establish the main theorem of the section, characterizing the domain measurablerepresentations of an inverse semigroup.
Theorem 4.23.
Let S be a countable and discrete inverse semigroup with identity ∈ S and α : S → I ( X ) be a representation of S on X . The following are then equivalent: (1) X is S -domain measurable. (2) X is not S -paradoxical. (3) X is S -domain Følner.Proof. (1) ⇒ (2). Suppose X is S -paradoxical. Then, choosing a domain measure µ and an S -paradoxical decomposition of X we would have1 = µ ( X ) ≥ µ ( A ) + · · · + µ ( A n ) + µ ( B ) + · · · + µ ( B m )= µ (cid:0) s A ⊔ · · · ⊔ s n A n (cid:1) + µ (cid:0) t B ⊔ · · · ⊔ t m B m (cid:1) = µ ( X ) + µ ( X ) = 2 , which gives a contradiction.(2) ⇒ (1). Consider the type semigroup Typ( α ) of the action. As X is not S -paradoxical we knowthat [ X ] and 2 · [ X ] are not equal in Typ( α ). It follows from Lemma 4.17(5) that ( n + 1) · [ X ] n · [ X ]and hence, by Tarski’s Theorem 4.18, there exists a semigroup homomorphism ν : Typ( α ) → [0 , ∞ ]such that ν ([ X ]) = 1. Then we define µ ( B ) := ν ([ B ]) which satisfies µ ( X ) = 1 and µ ( B ) = µ ( sB )for every B ⊂ D s ∗ s , proving that X is S -domain measurable.(3) ⇒ (1). Let { F n } n ∈ N be a sequence witnessing the S -domain Følner property of X and let ω be a free ultrafilter on N . Consider the measure µ defined by µ ( B ) := lim n → ω | B ∩ F n | / | F n | . MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 17 It follows from ω being an ultrafilter that µ is a finitely additive measure. Thus it remains to provethat µ ( B ) = µ ( sB ) for any B ⊂ D s ∗ s . Observe first that s acts injectively on B . Therefore we have | B ∩ F n | = | s ( B ∩ F n ) | ≤ | sB ∩ s ( F n ∩ D s ∗ s ) ∩ F n | + | ( sB ∩ s ( F n ∩ D s ∗ s )) \ F n |≤ | sB ∩ F n | + | s ( F n ∩ D s ∗ s ) \ F n | , and hence, normalizing by | F n | and taking ultralimits on both sides, we obtain µ ( B ) ≤ µ ( sB ). Theother inequality follows from a similar argument, noting that | s ∗ ( sB ∩ F n ) | = | sB ∩ F n | since s ∗ acts injectively on sB .(1) ⇒ (3). To prove this implication we will refine Namioka’s trick (see [41, Theorem 3.5]). ByLemma 4.20 and the fact that X is domain measurable we conclude that for every ε > F ⊂ S (which we assume to be symmetric, i.e., F = F ∗ ) there is a positive function h ∈ ℓ ( X )of norm 1 and with finite support such that || s ∗ h − s ∗ sh || < ε/ |F | for all s ∈ F . Moreover, byLemma 4.21, we may express the function h as a linear combination h = β | A | P A + · · · + β N | A N | P A N , where A ⊃ A ⊃ · · · ⊃ A N and N X i =1 β i = 1 . Consider now the set B s := ∪ Ni =1 ( A i ∩ D s ∗ s ) \ s ∗ ( A i ∩ D ss ∗ ). By Lemma 4.22, the function s ∗ h − s ∗ sh is non-negative on X \ B s and hence ε |F | > || s ∗ h − s ∗ sh || ≥ X x ∈ X \ B s s ∗ h ( x ) − s ∗ sh ( x ) = N X i =1 β i | A i | X x ∈ X \ B s (cid:16) s ∗ P A i ( x ) − s ∗ sP A i ( x ) (cid:17) ≥ N X i =1 β i | A i | X x ∈ s ∗ ( A i ∩ D ss ∗ ) \ A i ∩ D s ∗ s (cid:16) s ∗ P A i ( x ) − s ∗ sP A i ( x ) (cid:17) = N X i =1 β i | s ∗ ( A i ∩ D ss ∗ ) \ A i ∩ D s ∗ s || A i | = N X i =1 β i | s ∗ ( A i ∩ D ss ∗ ) \ A i || A i | . (4.5)Observe that the last inequality follows from the fact that the sets A i are nested. Indeed, writing Z i = A i ∩ D s ∗ s and T i = s ∗ ( A i ∩ D ss ∗ ) and denoting by Y c the complement in X of a subset Y , weneed to show that(4.6) Z ci ∩ T i ⊂ B cs = ∩ Nj =1 ( Z cj ∪ T j ) . Now, if i ≥ j then A i ⊂ A j and so T i ⊂ T j which implies that Z ci ∩ T i ⊂ Z cj ∪ T j . If i < j then A j ⊂ A i and so Z j ⊂ Z i which implies Z ci ⊂ Z cj and so Z ci ∩ T i ⊂ Z cj ∪ T j . This shows (4.6). Therest of the proof is similar to [41]. Denote by I = { , . . . , N } and consider the measure on I givenby µ ( J ) = P j ∈ J β j for every J ⊂ I and put µ ( ∅ ) := 0. For s ∈ F consider the set K s := { i ∈ I | | s ( A i ∩ D s ∗ s ) \ A i | < ε | A i |} . From Eq. (4.5) it follows that ε/ |F | > N X i =1 β i | s ∗ ( A i ∩ D ss ∗ ) \ A i | / | A i | ≥ ε X i ∈ I \ K s ∗ β i = ε µ ( I \ K s ∗ )and, thus, µ ( I \ K s ∗ ) < / |F | . From this and since F = F ∗ we obtain1 − µ ( ∩ s ∈F K s ) = µ ( I \ ∩ s ∈F K s ) = µ ( ∪ s ∈F I \ K s ) ≤ X s ∈F µ ( I \ K s ) < . Therefore the set ∩ s ∈F K s is not empty since its measure is non-zero; for any index i ∈ ∩ s ∈F K s wewill have that the corresponding set A i satisfies the domain Følner condition. (cid:3) Remark 4.24.
