Spectra in Some Inverse Semigroup Algebras

Abstract

Let S be the semigroup with generators $s_j ,s_j^* $(j = 1,2,..., N) and defining relations $s_j^* s_j = 1$, $s_j^* s_k = \theta (j \ne k)$, where 1 is the identity and θ the zero. We determine the spectrum of the average of these generators in l₁(S)/ℂθ, showing it to be an ellipse together with its interior. The major and minor diameters of the ellipse are 1 + 1/N and 1 — 1/N, respectively. So the interior is non-empty except in the degenerate case N = 1.