Journal article
C -algebras generated by partial isometries
Journal of Applied Mathematics and Computing, Vol.26(1), pp.1-48
02/2008
DOI: 10.1007/s12190-007-0009-0
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.
Details
- Title: Subtitle
- C -algebras generated by partial isometries
- Creators
- Ilwoo Cho - Department of Mathematics Saint Ambrose University 421 Ambrose Hall, 518 W. Locust Street Davenport IA 52308 USAPalle Jorgensen - Department of Mathematics University of Iowa McLean Hall Iowa City IA USA
- Resource Type
- Journal article
- Publication Details
- Journal of Applied Mathematics and Computing, Vol.26(1), pp.1-48
- Publisher
- Springer-Verlag; Berlin/Heidelberg
- DOI
- 10.1007/s12190-007-0009-0
- ISSN
- 1598-5865
- eISSN
- 1865-2085
- Language
- English
- Date published
- 02/2008
- Academic Unit
- Mathematics
- Record Identifier
- 9983985913002771
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