We mention here that the theory of type semigroups for representations of inversesemigroups includes the corresponding theory for partial actions of groups. Given a (discrete)group G and a non-empty set X , Exel defines the notion of a partial action of G on X (see, e.g.,[25]). In this context one can associate in a natural way the type semigroup Typ( X, G ) to thegiven partial action (see e.g. [2, Section 7]). Moreover, in [23] Exel associates to each group G aninverse semigroup S ( G ) such that the partial actions of G on X are in bijective correspondence withthe representations α : S ( G ) → I ( X ). (Note that representations of inverse semigroups are called actions in [23].) In this context, it can be shown, using the abstract definitions of these semigroups,that the type semigroup Typ( α ) introduced in Definition 4.12 is naturally isomorphic to the typesemigroup Typ( X, G ) of the corresponding partial action of G on X .4.4. Amenable inverse semigroups.
The goal of this section is to prove the analogue of The-orem 4.23 but considering amenable representations instead of the weaker notion of domain mea-surable ones. Therefore we will have to refine the reasoning of the previous section including thelocalization condition. In fact, let us first recall that by Proposition 4.6 the classical definition ofinvariant measure given by Day can be characterized by domain measurability and the condition µ ( A ∩ s ∗ sA ) = µ ( A ) , s ∈ S , A ⊂ S .
Note that, since A ∩ s ∗ sA = A ∩ s ∗ sS , we can call this property localization , for the measure µ isconcentrated in the domain of the projection s ∗ s ∈ E ( S ). We now extend this definition to thecontext of representations. Definition 4.25.
Let α : S → I ( X ) be a representation of the inverse semigroup S and let A ⊂ X be a subset. Then A is S -amenable when there is a measure µ : P ( X ) → [0 , ∞ ] such that:(1) µ ( A ) = 1.(2) µ ( B ) = µ ( α s ( B )) for all s ∈ S and B ⊂ D s ∗ s .(3) µ ( B ) = µ ( B ∩ D t ∗ t ) for all t ∈ S and B ⊂ X .We say that X is S -amenable when the latter holds for A = X .The following Lemma is just a simple observation, but it will be useful for later use. Lemma 4.26.
Every countable and inverse semigroup S has a decreasing sequence of projections { e n } n ∈ N that is eventually below every other projection, that is, e n ≥ e n +1 and for every f ∈ E ( S ) there is some n ∈ N such that f ≥ e n .Proof. Since S is countable we can enumerate the set of projections E ( S ) = { f , f , . . . } . TheLemma follows by letting e n := f . . . f n . (cid:3) The localization property of the measure can be included in the reasoning leading to Theorem 4.23,and this yields the following theorem.
Theorem 4.27.
Let S be a countable and discrete inverse semigroup with identity ∈ S and α : S → I ( X ) be a representation of S on X . Then the following conditions are equivalent: (1) X is S -amenable. (2) D e is not S -paradoxical for any e ∈ E ( S ) . (3) For every ε > and finite F ⊂ S there is a finite non-empty F ⊂ X such that F ⊂ D s ∗ s and | sF \ F | < ε | F | for all s ∈ F .Proof. Observe the equivalence between (1) and (2) follows from Theorem 4.23 and Lemma 4.26.Indeed, one can check that the proof of (1) ⇔ (2) in Theorem 4.23 works for any subset A ⊂ X , inparticular if A = D e . Thus, if D e is not paradoxical for any projection e then there are measures µ e on X such that µ e ( D e ) = 1. Now, by Lemma 4.26, let { e n } n ∈ N be a decreasing sequence ofprojections that is eventually below every other projection. A measure in X can be given by: µ ( B ) := lim n → ω µ e n ( B ∩ D e n ) , B ⊂ X , where ω is a free ultrafilter of N . It is routine to show that µ is then a probability measure on X satisfying the domain measurability and localization conditions mentioned above, i.e., µ ( A ) = µ ( sA )when A ⊂ D s ∗ s and µ ( B ) = µ ( B ∩ D s ∗ s ) for every s ∈ S , B ⊂ X . MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 19 In order to prove (3) ⇒ (1) observe that the condition (3) ensures the existence of a domainFølner sequence { F n } n ∈ N such that for every s ∈ S there is a number N ∈ N with F n ⊂ D s ∗ s for all n ≥ N . Consider then a free ultrafilter ω on N and the measure µ ( B ) := lim n → ω | B ∩ F n | / | F n | . It follows from Theorem 4.23 that µ is a domain measure, which, in addition, satisfies that µ ( D s ∗ s ) = lim n → ω | D s ∗ s ∩ F n | / | F n | = lim n → ω | F n | / | F n | = 1 , for all s ∈ S , i.e., the measure is localized.We will only sketch the proof (1) ⇒ (3) since it is just a refinement of the same reasoning as inTheorem 4.23. Let µ be an invariant measure on X . The corresponding mean m : ℓ ∞ ( X ) → C (seeProposition 4.19) satisfies m ( P s ∗ s ) = 1 for all s ∈ S . Then, any net h λ converging to m in normmust also satisfy || h λ (1 − P s ∗ s ) || → s ∈ S . In particular, this must also be the case forthe approximation h appearing in Lemma 4.20. To get the desired Følner set, we have to cut h sothat its whole support is within D s ∗ s for all s ∈ F . For this, consider F = { s , . . . , s k } and definethe function g := h P s ∗ s ...s ∗ k s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h P s ∗ s ...s ∗ k s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The function g has norm 1, is positive and has finite support, which is contained in D s ∗ s for all s ∈ F . Furthermore || h − g || ≤ ε . Thus, by substituting h by g in the proof of Theorem 4.23 andfollowing the same construction, we obtain a Følner set F within the support of g , that is, a Følnerset within the requirements of the theorem. (cid:3) Remark 4.28.
Following results in [29, Theorem 3.1] one has that for inverse semigroups amenabil-ity is equivalent to the Følner condition and a local injectivity condition on the Følner sets. Thepreceding theorem is an improvement of Gray and Kambites’ result applied to inverse semigroups.Amenability gives, in fact, that Følner sets can be taken within the corresponding domains and,thus, the local injectivity condition is guaranteed by the localization property of the measures.Note that Theorem 4.27 is not constructive in the sense that if X is not amenable, then one knowssome D e is paradoxical, but Theorem 4.27 does not tell which element e satisfies this condition.In the case S has a minimal projection e then one can improve the preceding Theorem. To do thiswe first remark the following simple and useful lemma. Lemma 4.29.
Let S be a discrete and countable inverse semigroup with a minimal projection e ∈ E ( S ) . Then e commutes with every s ∈ S .Proof. Note that e = e se s ∗ = s ∗ e s e by the minimality of e . Thus, we obtain e s = e se s ∗ s = e s e = s s ∗ e s e = s e , where, for the first equality we have multiplied the identity e = e se s ∗ from the right by s , andfor the last equality we have multiplied the identity e = s ∗ e se from the left by s . This proves theclaim. (cid:3) Proposition 4.30.
Let S be a discrete, countable inverse semigroup with a minimal projection e ∈ E ( S ) and let α : S → I ( X ) be a representation. Then the following conditions are equivalent: (1) X is S -amenable. (2) D e is not S -paradoxical. (3) D e is S -domain Følner.Proof. The implication (1) ⇒ (2) is a particular case of Theorem 4.27. For (2) ⇒ (3) note that itfollows from Lemma 4.29 that D e ⊂ X is an invariant subset for the action. Indeed, for any s ∈ Ss ( D e ∩ D s ∗ s ) = sD e ⊂ D e . The implication (2) ⇒ (3) therefore follows from Theorem 4.23 by considering, if necessary, theinduced action of S on D e . Finally, (3) ⇒ (1) can be proven in a similar fashion to that of (3) ⇒ (1) in Theorem 4.27. (cid:3) As an example of an inverse semigroup S with a minimal projection we consider the case where S satisfies the Følner condition but not the proper Følner condition (see Definition 3.1).
Proposition 4.31.
Let S be a countable and discrete inverse semigroup. Suppose S satisfies theFølner condition but not the proper Følner condition. Then S has a minimal projection.Proof. Following Theorem 3.8 there is an element a ∈ S such that | Sa | < ∞ . Suppose Sa = { s a, . . . , s k a } . Then we claim e := s ∗ s . . . s ∗ k s k aa ∗ ∈ S is a minimal projection. Indeed, for anyother projection f ∈ E ( S ) there is an i such that s i a = f a . In this case we have a ∗ f a = ( f a ) ∗ f a = ( s i a ) ∗ s i a = a ∗ s ∗ i s i a. Thus, multiplying by a from the left, aa ∗ f a = f a = aa ∗ s ∗ i s i a = s ∗ i s i a . Therefore f ≥ f aa ∗ = s ∗ i s i aa ∗ ≥ e , proving that e is indeed minimal. (cid:3) Note that, in order to produce an example of an inverse semigroup S that is Følner but not properFølner, as in the hypothesis of Proposition 4.31, S must have a minimal projection e ∈ E ( S ).Moreover, by Proposition 4.30, the domain D e must be domain-Følner. It can thus be shown that S := F ⊔ { } satisfies the Følner condition but not the proper one.5. Inverse semigroups, C ∗ -algebras and traces In this final part of the article we connect the analysis of the previous section with properties ofa C*-algebra R X generalizing the uniform Roe algebra R G of a group G presented in Section 2. Inparticular, we will show that domain measurability completely characterizes the existence of tracesin R X .Let S be a discrete inverse semigroup with identity and consider a representation α : S → I ( X )on a set X . As before, we will denote α s ( x ) simply by sx for any s ∈ S , x ∈ D s ∗ s . To construct theC ∗ -algebra R X consider first the representation V : S → B (cid:0) ℓ ( X ) (cid:1) given by V s δ x := (cid:26) δ sx if x ∈ D s ∗ s x / ∈ D s ∗ s , where { δ x } x ∈ X ⊂ ℓ ( X ) is the canonical orthonormal basis. V is a ∗ -representation of S in termsof partial isometries of ℓ ( X ). Define the unital *-subalgebra R X,alg in B ( ℓ ( X )) generated by { V s | s ∈ S } and ℓ ∞ ( X ). The C*-algebra R X is defined as the norm closure of R X,alg , i.e., R X := R X,alg = C ∗ (cid:16) { V s | s ∈ S } ∪ ℓ ∞ ( X ) (cid:17) ⊂ B ( ℓ ( X )) . Note that conjugation by V s implements the action of s ∈ S on subsets A ⊂ X . It is straightfor-ward to show the following intertwining equation for the generators of R X : P A V s = V s P s ∗ ( A ∩ D ss ∗ ) , s ∈ S , A ⊂ X , where, as before, P A ∈ B (cid:0) ℓ ( X ) (cid:1) is the orthogonal projection onto the closure of span { δ a | a ∈ A } .Note also that for any s ∈ S , f ∈ ℓ ∞ ( X ) (which we interpret as multiplication operators on ℓ ( X ))we have the following commutation relations between the generators of R X V s f = (cid:0) sf (cid:1) V s or , equivalently , f V s = V s (cid:0) s ∗ f (cid:1) . From this commutation relations, it follows that the algebra R X is actually the closure in operatornorm of the linear span of operators of the form V s P A , where s ∈ S , A ⊂ D s ∗ s . This fact will beused throughout the section. MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 21 Domain measures as amenable traces.
