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

  • Kristin Courtney (a1) and Tatiana Shulman (a2)

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.

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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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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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