Algebras on inverse semigroups with finite factorization: Rukolaine Idempotents

The results in this paper were motivated by the case when the inverse semigroup is the McAlister monoid MX on a set X. In [1] we considered some very large convolution algebras on MX , including some C ∗-algebras. Our main focus in [1] was deciding on the primitivity of the algebras, and one key tool was the use of generalized Rukolaine idempotents. To define these very large convolution algebras more generally we need to have a semigroup S with the finite factorization property, denoted here by FF . To be precise, let S be a semigroup without zero element θ. We say that S has property FF if, for each s ∈ S, the set F(s) = {(u, v) ∈ S × S : s = uv} is finite. For any field F, we may then consider the vector space Ξ[S] of all F-valued functions on S. When S has property FF , Ξ[S] is clearly closed under the usual convolution product, thus providing a very large superalgebra of the usual semigroup algebra FS. Now let S be a semigroup with zero element θ. We say that S has property FF if the set F(s) is finite for each non-zero element s ∈ S. The quotient space Ξ[S]/Fθ again gives a very large convolution algebra which we denote by Ξ0[S]. (This is analogous to the usual restricted semigroup algebras F0S and l0(S) when S has a zero element.) When convenient, we shall write a ∈ Ξ0[S] as a = ∑ {a(s)s : s ∈ S, s ̸= θ}. Obvious examples of semigroups with property FF are given by the free semigroups, or free commutative semigroups, of finite rank. For these examples, Ξ[S] is the algebra of formal power series in several variables (commutative or otherwise). These algebras have Jacobson radical of codimension 1 (since one can readily compute the quasi-inverse of any element which is not a scalar multiple of the identity element). When S is any inverse semigroup with property FF , the algebras Ξ[S] and Ξ0[S] have an abundance of idempotents which are excluded from the Jacobson radical; in fact we shall show that these algebras are always semisimple when the field is C. A key tool in proving this assertion is that the Barnes representations of FS and F0S extend to these superalgebras.