Extending Representations of Dense Subalgebras of $C^{\star }$-Algebras, and Spectral Invariance

Abstract

Let $A$ be a dense Fréchet subalgebra of the $C^{\star }$-algebra of compact operators $\mathcal{K}$ on a seprable Hilbert space. Assume that $A$ is spectral invariant in $\mathcal{K}$. We show that every algebraically cyclic subrepresentation of a topologically irreducible representation of $A$ is contained in a $\mathcal{K}$-module.

arXiv commentCase of arbitrary dense sub-algebras of the compacts added in version 3