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Elements of $C^{\ast }$-algebras Attaining their Norm in a Finite-dimensional Representation

Part of: Selfadjoint operator algebras General theory of linear operators

Published online by Cambridge University Press:  09 January 2019

Kristin Courtney  and
Tatiana Shulman
Kristin Courtney
Affiliation:
University of Virginia, Charlottesville, VA 22904, United States Email: kc2ea@virginia.edu
Tatiana Shulman
Affiliation:
Department of Mathematical Physics and Differential Geometry, Institute of Mathematics of Polish Academy of Sciences, Warsaw Email: tshulman@impan.pl
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Abstract

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We characterize the class of RFD $C^{\ast }$-algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the $C^{\ast }$-algebra is finite-dimensional, which is equivalent to the $C^{\ast }$-algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of $C^{\ast }$-algebras whose norms in finite-dimensional representations fit certain prescribed properties.

Keywords

AF-telescopesRFDprojective

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 47A67: Representation theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 1 , February 2019 , pp. 93 - 111
Copyright
© Canadian Mathematical Society 2017 

Footnotes

The research of author T. S. was supported by the Polish National Science Centre grant under the contract number DEC-2012/06/A/ST1/00256 and by the Eric Nordgren Research Fellowship Fund at the University of New Hampshire.

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