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# Elements of $C^*$-algebras attaining their norm in a finite-dimensional representation

Published:2017-12-04

• Kristin Courtney,
University of Virginia , Charlottesville, VA 22904, USA
• Tatiana Shulman,
Department of Mathematical Physics and Differential Geometry, Institute of Mathematics of Polish Academy of Sciences, Warsaw, Poland
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## Abstract

We characterize the class of RFD $C^*$-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 $C^*$-algebra is finite-dimensional, which is equivalent to the $C^*$-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 $C^*$-algebras whose norms in finite-dimensional representations fit certain prescribed properties.
 Keywords: AF-telescope, RFD, projective
 MSC Classifications: 46L05 - General theory of $C^*$-algebras 47A67 - Representation theory

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