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 §1 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 §2 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 §3 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 ∈ F implies ef ∈ F) with ↑e ⊆ F for every e ∈ F, where ↑e = {u ∈ E : u ≥ e}. We write 2 for the usual two element semilattice {0, 1}. It is well known that F : E → 2 is a homomorphism if and only if {u ∈ E : F (u) = 1} 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 ∈ S, e ∈ E(S), we have s∗es ∈ E(S) and so we may define sF on E(S) by sF (e) = F (s∗es)(e ∈ E(S)). Note that sF is a homomorphism into 2 since, for u, v ∈ 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 ∈ 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 ∈ 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 ∈ S. Clearly, t(sF ) = (ts)F . Let αjsjF = 0 with αj ∈ C, sj ∈ S. For u ∈ E(S), we have

X X ∗ ∗ X ∗ αjtsjF (u) = αjF (sj t utsj) = αjsjF (t ut) = 0

2 since t∗ut ∈ 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 ∈ S. For this, it is enough to show that ht∗σF, sF i = hσF, tsF i

for all non-zero σF, sF ∈ 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 ∈ 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 ∈ CE(S) as follows. Since sF 6= sjF , there exists pj ∈ 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 §2 that the construction gives many interesting, even faithful, irreducible representations when S is not a Cliﬀord semigroup.

3 Suppose now that S has zero element θ. Let F : E(S) → 2 be a homomorphism. Note that Lemma 1.1 holds equally in this context. If F (θ) = 1, then F (e) = 1 for all e; we discard this F . Otherwise F (θ) = 0 and F induces a multiplicative linear functional on CE(S)/Cθ. It is straightforward to verify that there is a corresponding irreducible ∗ -representation πF of the contracted semigroup algebra C0S on the inner product space VF with the action discussed above. Theorem 1.6. Let S have zero element θ. For any multiplicative linear functional F on ∗ CE(S)/Cθ, πF is an irreducible -representation of C0S on the inner-product space VF .

Now let F be any field and replace CS by the algebra FS. In the above construction we now work over the field F instead of the field C. The F-vector space VF is no longer an inner-product space and so the issue of a ∗-representation does not arise. All the other arguments apply, and with the same notation, we obtain the following theorem.

Theorem 1.7. For any multiplicative linear functional F on FE(S), πF is an irreducible representation of FS on the vector space VF . The similar result holds for the contracted semigroup algebra F0S. A routine argument (see, for example, Crabb and Munn [5]) shows that the construction for CS extends to the Banach algebra `1(S). The representation of CS clearly 1 1 extends to a bounded representation of ` (S) on the ` -completion of VF , say XF , which is also an inner-product space. We again denote this representation by πF .

Theorem 1.8. For any multiplicative linear functional F on CE, πF is an irreducible ∗ 1 -representation of ` (S) on the inner-product space XF . The similar result holds for the 1 contracted semigroup algebra `0(S). Proof. We prove that the representation is irreducible. It is enough (see, for example, [5]) to show that πF is topologically irreducible and has one (strictly) cyclic vector. It is clear that F is a (strictly) cyclic vector. Let v ∈ XF with v 6= 0, and let > 0. It is enough to 1 P∞ prove that there exists a ∈ ` (S) with kav − F k < . Let v = αsF + j=1 αjsjF with P P∞ α 6= 0, |αj| < ∞ and all basis vectors distinct. Choose k so that j=k+1 |αj| < |α|. Let Q = q1 . . . qk be as deﬁned in the proof of Lemma 1.4. By Lemma 1.1 (ii), for 1 ≤ i ≤ k < j, qisjF is either sjF or 0. So αjQsjF = βjsjF where βj is αj or 0. Hence ∗ P∞ ∗ ∗ s Qv = αF + j=k+1 βjs sjF . It follows that k(1/α)s Qv − F k < , as required.

2. Some examples We begin with the complex field and we consider first the usual semigroup algebras; when S has no zero element we consider CS, and when S has a zero element we consider C0S. The first author and Munn have produced several interesting examples of primitive semigroup algebras; see [3, 5]. We show here how our construction illuminates and simplifies many of these examples.

