Morita Equivalence of Brandt Semigroup Algebras

TL;DR

This paper proves Morita equivalence between Brandt semigroup algebras for different index sets over the same group and explores implications for Hochschild cohomology and amenability.

Contribution

It establishes Morita equivalence of Brandt semigroup algebras for different index sets and analyzes consequences for cohomology and amenability.

Findings

Brandt semigroup algebras are Morita equivalent for different index sets.

Hochschild cohomology groups are trivial for amenable groups.

Approximate amenability is not preserved under Morita equivalence.

Abstract

We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $\ell(B(I,G))$ and $\ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, we show that if $G$ is an amenable group, then for a wide class of Banach $\ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(\ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant.