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

  • 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 AF-telescope, RFD, projective
MSC Classifications: 46L05, 47A67 show english descriptions General theory of $C^*$-algebras
Representation theory
46L05 - General theory of $C^*$-algebras
47A67 - Representation theory
 

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