Elements of $C^*$-algebras Attaining Their Norm in a Finite-Dimensional Representation

TL;DR

This paper characterizes RFD $C^*$-algebras by their elements attaining norms in finite-dimensional representations and explores conditions for all elements to do so, linking to the structure of irreducible representations.

Contribution

It provides a characterization of RFD $C^*$-algebras based on norm-attaining elements and their relation to finite-dimensional irreducible representations.

Findings

RFD $C^*$-algebras have a dense subset of elements attaining their norm in finite-dimensional representations.

All elements attain their norm in finite-dimensional representations iff all irreducible representations are finite-dimensional.

Existence of elements with prescribed norm properties in more general $C^*$-algebras.

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.