Before proving the next theorem we first need tointroduce some notation and some preparing lemmas. The first result defines a canonical conditionalexpectation from B (cid:0) ℓ ( X ) (cid:1) onto ℓ ∞ ( X ). The proof is virtually the same as in the case of groups(see, e.g., [12]). Lemma 5.1.
The linear map E : B (cid:0) ℓ ( X ) (cid:1) → ℓ ∞ ( X ) given by E ( T ) = X x ∈ X P { x } T P { x } , is a conditional expectation, where the sum is taken in the strong operator topology. To analyze the dynamics on Følner sets of inverse semigroups it is convenient to introduce thefollowing equivalence relation in X . Let α : S → I ( X ) be a representation of S on X . Given a pair u, v ∈ X , we write u ≈ v if there is some s ∈ S such that u ∈ D s ∗ s and su = v . The relation ≈ isan equivalence relation: in fact, since S is unital, u ≈ u , and if u ≈ v then v ≈ u by considering v = su ∈ D ss ∗ . For transitivity, if u ≈ v ≈ w then u ∈ D s ∗ t ∗ ts , where s, t witness u ≈ v and v ≈ w respectively. We will see in Lemma 5.3 that if a set F ⊂ X has only one ≈ -class, then the corner P F R X P F has dimension | F | as a vector space.The next lemma guarantees the existence of transitive domain Følner sets, i.e., of domain Følnersets F such that F/ ≈ is a singleton. Lemma 5.2.
Let α : S → I ( X ) be a representation of S on X . If A ⊂ X is domain Følner then forevery ε > and finite F ⊂ S , there is a ( ε, F ) -domain Følner F ⊂ A with exactly one ≈ -equivalenceclassProof. Since A is domain Følner, for any ε > F ⊂ S there is a finite F ⊂ A such that | s ( F ∩ D s ∗ s ) \ F | < ε |F | | F | , for all s ∈ F . Decomposing F into its ≈ -classes we get F = F ⊔ · · · ⊔ F L , where u ≈ v if and only if u, v ∈ F i for some i . To prove the claim it is enough to prove that some F j must be ( ε, F )-domain Følner.Indeed, suppose for all j = 1 , . . . , L there is an s j ∈ F such that (cid:12)(cid:12)(cid:12) s j (cid:16) F j ∩ D s ∗ j s j (cid:17) \ F j (cid:12)(cid:12)(cid:12) ≥ ε | F j | . Observe that the choice of s j is not unique, but we can consider a particular fixed choice. Arrangethen the indices according to the following: for s ∈ F , consider Λ s := { j ∈ { , . . . , L } | s j = s } .Note that some Λ s might be empty. Define F s := ⊔ i ∈ Λ s F i and observe | s ( F s ∩ D s ∗ s ) \ F s | = X j ∈ Λ s (cid:12)(cid:12)(cid:12) s j (cid:16) F j ∩ D s ∗ j s j (cid:17) \ F j (cid:12)(cid:12)(cid:12) ≥ ε X j ∈ Λ s | F j | = ε | F s | . Taking the sum over all s ∈ F we get ε | F | > X s ∈F | s ( F ∩ D s ∗ s ) \ F | = X s,t ∈F | s ( F t ∩ D s ∗ s ) \ F t |≥ X s ∈F | s ( F s ∩ D s ∗ s ) \ F s | ≥ ε X s ∈F | F s | = ε | F | . This is a contradiction and, thus, some F j must witness the domain Følner condition. (cid:3) The next lemma computes the dimension of a certain corner of the algebra R X . Lemma 5.3.
Let α : S → I ( X ) be a representation of S on X . Let F , F ⊂ X be finite sets suchthat F ∪ F has only one ≈ -class. Then W := P F R X P F has linear dimension | F | | F | .Proof. To prove the claim it suffices to show that for every u i ∈ F i , i = 1 , M u ,u δ x = (cid:26) δ u if x = u is contained in W . Since u ≈ u there must be an element s ∈ S such that u ∈ D s ∗ s and su = u .It is straightforward to prove that in this case M u ,u = P F P u V s P u P F ∈ P F R X P F = W , hence dim W = | F | | F | . (cid:3) We are now in a position to show the main theorem of the section, which characterizes the domainmeasurability of the action in terms of amenable traces of the algebra R X . Theorem 5.4.