Example 2.1. Let PX be the polycyclic (or Cuntz) semigroup on a non-empty set X. Thus PX is the quotient of the free inverse monoid FIMX , with zero attached, by the relations xx∗ = 1, yx∗ = θ for all x, y ∈ X, y 6= x. (We remark that the deﬁning relations

4 are reversed in the C∗-setting, so that the generators correspond to isometries on Hilbert space rather than co-isometries.) When X is a singleton, PX is just the well known bicyclic semigroup. Domanov [7] proved that F0PX is primitive for any X and any ﬁeld F. Another proof was given by Munn [11]. Our construction easily gives a faithful ∗ irreducible -representation for C0PX . Since the bicyclic semigroup case is very well understood, and some of the results below are diﬀerent for that case, we shall assume from now on that |X| ≥ 2 for PX . We require the following three properties of PX (see Lawson [10, §9.3] for proofs).

∗ P 1. Each non-zero element of PX may be uniquely represented in the form u v with u, v ∈ FMX (the free monoid on X) ∗ P 2. Each non-zero idempotent of PX may be uniquely represented in the form v v with v ∈ FMX ∗ ∗ P 3. The product u uv v is non-zero if and only if, for some w ∈ FMX , we have u = wv or v = wu, in which case the product is z∗z where z is the longer of u, v.

We remark that it follows easily from P 3 that uv∗ is non-zero if and only if u = wv or v = wu (for some w ∈ FMX ).

Note that the identity 1 is given by the empty word in FMX . Let F be the multiplicative linear functional corresponding to the filter {1}. For any ∗ ∗ u ∈ FMX , it follows from P 3 that ↑u u is the finite chain consisting of all z z where z is a suffix of u (including z = 1). Thus, for u = u2u1, we have

∗ ∗ ∗ ∗ ↑u u = {1, u1u1, u1u2u2u1}.

It is convenient to denote the corresponding multiplicative linear functional by (u1, u2). More generally, for u = un . . . u2u1, we denote the corresponding multiplicative linear functional by (u1, u2, . . . , un). With this notation we have F = (). For any x ∈ X, we have x∗xF = 0 by Lemma 1.1 (ii), and hence xF = xx∗xF = 0. It follow that vF = 0 ∗ ∗ ∗ for all v ∈ FMX . We have x F (e) = F (xex ) = 1 if and only if e = 1 or e = x x. Thus ∗ ∗ x F = (x). More generally, with u = un . . . u2u1, we have u F = (u1, u2, . . . , un). These latter vectors give our usual orthonormal basis for VF . We now consider the action of πF on these basis vectors. For x ∈ X, we have

∗ ∗ ∗ ∗ x (u1, . . . , un) = x u () = (ux) () = (x, u1, . . . , un).

∗ We also have x(u1, . . . , un) = xu () and this is 0 unless x = u1 in which case we have

∗ u1(u1, . . . , un) = (un . . . u2) () = (u2, . . . , un).

P ∗ To prove that πF is faithful, let a ∈ C0PX with a 6= 0, a = αu∗vu v where each αu∗v 6= 0. Let ξ be an element v of minimal length which appears in the above sum (ξ could be 1); note that any corresponding coeﬃcient αu∗ξ is non-zero. Then, since ∗ P ∗ ∗ ∗ (ξ) = ξ (), a(ξ) = αu∗vu vξ (). It follows from P 3 that vξ is non-zero if and only if ∗ v = zξ for some z ∈ FMX , in which case vξ = z. Hence, since z() = 0 if z 6= 1,

X ∗ X a(ξ) = αu∗zξu z() = αu∗ξ(u) 6= 0.

5 ∗ We have proved that C0PX is -primitive for any X. The same argument applies for 1 `0PX .

