Summary of Spectral Invariance Results

TL;DR

This paper summarizes recent results on spectral invariance of dense subalgebras in C*-algebras linked to dynamical systems, focusing on polynomial growth Lie groups and tempered actions.

Contribution

It extends spectral invariance results to Schwartz function subalgebras in the context of polynomial growth Lie groups acting on locally compact spaces.

Findings

Spectral invariance holds for Schwartz subalgebras in certain C*-crossed products.

The results apply to compactly generated polynomial growth Type R Lie groups.

A natural smooth crossed product is dense and spectral invariant in the C*-crossed product.

Abstract

The author's recent results on spectral invariant dense subalgebras of C*-algebras associated with dynamical systems are summarized. If G is a compactly generated polynomial growth Type R Lie group, and the action of G on S(M) (Schwartz functions on a locally compact G-space M) is tempered in a certain sense, then there is a natural smooth crossed product S(G X M) which is dense and spectral invariant in the C*-crossed product C*(G X M).