Let S be a countable and discrete inverse semigroup with identity ∈ S , and let α : S → I ( X ) be a representation of S on X . Then the following are equivalent: (1) X is S -domain measurable. (2) X is not S -paradoxical. (3) X is S -domain Følner. (4) R X,alg is algebraically amenable. (5) R X has an amenable trace (and hence is a Følner C ∗ -algebra). (6) R X is not properly infinite. (7) [0] = [1] in the K -group of R X .Proof. The equivalences (1) ⇔ (2) ⇔ (3) follow from Theorem 4.23.(1) ⇒ (5). Consider the conditional expectation E : B (cid:0) ℓ ( X ) (cid:1) → ℓ ∞ ( X ) introduced in Lemma 5.1.Since X is S -domain measurable, by Proposition 4.19, there is a mean m : ℓ ∞ ( X ) → C satisfying m ( sf ) = m ( s ∗ sf ). We claim that then φ ( T ) := m ( E ( T )) is a hypertrace on R X . Indeed, observethat linearity, positivity and normalization follow from those of m and E . Hence we only haveto prove the hypertrace property for the generators of R X . Note that since E is a conditionalexpectation we have φ ( f T ) = φ ( T f ) for any f ∈ ℓ ∞ ( X ), T ∈ B (cid:0) ℓ ( X ) (cid:1) . To show the same relationfor the generator V s note first that for any s, t ∈ S the following relation holds: E ( V s T ) ( x ) = (cid:26) T s ∗ x,x if x ∈ D ss ∗ x / ∈ D ss ∗ E ( T V s ) ( y ) = (cid:26) T y,sy if y ∈ D s ∗ s y / ∈ D s ∗ s . It follows from the action introduced in Eq. (4.2) that s E ( T V s ) = E ( V s T ) and E ( T V s ) = E ( T V s ) P s ∗ s and thus φ ( V s T ) = m ( E ( V s T )) = m ( s · E ( T V s )) = m ( E ( T V s ) P s ∗ s ) = m ( E ( T V s )) = φ ( T V s ) , where we used the invariance of the mean in the third equality.(5) ⇒ (6). Suppose that R X is properly infinite and has a hypertrace φ . Then we obtain acontradiction from1 = φ (1) ≥ φ ( W W ∗ ) + φ ( W W ∗ ) = φ ( W ∗ W ) + φ ( W ∗ W ) = φ (1) + φ (1) = 2 , where W , W are the isometries witnessing the proper infiniteness of 1 ∈ R X .(6) ⇒ (2). Suppose that s i , t j , A i , B j , i = 1 , . . . , n , j = 1 , . . . , m , implement the S -paradoxicalityof X , that is, A i ⊆ D s ∗ i s i , B j ⊆ D t ∗ j t j , and X = s A ⊔ . . . ⊔ s n A n = t B ⊔ . . . ⊔ t m B m ⊃ A ⊔ . . . ⊔ A n ⊔ B ⊔ . . . ⊔ B m . Consider now the operators W := V s ∗ P s A + · · · + V s ∗ n P s n A n and W := V t ∗ P t B + · · · + V t ∗ m P t m B m . These are both partial isometries, since V s ∗ i P s i A i and V t ∗ j P t j B j are partial isometries with pairwiseorthogonal domain and range projections. Furthermore, W ∗ W is the projection onto the union ofthe domains of V s ∗ i P s i A i , which is the whole space ℓ ( X ). The same argument proves that W ∗ W = 1.Therefore, to prove the claim we just have to show that W W ∗ and W W ∗ are orthogonal projections. MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 23 But these correspond to projections onto ⊔ ni =1 A i and ⊔ mj =1 B j , respectively, which are disjoint setsin X .(3) ⇒ (4). By Lemma 5.2, given ε > F ⊂ S , there is a finite non-empty F ⊂ X withexactly one ≈ -class and such that | s ( F ∩ D s ∗ s ) \ F | < ε | F | , for all s ∈ F . Consider the space W := P F R X P F = P F R X,alg P F and observe V s W = { V s P F T P F | T ∈ R X,alg } = (cid:8) P s ( F ∩ D s ∗ s ) V s T P F | T ∈ R X,alg (cid:9) ⊂ (cid:8) P s ( F ∩ D s ∗ s \ F ) V s T P F | T ∈ R X,alg (cid:9) + (cid:8) P F P s ( F ∩ D s ∗ s ) V s T P F | T ∈ R X,alg (cid:9) . Therefore V s W + W ⊂ P s ( F ∩ D s ∗ s ) \ F (cid:16) R X,alg (cid:17) P F + P F ( R X,alg ) P F , and, by Lemma 5.3,dim ( W + V s W ) ≤ | F | + | F | | s ( F ∩ D s ∗ s ) \ F | ≤ (1 + ε ) dim ( W ) , which proves the algebraic amenability of R X,alg .(4) ⇒ (5). This follows from one of the main results of [5], which states that if a pre-C ∗ -algebrais algebraically amenable then its closure has an amenable trace (see [5, Theorem 3.17]) and henceis a Følner C*-algebra (cf., Definition 2.2 (1)).(5) ⇒ (7). Any trace φ on R X , by the universal property of the K group, induces a grouphomomorphism φ : K ( R X ) → R such that φ ([ P ]) = φ ( P ) for any projection P ∈ R X . In thiscase φ ([1]) = φ (1) = 1 while 0 = φ (0) = φ ([0]). In particular, [1] = [0] in the K group of R X .(7) ⇒ (2). If X is S -paradoxical, then it follows from Lemma 4.17(3) and the same argument asin the implication (6) ⇒ (2) that [1] = [1] + [1] in K ( R X ). Therefore [1] = [0] in K ( R X ). (cid:3) Theorem 5.4 can be generalized, along the lines of [49], to hold for any set A ⊂ X . Corollary 5.5.