Example 2.2. Let S be the free inverse monoid on a set X, S = FIMX . Crabb and Munn [3] prove that CS is primitive if and only if X is infinite (it is not even prime when X is finite). They do not give an explicit faithful irreducible representation but they construct a co-maximal left ideal; Crabb [1] later constructed a faithful irreducible ∗-representation. Here, we construct essentially the same ∗-representation by a suitable choice of multiplicative linear functional on CE(S). We use the Scheiblich viewpoint on FIMX (see [12, 13]). We write GX for the free group on X, and EX for the set of all left-closed finite subsets A of X. (Recall that A is left-closed if, for each g ∈ A, every prefix of the word g (including 1) is also in A.) Thus, FIMX is the set of all pairs (A, g) with A ∈ EX and g ∈ A; and the multiplication is given by

(A, g)(B, h) = (A ∪ gB, gh).

The semilattice E(S) is given by all elements (A, 1), and multiplication of idempotents corresponds to the union of the left-closed sets. For g ∈ GX , the content of g, con(g), consists of all x ∈ X for which x or x−1 occurs in the reduced form of g. For a subset D of GX , con(D) is the union of all con(g) with g ∈ D. Let X be infinite. As in [3], we take S to be a set whose elements are countably infinite pairwise-disjoint subsets of X with union X, and such that |S| = |X|. We easily obtain a bijection ψ : EX → S. For each A ∈ EX , we write φ(A) = ψ(A)\con(A); thus the φ(A) give pairwise-disjoint countably infinite subsets of X. The subset H of GX is defined by   [ [ H = {1} ∪  xA A∈ EX x∈φ(A) and is easily seen to be a left-closed (infinite) subset of GX . Let H be the filter in E(S) generated by the idempotents ({1}∪xA, 1) for all of the above xA. Since H is left-closed, it is easily verified that the corresponding multiplicative linear functional F on CE(S) is determined by the constraint that F ((B, 1)) = 1 if and only if B ⊆ H. We now show ∗ that the corresponding irreducible -representation πF is faithful. Suppose towards a contradiction that ker(πF ) is not the zero ideal. We first recall some notation and a result from Crabb and Munn [3]. Let Y be a finite subset of X and let A ∈ EY . The idempotent σY (A) is defined to be the product of all the factors (A, 1) − (B, 1) where B ∈ EY ,A ⊆ B, |B| = |A| + 1. It follows from [3, Lemma 3] that there exists A ∈ EX and a finite subset Y of X so that con(A) ⊆ Y and P σY (A) ∈ ker(πF ). Thus the kernel contains k = (A, 1) + δj(Bj, 1) where each δj is 1 or −1 and each Bj (strictly) contains A. Now choose x ∈ φ(A) but not in any con(Bj). Let s = (A ∪ x−1, x−1), so that s∗ = (xA ∪ 1, x) and s∗s = (xA ∪ 1, 1). We have (A, 1)s = s. Since F ((xA ∪ 1, 1)) = 1, it follows from Lemma 1.1 (i) that the first term of ksF is −1 −1 non-zero. Let B be any of the Bj. We calculate that (B, 1)s = (B ∪ x , x ) = t, say, and then t∗t = (xB ∪ 1, 1). Since x∈ / con(B), each element of xB has first letter x. The only elements of H with first letter x are those in xA, since the φ(A) are disjoint. Since |B| > |A|, it follows that xB 6⊆ H. We thus have F (t∗t) = 0 and all the other terms of ksF are zero. This gives the contradiction that ksF 6= 0. Thus CFIMX is *-primitive.

6 For faithfulness in the `1 case, see Crabb [1], where the representation is the same as in this construction.

Example 2.3. Let S be the McAlister monoid on a set X, MX . Thus MX is the Rees quotient of FIM by the ideal generated by all elements xy∗, x∗y with x, y ∈ X, x 6= y. X −→ We follow the description of M given in [2]. Thus the elements are given by a b c and ←− X a b c where a, b, c ∈ FMX , the free monoid on the set X. In particular, idempotents occur when b is the empty set, and are denoted by a c. The McAlister monoid has a zero element and so we are considering the algebra C0MX . When X is finite, the algebra is not even prime; see Munn [11]. When X is uncountable, the algebra is not primitive; see [2]. When X is countably infinite, Crabb and Munn [5] proved that the algebra is ∗-primitive by constructing a faithful irreducible star representation. Here we present a shorter argument that gives essentially the same ∗-representation. Since X is countably infinite, so also is FMX . Let {wn} be an enumeration of all the non-empty words in FMX and let {xn} be the sequence in X obtained by reading off all the letters in the concatenation w1w2w3 .... Let en = x1x2 . . . xn. It is clear that F = {en} is a filter in E(S) (it is even a chain). Let F be the corresponding multiplicative ∗ linear functional, so that πF is an irreducible -representation. We show now that πF is faithful. We refer the reader to [2, Lemma 4.5]. The argument given there holds for any 1 uncountable X and it applies equally for the algebras C0MX and `0MX to show that any 1 non-zero ideal I of C0MX or `0MX , in particular ker(πF ) if πF is not faithful, contains some en. But enF = F , and so the proof is complete.

