Extending representations of normed algebras in Banach spaces

TL;DR

This paper investigates how representations of normed algebras on Banach spaces can be extended to larger algebras, analyzing centralizer algebras and their properties in various algebraic contexts.

Contribution

It provides new conditions under which representations of normed algebras extend to larger algebras and explores their properties in ordered and involutive settings.

Findings

Representation extension is possible when A is a left ideal of a larger algebra.

Left centralizer algebra embeds as a topological algebra as the left normalizer.

Double centralizer algebra embeds as a topological algebra as the normalizer of the image.

Abstract

Let X be a non-degenerate left Banach module over a normed algebra A having a bounded approximate left identity. We show that, if A is a left ideal of a larger algebra, then this representation can be extended to a representation of the larger algebra. Based on this result, we study in detail the existence and properties of representations of the various centralizer algebras of A which are compatible with the original representation of A. As a special case we obtain that, if A embeds as a topological algebra into the bounded operators on X, then the left centralizer algebra of $A$ embeds as a topological algebra as the left normalizer of the image, and the double centralizer algebra of A embeds as a topological algebra as the normalizer of the image. We also consider ordered and involutive contexts, and cover the right-sided cases, which are not always the obvious analogues of the left-sided cases, in detail as well.