C
*-algebras generated by partial isometries |
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Authors: | Ilwoo Cho Palle Jorgensen |
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Institution: | 1. Department of Mathematics, Saint Ambrose University, 421 Ambrose Hall, 518 W. Locust Street, Davenport, IA, 52308, USA 2. Department of Mathematics, University of Iowa, McLean Hall, Iowa City, IA, USA
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Abstract: | We study the interconnection between directed graphs and operators on a Hilbert space. The intuition supporting this link is the following feature shared by partial isometries (as operators on a Hilbert space) on the one hand and edges in directed graphs on the other. A partial isometry a is an operator in a Hilbert space H, i.e., a:H→H which maps a (closed) subspace in H isometrically onto a generally different subspace. The respective subspaces are called the initial space and the final space of a. Denoting the corresponding (orthogonal) projections by p i and p f , note that a partial isometry a may be thought of as an edge from one vertex to another (which are not necessarily distinct) in a directed graph. And conversely, every directed graph has such a representation. Since neither the partial isometries nor the directed edges in a fixed model allow unrestricted composition, the algebraic construct which is useful is that of a groupoid. In this paper we develop this as a representation theory, and we explore the connection between realizations in the context of C *-algebras. The building blocks in our theory are certain matricial C *-algebras which we define. We then prove how they serve to localize our global representations. |
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