We remark that the above sequence {xn} may be replaced by any sequence {yn} in X for which the inﬁnite word y1y2 ... contains every word wn ∈ FMX as a subword.

3. Contracted semigroup algebras on PX

There has been much interest in algebras associated with the polycyclic semigroup PX , both for X finite and X infinite (see, for example, [4] in the `1 setting, and [6] in the ∗ C -setting). We begin with the contracted semigroup algebra AX = C0PX . Our main aim here is to present short elementary proofs, even when results are known. It is easy to list all the filters in the semilattice E(PX ) with their associated irreducible *-representations. In §2 we considered only the finite filters of E(PX ) and these gave a basis for VF (here denoted by V ) on which we had the faithful irreducible *-representation πF (here denoted by π). Every other filter in E(PX ) is infinite and corresponds to a sequence (ξ1, ξ2,...) of elements of X. When X is finite, every faithful irreducible ∗-representation is spatially equivalent to π. Every irreducible ∗-representation corresponding to an infinite filter has the same kernel KX given by all a ∈ C0PX with π(a) of finite rank, equivalently the ideal P ∗ AX (1 − p)AX where p = {x x : x ∈ X}. When X is infinite, C0PX is a simple algebra and so all irreducible representations are faithful. We show very easily that there exist uncountably many faithful irreducible ∗-representations which are not spatially equivalent. When X is finite, the same result holds for the quotient algebra C0PX /KX (which may be regarded as a pre-Cuntz algebra). We also consider the corresponding results for the 1 Banach algebra AX = `0(PX ) where some special arguments are needed to obtain the corresponding results. For brevity, we consider both AX and AX in the same theorem;

7 we give ﬁrst the purely algebraic proof for AX and then the needed modiﬁcation for the Banach algebra AX .

We recall from example 2.1, that for any non-zero ξ ∈ FMX we have the corresponding ∗ finite filter ↑ξ ξ. With ξ = ξn . . . ξ2ξ1, we denoted the corresponding multiplicative linear functional by F = (ξ1, . . . , ξn). It is clear that we obtain an infinite filter (again a chain) by extending this finite sequence to an arbitrary sequence {ξn} in X. We denote the corresponding multiplicative linear functional by F = (ξ1, . . . , ξn,...). (Recall that we have discarded the trivial filter ↑θ in this context.) These are the only infinite filters in E(PX ), for if e ∈ E(PX ) is outside such an infinite chain then ef = θ for any f in the chain. We thus have a complete description of all the filters in E(PX ). The action on (ξ1, . . . , ξn,...) is similar to the finite filter case with one critical differ- ence. An easy computation gives x(ξ1, . . . , ξn,...) = 0 unless x = ξ1 in which case

ξ1(ξ1, . . . , ξn,...) = (ξ2, . . . , ξn,...).

In the finite filter case with F = (ξ1, . . . , ξn), we have π(v)F = 0 whenever |v| > n; in the infinite case we have πF (v)F 6= 0 for any prefix v of the infinite sequence ξ = ξ1ξ2 . . . ξn ..., in which case, πF (v)F corresponds to the sequence ξ with the prefix v removed. Also,

∗ x (ξ1, . . . , ξn,...) = (x, ξ1, . . . , ξn,...).

