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

TL;DR

This paper investigates the properties of dense Fréchet subalgebras of compact operators, demonstrating that spectral invariance ensures algebraically cyclic subrepresentations are contained within a a-kernel module, thus extending representation theory.

Contribution

It establishes that spectral invariance in dense subalgebras of compact operators guarantees inclusion of algebraically cyclic subrepresentations in a-modules, extending the understanding of their structure.

Findings

Algebraically cyclic subrepresentations are contained in a-modules.

Spectral invariance ensures the extension of representations.

Results apply to dense Fre9chet subalgebras of compact operators.

Abstract

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