Mathematics – Functional Analysis
Scientific paper
2009-04-21
In ``Operator Structures and Dynamical Systems'' (M. de Jeu, S. Silvestrov, C. Skau, J. Tomiyama (Eds.)), Contemporary Mathema
Mathematics
Functional Analysis
22 pages. Minor changes in presentation. To appear in the proceedings (Contemporary Mathematics) of "Operator Structures and D
Scientific paper
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.
Dirksen Sjoerd
Jeu Marcel de
Wortel Marten
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