We write τn(ξ) for the tail of ξ given by τn(ξ) = (ξn+1, ξn+2,...). An arbitrary basis element of VF is thus of the form (u, τn(ξ)) where u ∈ FMX and τn(ξ) is the longest tail of ξ in the infinite word uτn(ξ). Suppose now that X is finite, say X = {x1, . . . , xm}. Let the first element of ξ be ∗ Pm ∗ xj. Then xiξ = 0 for i 6= j and (1 − xj xj)ξ = 0. Let p = i=1 xi xi. Thus 1 − p is in the kernel of πF . We readily check that π(1 − p) is the rank one orthogonal projection onto C(). Since π is irreducible on V , it follows that π(KX ) consists of all the finite rank operators on V , where KX = AX (1−p)AX . For the Banach algebra AX , we define KX to be the closed ideal of AX consisting of all a ∈ AX for which π(a) is a compact operator 1 on the representing ` space XF .

Theorem 3.1. Let X be ﬁnite and let F = (ξ1, . . . , ξn,...). For AX , we have ker(πF ) = KX ; for AX , we have ker(πF ) = KX .

Proof. For the case of AX , it is enough to prove that ker(πF ) ⊆ KX . Let a ∈ ker(πF ), P ∗ a = αs∗ts t. Let N be an upper bound for the length of all t which appear in a. Let v be any basis vector in V (the representation space for π) with |v| > N. Choose a basis ∗ vector w in VF which begins with v. For η a basis vector in VF , consider all s t in a ∗ with s tw = η. Then the sum of the corresponding αs∗t over this collection is zero. Since |t| ≤ N, we see that s∗tv has a fixed value in V if and only if s∗tw has a fixed value in VF . It follows that π(a)v = 0. This forces π(a) to have finite rank and so completes the proof for AX . We now consider the modification needed for the case of AX . It is well known that π(a) is compact if and only if π(a)v → 0 as |v| → ∞, where v is a basis element of V . P ∗ P Let a ∈ ker(πF ), a = αs∗ts t, with |αs∗t| < ∞. Given N, let aN be the partial sum of a consisting of all terms with |s| ≤ N, |t| ≤ N. Let bN = a − aN . Then bN → 0 as

8 N → ∞ (equivalently, as |v| → ∞). Let v ∈ V with |v| > N. Choose a basis vector w in VF which begins with v. Since πF (a)w = 0, it follows that kπF (aN )wk < N where ∗ ∗ N → 0 as N → ∞. For η a basis vector in VF , consider all s t in aN with s tw = η. Since each |t| ≤ N, we see that s∗tv has a fixed value in V if and only s∗tw has a fixed value in VF . For corresponding fixed terms, the sum of the coefficients determines norms. It follows that kπ(aN )vk = kπF (aN )wk < N .

But kπ(bN )vk → 0 as N → ∞. It follows that π(a)v → 0 as |v| → ∞. Conversely, suppose that π(a)v → 0 as |v| → ∞. Split a = aN + bN as above, where ∗ aN is a ﬁnite sum and kbN k < N . Let M be the longest t which appears in any term s t in aN . Choose P so that kπ(a)vk < N for |v| > P . Let w be any basis vector of XF . Choose an initial segment v of w so that |v| > max(M,P ). As above, we deduce that

kπF (aN )wk = kπ(aN )vk = kπ(a)v − π(bN )vk ≤ kπ(a)vk + kbN k < 2N .

It follows that πF (a)w = 0 for all w and hence πF (a) = 0.

∗ Theorem 3.2. Let X be ﬁnite. Every faithful irreducible -representation of AX or AX is spatially equivalent to π.

∗ Proof. Let ρ be a faithful irreducible -representation of AX on an inner product space W . Since 1 − p is a non-zero idempotent we have ρ(1 − p)w = w for some non-zero w ∈ W . Then

m X hw, wi = hρ(1 − p)w, wi = hw, wi − hρ(xi)w, ρ(xi)wi i=1

and so ρ(xi)w = 0 for i = 1, . . . , m. It follows that ρ(v)w = 0 for all v ∈ FMX , and hence ∗ ρ(u v))w = 0 except when v = 1. For all u ∈ FMX , we have

hρ(u∗)w, ρ(u∗)wi = hw, ρ(uu∗)wi = hw, wi.

Since ρ is irreducible, the representation is determined by all ρ(u∗)w. These are orthogonal since ∗ ∗ ∗ ∗ hρ(u1)w, ρ(u2)wi = hw, ρ(u1u2)wi = hρ(u2u1)w, wi = 0

if u1 6= u2. Hence ρ is spatially equivalent to π with implementing map T : W → V which sends orthonormal basis element ρ(u∗)w to orthonormal basis element (u) = π(u∗)(). The argument extends easily to the case of AX .

Theorem 3.3. Let X be ﬁnite with |X| > 1. Both the pre-Cuntz algebras AX /K and AX /K are simple.

Proof. We begin with AX /K. This has generators S1,...,Sn subject to the relations ∗ ∗ ∗ ∗ SjSj = I,SiSj = 0(i 6= j),I = S1 S1 + ··· + SnSn. Let J be a non-zero ideal of An, and let T ∈ J with T 6= 0. For µ = (µ1, µ2, . . . , µk) with 1 ≤ µj ≤ n, we write

Sµ = Sµ1 Sµ2 ...Sµk . We have

∗ ∗ T = α0(Sµ0 ) Sν0 + α1(Sµ1 ) Sν1 + ···

9 ∗ with non-zero coeﬃcients αj. Note that SµSν = I if and only if µ = ν. Since ∗ (1/α0)Sµ0 T (Sν0 ) ∈ J, we may thus assume, without loss of generality, that the ﬁrst term in T is just I so that

∗ T = I + α1(Sµ1 ) Sν1 + ··· .

∗ ∗ Suppose ﬁrst that the term (Sµ1 ) appears and begins with Si . Choose m 6= i. Then ∗ SmTSm ∈ J and ∗ ∗ ∗ SmTSm = I + α2Sm(Sµ2 ) Sν2 Sm + ··· . (†)

Otherwise T = I + α1Sν1 + ··· . Suppose that the last term of Sν1 is Sj. Choose m 6= j. ∗ Then SmTSm ∈ J and again (†) holds. In either case we eliminate one term from the ∗ sum, and we may rewrite each term in standard form (Sµ) Sν . Continue this procedure, and after finitely many steps we find I ∈ J, as required. For the case of AX /K, we argue as above. Eventually we find in J an element I + R where kRk < 1 and so I + R is invertible, and again the proof is complete.

Suppose now that X is inﬁnite.

Theorem 3.4. For X inﬁnite, both AX and AX are simple.

Proof. Let J be a non-zero ideal in AX . Since π is irreducible on AX , it is also irreducible when restricted to J. Hence there exists a ∈ J with π(a)() = (). Let a = a0 + b0, where P ∗ P a0 = αu∗ u and b0 = x∈X axx for suitable ax ∈ AX . Then X () = π(a)() = αu∗ (u) + 0 and so a0 = 1. Since X is infinite we may choose x ∈ X which does not appear in b0. ∗ ∗ Then xax = xx + 0 = 1 ∈ J. The proof is complete for AX . For AX , argue initially as above, except that a0, b0 may now be (countably) infinite sums. Again we find that a0 = 1. Only a countable set of elements of X appear in b0. When X is uncountably infinite, we may choose z ∈ X disjoint from this countable set. ∗ We then have 1 = zaz ∈ J and hence BX is simple. Suppose finally that X is countably P infinite. We clearly have kaxk < ∞ and so we may choose some x ∈ X with kaxk < 1. ∗ ∗ ∗ Then J contains xax = 1+xaxxx +0, which is invertible since kxaxxx k = kaxk < 1.

Theorem 3.5. For X inﬁnite, each of AX and AX has an uncountable collection of faithful irreducible ∗-representations which are not spatially equivalent.

