Extending representations of normed algebras in Banach spaces

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.