Let S be a countable and discrete inverse semigroup with identity ∈ S , andlet α : S → I ( X ) be a representation of S on X . Let A ⊂ X , then the following conditions areequivalent: (1) A is S -domain measurable. (2) A is not S -paradoxical. (3) There is a tracial weight ψ : R + X → [0 , ∞ ] such that ψ ( P A ) = 1 . (4) P A ∈ R X is not a properly infinite projection.Proof. Most of the proof is similar as in the reasoning leading to Theorem 5.4. The only differenceregards condition (3), and the replacement of the trace with a weight. To prove that (3) ⇒ (1),consider the measure µ ( B ) = ψ ( P B ). This µ will then be a measure on X such that µ ( A ) = 1.Furthermore, invariance follows from ψ being tracial:(5.1) µ ( B ) = ψ ( V s ∗ s P B ) = ψ ( V s P B V s ∗ ) = ψ ( P sB ) = µ ( α s ( B ))for every B ⊂ D s ∗ s .To prove (1) ⇒ (3) we adapt the ideas in [49, Proposition 5.5]. Given a domain measure µ normalized at A , denote by P fin ( X ) the (upwards directed) set of K ⊂ X with finite measure, i.e., µ ( K ) < ∞ . Given K ∈ P fin ( X ) consider the finite measure µ K ( B ) = µ ( K ∩ B ) and extend it, asin Proposition 4.19, to a functional m K . Given a non-negative f ∈ ℓ ∞ ( X ) define m ( f ) := sup K ∈P fin ( X ) m K ( f ) . Then m is R + -linear, lower-semicontinuous, normalized at P A , and satisfies that m ( sf ) = m ( s ∗ sf ),for every s ∈ S, f ∈ ℓ ∞ ( X ). Finally, the weight given by ψ := m ◦ E : R + X → [0 , ∞ ] is a tracialweight. (cid:3) In our last result of the section we point out that one can translate the Følner sequences of X onto Følner sequences of projections in R X (cf., Definition 2.2(2)). To prove it we first recall thefollowing known result, which states that Følner sequences are preserved under C*-closure (see, e.g.,[10]). Lemma 5.6.
Let
T ⊂ B ( H ) be a set of operators on a separable Hilbert space H . Suppose T has aFølner sequence { P n } ∞ n =1 . Then { P n } ∞ n =1 is a Følner sequence for the C ∗ -algebra generated by T . Proposition 5.7.
Let S be a countable and discrete inverse semigroup and α : S → I ( X ) be arepresentation. If X is domain measurable, then there is a sequence of projections { P n } ∞ n =1 ⊂ R X which is a Følner sequence of projections for each operator T ∈ R X .Proof. Choose a S -domain Følner sequence { F n } ∞ n =1 of X , that exists since X is domain measurable(see Theorem 5.4). Consider the orthogonal projection P n onto span { δ f | f ∈ F n } ⊂ ℓ ( X ). Clearly, P n lies within R X , so, by Lemma 5.6, it is enough to show that it is a Følner sequence for allgenerating elements V s P A , s ∈ S , A ⊂ X . For this we compute( V s P A P n ) ( δ x ) = (cid:26) δ sx , if x ∈ D s ∗ s ∩ F n ∩ A , otherwise( P n V s P A ) ( δ x ) = (cid:26) δ sx , if x ∈ A ∩ s ∗ ( F n ∩ D ss ∗ )0 , otherwise . Thus we have the following estimates in the Hilbert-Schmidt norm: || V s P A P n − P n V s P A || ≤ |{ x ∈ F n ∩ D s ∗ s | sx F n }| + | s ∗ ( F n ∩ D ss ∗ ) \ F n | = | s ( F n ∩ D s ∗ s ) \ F n | + | s ∗ ( F n ∩ D ss ∗ ) \ F n | . Noting that || P n || = | F n | the result follows from normalizing by | F n | and taking limits on bothsides of the inequality. (cid:3) Traces and amenable traces.
The aim of this section is to prove that either all traces in R X are amenable or this algebra has no traces at all. Similar results hold for nuclear C*-algebras (see[12, Proposition 6.3.4]) and for uniform Roe algebras over metric spaces (see [5, Corollary 4.15]).We begin by proving that traces of R X factor through ℓ ∞ ( X ) via the conditional expectation E (seeLemma 5.1). Lemma 5.8.
Let α : S → I ( X ) be a representation and let φ : R X → C be a trace. Then φ ( T ) = φ ( E ( T )) for every T ∈ R X , where E denotes the canonical conditional expectation onto ℓ ∞ ( X ) .Proof. Since the closure of the linear span of the elements of the form V s P A , s ∈ S , A ⊂ D s ∗ s , isdense in R X it is enough to show the claim for these elements.First suppose that A has no fixed points under s , i.e., { a ∈ A | sa = a } = ∅ . Consider the graphwhose vertices are the elements of A and such that two vertices a, b are joined by an edge if and onlyif b = sa . Since the action of s is injective on A it is clear that every vertex has at most degree 2 andno loops. Therefore it can be colored by 3 colors and, thus, there is a partition A = B ⊔ B ⊔ B such that if a ∈ B i then sa B i . This allows us to decompose V s P A as V s P A = V s P B + V s P B + V s P B . Taking traces on each side of the equality gives φ ( V s P A ) = φ ( P B V s P B ) + φ ( P B V s P B ) + φ ( P B V s P B ) = 0 . But in this case we also have E ( V s P A ) = 0 and the equality φ = m ◦ E follows.Second, for arbitrary s ∈ S and A ⊂ D s ∗ s one can decompose A = B ⊔ C , where B := { a ∈ A | sa = a } is the set of fixed points and C := { a ∈ A | sa = a } . In this case V s P A = V s P B + V s P C . By the above paragraph it follows that φ ( V s P C ) = 0, while V s P B is a projection in ℓ ∞ ( S ). Thus φ ( V s P A ) = φ ( V s P B ) + φ ( V s P C ) = φ ( E ( V s P B )) = φ ( E ( V s P A )) , which concludes the proof. (cid:3) MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 25 The following theorem is a consequence of Theorem 5.4 and the latter lemma.
Theorem 5.9.
Let S be a countable and discrete inverse semigroup with identity ∈ S , and let α : S → I ( X ) be a representation. Consider a positive linear functional φ on R X . Then the followingconditions are equivalent: (1) φ is an amenable trace on R X . (2) φ is a trace on R X . (3) φ = φ | ℓ ∞ ( X ) ◦E and the measure µ ( A ) := φ ( P A ) satisfies domain measurability, i.e., µ ( A ) = µ ( sA ) for all s ∈ S , A ⊂ D s ∗ s .Proof. The fact that (1) ⇒ (2) is obvious. For (2) ⇒ (3) note that by Lemma 5.8 it remains onlyto prove that µ ( A ) = µ ( sA ) for every s ∈ S , A ⊂ D s ∗ s . This follows from φ being a trace andEq. (5.1). Finally, (3) ⇒ (1) is proved as the implication (1) ⇒ (5) in Theorem 5.4. (cid:3) We summarize next some important consequences of the previous theorems.
