A Construction for Irreducible Representations Is Applied to Various Inverse Semigroup Algebras

A construction for irreducible representations is applied to various inverse semigroup algebras M. J. Crabb, J. Duncan, C. M. McGregor 0. Introduction In recent years there has been a growing number of constructions of faithful irreducible representations for the semigroup algebra FS of some specific inverse semigroups S. As usual, when S has a zero element θ, the contracted semigroup algebra F0S = FS=Fθ is considered. When F = C, the interest has been on faithful irreducible ∗-representations 1 1 of CS (or C0S), and also ` (S) (or `0(S)), on an inner-product space. In the light of the complicated nature of such constructions in the context of free groups, free semigroups and other cases, it is not surprising that some of the recent constructions for certain classes of inverse semigroups were rather difficult. For any inverse semigroup S, we give here a general construction for irreducible repre- ∗ sentations (or -representations) of FS and F0S (or CS and C0S), with easy extensions to the `1-settings. While our approach to the construction is new and very straightforward, in Steinberg [14] a radically different approach produces essentially the same representations. There is also some overlap with work of Exel and recent work on Leavitt (path) algebras; see, for example, Exel [8] and Goncalves [9]. We are grateful for private com- munications pointing out these connections. Our representations are parametrized by the filters of the semilattice of idempotents E(S) of S, equivalently by the multiplicative linear functionals on the commutative algebra FE(S). When S is a group, the construction gives only the trivial one-dimensional representation (associated with the augmentation functional); but for most inverse semigroups, the construction gives a very rich collection of irreducible representations. Moreover, it illuminates the known constructions of faithful irreducible representations for several important classes of inverse semigroups; see, for example, [1, 3, 5]. In x1 we present the (surprisingly elementary) construction for these irreducible ∗ -representations of CS and C0S, and the corresponding irreducible representations for 1 FS and F0S. We then give the necessary modifications for the Banach algebra ` -setting. In x2 we apply the construction to a series of known examples in [1, 5] thereby presenting simpler proofs of known results on primitivity and ∗-primitivity. In x3 we elaborate the discussion for the case when S is the polycyclic semigroup PX on a set X. Quotients ∗ 1 ∗ of the -algebras C0PX and `0(PX ) are pre-algebras of the important Cuntz C -algebras. Our emphasis here is on giving very elementary proofs of several results. For example, the proof that the Cuntz C∗-algebras are simple requires substantial effort; for the corresponding result in the purely algebraic or `1-setting, the proof requires only a few lines (thanks to the explicit description of generic elements in these other algebras). For these algebras, we also give, very easily, uncountable families of faithful irreducible ∗-representations no two of which are spatially equivalent. 1 1. The construction Let S be an inverse semigroup, and let E(S) be its semilattice of idempotents. We give a general construction (see also Steinberg [14, Proposition 7.19]) for some irreducible ∗- representations of the semigroup algebra CS (and the contracted semigroup algebra C0S, when S has a zero element), and then for the `1-completions of these algebras. When we work over an arbitrary field F, the construction gives an irreducible representation. We recall that, for any semilattice E, a filter F is a (non-empty) subsemilattice of E (so e; f 2 F implies ef 2 F) with "e ⊆ F for every e 2 F, where "e = fu 2 E : u ≥ eg. We write 2 for the usual two element semilattice f0; 1g. It is well known that F : E ! 