∗ Proof. Let πs, πt be faithful irreducible -representations of AX on Vs,Vt corresponding to sequences s, t in X. We show that πs, πt are spatially equivalent if and only if the sequences s, t have a common tail, and the result is then clear. Suppose ﬁrst that πs, πt are spatially equivalent. Then there exists an isomorphism P : Vt → Vs such that πs(a)P = P πt(a) for all a. Take aN = tN . . . t2t1. Then, for all N, we have πt(aN )t 6= 0 and so P πt(aN )t 6= 0. Thus πs(aN )P t 6= 0 for all N. Recall ∗ Pk ∗ that the basis elements of Vs are of the form πs(u v)s so P t = i=1 αiπs(ui vi)s, say. ∗ ∗ Since πs(aN )πs(u v)s = 0 unless the sequence πs(u v)s begins t1, t2, . . . , tN , it follows ∗ that, for each N, πs(ui vi)s begins t1, t2, . . . , tN for some 1 ≤ i ≤ k. Hence there must

10 ∗ exist 1 ≤ j ≤ k such that πs(uj vj)s begins t1, t2, . . . , tN for arbitrarily large N. Then ∗ uj vjs = t so that s and t have a common tail. Conversely, suppose that s, t have a common tail τ. For some ﬁnite sequence p we have (s) = (p, τ). It follows that Vs ⊆ Vτ . We also have τ ∈ Vs and so Vτ ⊆ Vs. Thus Vs = Vτ . A similar argument gives Vt = Vτ , as required. For the case of AX , begin the proof as above, noting that P : Vt → Vs is a bicontinuous isomorphism. Since kπt(aN )tk = 1 for all N, there is some δ > 0 such that kπs(aN )P tk ≥ δ for all N. It follows that P t has some u∗vs = t in its support; for otherwise we must have kπs(aN )P tk → 0. Thus s, t have a common tail. The rest of the proof follows as before.

Theorem 3.6. For X ﬁnite, each of the pre-Cuntz algebras AX /K and AX /K has an uncountable collection of faithful irreducible ∗-representations which are not spatially equivalent.

Proof. By Theorem 3.1, each inﬁnite sequence s induces a faithful irreducible ∗-representation of AX /K or AX /K. Now argue as in Theorem 3.5.

References 1. M.J. Crabb. ‘Algebras associated with a free inverse monoid’. J. Aust. Math. Soc. 86 (2009), 27–31.

2. M.J. Crabb, J. Duncan, C.M. McGregor. ‘Primitivity theorems for convolution algebras on McAlister monoids’. Math. Proc. Roy. Irish Acad. 114A (2014), 149–163.

3. M.J. Crabb, W.D. Munn. ‘On the algebra of a free inverse monoid’. J. Algebra 184 (1996), 297–303.

4. M.J. Crabb, W.D. Munn. ‘On the contracted `1-algebra of a polycyclic monoid’. Port. Math. (N.S.) 63 (2006), 327–334.

5. M.J. Crabb, W.D. Munn. ‘The contracted `1-algebra of a McAlister monoid’. Math. Proc. Roy. Irish Acad. 107 (2007), 31–41.

6. K.R. Davidson. ‘C∗-algebras by example’. (Providence, RI: American Mathematical Society, Fields Institute Monographs, 6, 1996).

7. O.I. Domanov. ‘Semisimplicity and identities of inverse semigroup algebras’. Rings and modules, Mat. Issled 38 (1976), 123–137.

8. R. Exel. ‘Inverse semigroups and combinatorial C*-algebras’. Bull. Braz. Math. Soc. 39(2) (2008), 191–313.

9. D. Goncalves, D. Royer. ‘On the representations of Leavitt path algebras’. J. Algebra 333 (2011), 258–272.

10. M.V. Lawson. Inverse semigroups (World Scientiﬁc Singapore, 1998).

11 11. W.D. Munn. ‘On McAlister’s monoid and its contracted algebra’. (Proceedings of the Conference on Semigroups and Languages, Lisbon, 2005).

12. H.E. Scheiblich. ‘Free inverse semigroups’. Semigroup Forum 4 (1972), 351–359.

13. H.E. Scheiblich,. ‘Free inverse semigroups’. Proc. Amer. Math. Soc. 38 (1973), 1–7.

14. B. Steinberg. ‘A groupoid approach to discrete inverse semigroup algebras’. Adv. in Math. 223 (2010), 689–727.