Corollary 5.10.
Let S be a countable and discrete inverse semigroup with identity ∈ S , and let α : S → I ( X ) be a representation. Then: (1) X is S -domain measurable if and only if there is a trace on R X , in which case every traceon R X is amenable. (2) There is a canonical bijection between the space of measures on X such that µ ( A ) = µ ( sA ) when s ∈ S , A ⊂ D s ∗ s and the space of traces of R X . Traces in amenable inverse semigroups.
The goal of this last section is to state the ana-logue of Theorem 5.4 but considering amenable semigroups instead of the weaker notion of domainmeasurable ones. That is, we will give additional C*-algebraic characterizations of amenable inversesemigroups than those in Theorem 4.27. The proof of the following result is straightforward (seethe proof of Theorem 5.4).
Theorem 5.11.
Let S be a countable and discrete inverse semigroup with identity ∈ S and let α : S → I ( X ) be a representation. Then the following conditions are equivalent: (1) X is S -amenable. (2) D e is not S -paradoxical for any e ∈ E ( S ) . (3) R X has a trace φ such that φ ( V e ) = 1 for all e ∈ E ( S ) . (4) No projection V e ∈ R X is properly infinite. In the appendix to [19], Rosenberg showed that any countable discrete group G with a qua-sidiagonal left regular representation is amenable (see [12]). This result implies that if C ∗ r ( G ) isquasidiagonal then G is amenable (cf., [12, Corollary 7.1.17]). That the reverse implication alsoholds was recently shown in [54, Corollary C]. Quasidiagonality of a C*-algebra can be defined interms of a net of unital completely positive (u.c.p.) maps (see, for example, [12, Definition 7.1.1]and [57, 58]): Definition 5.12.
A unital separable C*-algebra A is called quasidiagonal if there exists a sequenceof u.c.p. maps ϕ n : A → M k n ( C ) which is both asymptotically multiplicative, i.e., k ϕ n ( AB ) − ϕ n ( A ) ϕ n ( B ) k → A, B ∈ A , and asymptotically isometric, i.e., k A k = lim n →∞ k ϕ n ( A ) k , A ∈ A .We conclude this article by showing that Rosenberg’s implication still holds for some special classof inverse semigroups. We also prove that the reverse implication is false (the so called Rosenbergconjecture in the case of groups, see [12, 54]). Recall that the reduced C ∗ -algebra of an inversesemigroup is the C ∗ -algebra generated by the left regular representation: C ∗ r ( X ) := C ∗ (cid:0) { V s } s ∈ S (cid:1) ⊂ R X . Note that the following theorem can only be true for the reduced C ∗ -algebras (either in this contextor for groups), since the uniform Roe algebras R X (and R G ) are almost never finite, and thus almostnever quasi-diagonal (see [12, Proposition 7.1.15]). Theorem 5.13.
Let S be a countable and discrete inverse semigroup with identity ∈ S , and let α : S → I ( X ) be a representation. Then: (1) If C ∗ r ( X ) is quasidiagonal then X is S -domain measurable. (2) If C ∗ r ( X ) is quasidiagonal and S has a minimal projection then X is S -amenable. (3) There are amenable inverse semigroups with and without minimal projections with non-quasidiagonal reduced C ∗ -algebras.Proof. As is customary in the literature, we will denote by tr k n the normalized trace in M k n ( C ),while Tr will stand for the usual trace, i.e., tr k n ( · ) = Tr ( · ) /k n .The proof of (1) is a particular case of Proposition 7.1.6 in [12]. Indeed, let ϕ n : C ∗ r ( X ) → M k n ( C )be a sequence of u.c.p. maps that are asymptotically multiplicative and isometric. Then any clusterpoint τ of { tr k n ◦ ϕ n } n ∈ N is an amenable trace of C ∗ r ( X ). Thus let Φ be any hypertrace extending τ , and consider the measure µ ( A ) := Φ ( P A ). It follows from Eq. (5.1) that µ is a domain measure.Let e ∈ E ( S ) be the minimal projection and consider ϕ n : C ∗ r ( X ) → M k n ( C ) a sequence ofunital completely positive (u.c.p.) maps that are asymptotically multiplicative and asymptoticallyisometric (see Definition 5.12). To prove (2) we will construct a new sequence of u.c.p. maps φ n that are asymptotically multiplicative, asymptotically isometric and with asymptotically normalizedtrace in V e . For this, recall that, by Lemma 4.29, e commutes with every s ∈ S .Observe that ϕ n ( V e ) is a positive matrix, whose norm is greater than 1 − ε n and such that || ϕ n ( V e ) − ϕ n ( V e ) ϕ n ( V e ) || < ε n for some ε n > ε n → n → ∞ . We will assumethat ε n < / n . A routine exercise shows that the spectrum of ϕ n ( V e ) is contained in[0 , δ n ) ∪ (1 − δ n ,
