On one-sided primitivity of Banach algebras

TL;DR

This paper investigates the algebraic structure of certain Banach algebra completions of a semigroup algebra, demonstrating their primitivity and analyzing the properties of their modules, with implications for understanding algebraic representations.

Contribution

It introduces two specific Banach algebra completions of a semigroup algebra and proves their left-primivity and module properties, providing new insights into their algebraic structure.

Findings

Both completions are left-primitive with separating families of infinite-dimensional irreducible right modules.

The semigroup algebra itself is left-primitive but not right-primitive.

All irreducible right modules of the semigroup algebra are finite-dimensional.

Abstract

Let $S$ be the semigroup with identity, generated by $x$ and $y$, subject to $y$ being invertible and $yx=xy^2$. We study two Banach algebra completions of the semigroup algebra $\mathbb{C}S$. Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an alternative proof that $\mathbb{C}S$ is left-primitive but not right-primitive. We show further that, in contrast to the completions, every irreducible right module for $\mathbb{C}S$ is finite dimensional and hence that $\mathbb{C}S$ has a separating family of such modules.