On the existence of an ultra central approximate identity for certain semigroup algebras

TL;DR

This paper characterizes when certain semigroup algebras possess an ultra central approximate identity, revealing that for Brandt semigroups over infinite sets, such identities do not exist.

Contribution

It provides a characterization of ultra central approximate identities in semigroup algebras, especially for inverse and Brandt semigroups, highlighting conditions on the index set.

Findings

Ultra central approximate identities exist for $ ext{ell}^1(S)$ when $S$ is a uniformly locally finite inverse semigroup.

For Brandt semigroups over infinite sets, $ ext{ell}^1(S)$ lacks an ultra central approximate identity.

The existence of such identities is equivalent to the finiteness of the index set in Brandt semigroups.

Abstract

In this paper we characterize the existance of an ultra central approximate identity for $\ell^{1}(S)$, where $S$ is a uniformly locally finite inverse semigroup. As an application, for the Brandt semigroup $S=M^{0}(G,I)$ over a non-empty set $I$, we show that $\ell^{1}(S)$ has an ultra central approximate identity if and only if $I$ is finite.