1] where δ n = (1 − √ − ε n ). Let r n be the number of eigenvalues of ϕ n ( V e )in (1 − δ n , r n ≥ n ∈ N , since || ϕ n ( V e ) || ≥ − ε n . Let W n ⊂ C k n be the subspace generated by the eigenvectors of ϕ n ( V e ) of eigenvalues close to 1. Finally, let Q n : C k n → C r n be a linear map such that Q n | W n is an isometry onto C r n and Q | W ⊥ n = 0, i.e., Q n : C k n → C r n , Q n Q ∗ n = 1 r n and Q ∗ n Q n = P W n . For each n ∈ N let m n ∈ N be large enough such that(5.2) m n r n k n + m n r n (1 − δ n ) ≥ − δ n . Consider the maps:(5.3) φ n : C ∗ r ( X ) → M k n + m n r n ( C ) , A ϕ n ( A ) ⊕ (cid:0) Q n ϕ n ( A ) Q ∗ n ⊗ m n (cid:1) . By construction it is clear that the maps φ n are unital, completely positive and asymptotically iso-metric. They are also asymptotically multiplicative. For this first observe that, using the minimalityof e and Lemma 4.29, V e commutes with every A ∈ C ∗ r ( X ). Thus, by summing and substracting ϕ n ( V e ) ϕ n ( A ), ϕ n ( A ) ϕ n ( V e ) and ϕ n ( V e A ), we have || Q ∗ n Q n ϕ n ( A ) − ϕ n ( A ) Q ∗ n Q n || ≤ || ϕ n ( A ) || || ϕ n ( V e ) − Q ∗ n Q n || + || ϕ n ( V e ) ϕ n ( A ) − ϕ n ( V e A ) || + || ϕ n ( A ) ϕ n ( V e ) − ϕ n ( AV e ) ||≤ || A || || ϕ n ( V e ) − Q ∗ n Q n || + || ϕ n ( V e ) ϕ n ( A ) − ϕ n ( V e A ) || + || ϕ n ( A ) ϕ n ( V e ) − ϕ n ( AV e ) || n →∞ −−−→ . This asymptotic commutation gives the asymptotic multiplicativity of φ n by a straightforward com-putation since, for any A, B ∈ C ∗ r ( X ) || φ n ( AB ) − φ n ( A ) φ n ( B ) || ≤ || ϕ n ( AB ) − ϕ n ( A ) ϕ n ( B ) || + || Q n ϕ n ( AB ) Q ∗ n − Q n ϕ n ( A ) Q ∗ n Q n ϕ n ( B ) Q ∗ n ||≤ || ϕ n ( AB ) − ϕ n ( A ) ϕ n ( B ) || + || B || || Q ∗ n Q n ϕ n ( A ) − ϕ n ( A ) Q ∗ n Q n || n →∞ −−−→ . MENABILITY AND PARADOXICALITY IN SEMIGROUPS AND C ∗ -ALGEBRAS 27 Furthermore, the maps φ n have the desired asymptotically normalized trace property at V e :1 ≥ tr ( φ n ( V e )) = tr ( ϕ n ( V e )) + m n tr ( Q n ϕ n ( V e ) Q ∗ n ) ≥ m n k n + m n r n Tr ( Q n ϕ n ( V e ) Q ∗ n )= m n k n + m n r n Tr ( Q ∗ n Q n ϕ n ( V e )) ≥ m n r n k n + m n r n (1 − δ n ) ≥ − δ n n →∞ −−−→ , where the last inequality follows from the choice of m n , see Eq. (5.2). By the discussion in (1) andTheorem 5.11, any cluster point Φ of { tr k n + m n r n ◦ φ n } n ∈ N will give an amenable trace of C ∗ r ( X )normalized at V e . Indeed, observe that µ ( D e ) = Φ ( V e ) = lim n →∞ tr k n + m n r n ( φ n ( V e )) = 1 . We conclude that µ is a domain-measure localized at D e and, by Proposition 4.30 and Theorem 5.11, X is then S -amenable.Finally, for (3), consider the bicyclic inverse monoid S = h a, a ∗ | a ∗ a = 1 i . It is routine toshow that E ( S ) = (cid:8) , aa ∗ , a ( a ∗ ) , . . . (cid:9) and, thus, S has no minimal projection. Moreover, it isamenable (see [21, pp. 311]) and C ∗ r ( S ) is not quasidiagonal (since it is not even finite, see [12,Proposition 7.1.15]).For the case when S does have a minimal projection, consider the semigroup T = F ⊔ { } ,where 0 is a zero element. Since T has a zero element it follows that T is amenable (take µ atomicwith total mass at 0) and has a minimal projection (namely 0). Furthermore, note that C ∗ r ( S ) isnot quasidiagonal, since it contains the reduced group C*-algebra of F , which is not quasidiagonal(see [12, Proposition 7.1.10] and [54, Corollary C]). (cid:3) Remark 5.14.
The condition of the existence of a minimal projection e in Theorem 5.13(2) isassumed to guarantee that it commutes with every element s ∈ S . In particular, the proof of (2) inTheorem 5.13 can also be applied to the more general setting of a unital and separable C ∗ -algebra A and a projection P ∈ A , with P commuting with every A ∈ A . Indeed, suppose that A is alsoquasidiagonal. Then, by the same argument, the construction of Eq. (5.3) gives a quasidiagonalapproximation of A with asymptotic normalized trace at P . In both cases, however, the conditionthat AP = P A for every A ∈ A is essential.We conclude with some natural questions. Suppose α : S → I ( X ) is a representation of an inversesemigroup S on a set X . Q1:
Suppose that C ∗ r ( X ) is quasidiagonal. Is X then S -amenable? Q2:
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E-mail address : [email protected] Department of Mathematics, University Carlos III Madrid, Avda. de la Universidad 30, 28911Legan´es (Madrid), Spain and Instituto de Ciencias Matem´aticas - ICMAT (CSIC - UAM - UC3M - UCM).
E-mail address : [email protected] Department of Mathematics, University Carlos III Madrid, Avda. de la Universidad 30, 28911Legan´es (Madrid), Spain and Instituto de Ciencias Matem´aticas - ICMAT (CSIC - UAM - UC3M - UCM).
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