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

Courtney, Kristin ; Shulman, Tatiana

Canadian journal of mathematics, Tome 71 (2019) no. 1, pp. 93-111

Résumé

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.

Détail

DOI : 10.4153/CJM-2017-040-x

Mots-clés : AF-telescopes, RFD, projective

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