A construction for irreducible representations is applied to various inverse semigroup algebras

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 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 representations (or ∗-representations) of FS and F0S (or CS and C0S), with easy extensions to the `-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 communications 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 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 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 `-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.