2 is a homomorphism if and only if fu 2 E : F (u) = 1g is a filter. Any such homomorphism F has a natural extension to a multiplicative linear functional on CE, also denoted by F . We begin with the algebra CS where S does not have a zero element. Let F be any filter in E(S) with corresponding multiplicative linear functional F on CE(S). For s 2 S; e 2 E(S), we have s∗es 2 E(S) and so we may define sF on E(S) by sF (e) = F (s∗es)(e 2 E(S)): Note that sF is a homomorphism into 2 since, for u; v 2 E(S), sF (u)sF (v) = F (s∗us)F (s∗vs) = F (s∗uss∗vs) = F (s∗uvss∗s) = F (s∗uvs) = sF (uv): Let VF be the complex linear span of all the non-zero multiplicative linear functionals sF . We show first that VF is an inner-product space and that the action t : sF 7! tsF is well-defined and has a natural extension to a well-defined irreducible ∗-representation of CS on VF . Lemma 1.1. (i) Let G be a filter in E(S) with corresponding multiplicative linear functional G, and let u 2 E(S). Then uG = G or uG = 0 according as G(u) = 1 or G(u) = 0. (ii) We have sF 6= 0 if and only if F (s∗s) = 1, and sF = 0 if and only if F (s∗s) = 0. Proof. (i) For e 2 E(S) we have uG(e) = G(u∗eu) = G(ue) = G(u)G(e). The result follows. 6= 0 6= 0 = 1 (ii) Using (i) we have sF , s∗sF , F (s∗s) : = 0 = 0 = 0 There is a natural inner-product structure on the complex linear span of the non-zero multiplicative linear functionals sF by taking these to be an orthonormal basis. Note from Lemma 1.1 that sF = F for every s in the filter F. More generally, by Lemma 1.3 below, we have sF = tF 6= 0 if and only if s∗tF = F . Lemma 1.2. The action t : sF 7! tsF has a well defined natural extension to an action of CS on VF . P Proof. Let t 2 S. Clearly, t(sF ) = (ts)F . Let αjsjF = 0 with αj 2 C; sj 2 S. For u 2 E(S), we have X X ∗ ∗ X ∗ αjtsjF (u) = αjF (sj t utsj) = αjsjF (t ut) = 0 2 since t∗ut 2 E(S). This shows, by a special case, that the original action is well-defined, and then that the action by t lifts to a well-defined action on VF . The usual linear extension gives a representation of CS on VF ; we denote it by πF . Note that the action by πF is always given by the same formula; only the representing space VF changes. When it is helpful, we simplify the notation by writing πF (a)v as av. ∗ Lemma 1.3. The representation πF is a -representation. ∗ ∗ Proof. It is enough to show that πF (t ) = πF (t) for each t 2 S. For this, it is enough to show that ht∗σF; sF i = hσF; tsF i for all non-zero σF; sF 2 VF . Since the non-zero elements sF form an orthonormal basis, each side of the above equation can take only the values 0 or 1. It is thus enough to show that both sides of the above equation take the value 1 simultaneously. This amounts to showing that t∗σF = sF if and only if σF = tsF (with all elements non-zero). Suppose that σF = tsF . Then t∗σF = t∗tsF . By Lemma 1.1 (i) we have F (s∗t∗ts) = 1 and hence t∗tsF = sF , by Lemma 1.1 (ii). This gives t∗σF = sF . Since t; s; σ are arbitrary in S, the reverse implication holds equally. Lemma 1.4. The representation πF is irreducible on VF . Proof. It is obvious that the vector F is cyclic for the representation πF . We have to show the same for any non-zero vector v 2 VF ; for this it is enough to show that we can map v to F . Let v = αsF + α1s1F + ··· + αkskF , where the coefficients are all non-zero and the basis vectors in VF are all distinct. For each j = 1; : : : ; k, we choose qj 2 CE(S) as follows. Since sF 6= sjF , there exists pj 2 E(S) such that, either sF (pj) = 1 and sjF (pj) = 0 in which case we take qj = pj, or sF (pj) = 0 and sjF (pj) = 1 in which case we take qj = 1 − pj. In either case, by Lemma 1.1 (ii), qjsF = sF and qjsjF = 0. Let Q = q1 : : : qk. Then, since the qj commute, Qv = αsF . Using Lemma 1.1 (ii) again, we obtain s∗Qv = αs∗sF = αF . We have proved the following theorem. Theorem 1.5. For any multiplicative linear functional F on CE(S), πF is an irreducible ∗ -representation of CS on the inner-product space VF . When F is constantly 1 on E(S), the representation is one-dimensional and is just the representation corresponding to the augmentation functional on CS. When S is a group, this is the only irreducible representation given by our construction. More generally, let S be a Clifford semigroup, i.e. an inverse semigroup in which each idempotent is central, so that S is just a union of groups. In this case, for any multiplicative linear functional on CE(S), we have sF (e) = F (s∗es) = F (s∗se) = F (s∗s)F (e) and hence sF = 0 or sF = F , so that every irreducible representation πF is one- dimensional. Nonetheless, we shall see in x2 that the construction gives many interesting, even faithful, irreducible representations when S is not a Clifford semigroup.

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