[Paper Review] Summary of Spectral Invariance Results

Larry B. Schweitzer|arXiv (Cornell University)|Jan 23, 1993

Advanced Operator Algebra Research5 references3 citations

TL;DR

This paper establishes spectral invariance for smooth crossed products associated with polynomial growth Lie groups acting on locally compact spaces. For a compactly generated, polynomial growth Type R Lie group G acting on a G-space M, the smooth crossed product S(G ⋉ M) is shown to be spectral invariant in the C*-crossed product C*(G ⋉ M), under a tempered action condition, extending spectral invariance results to dynamical systems with group symmetry.

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).

Motivation & Objective

• To extend spectral invariance results to dynamical systems with group actions.
• To establish conditions under which smooth crossed products are spectral invariant in C*-crossed products.
• To analyze the structure of S(G ⋉ M) for compactly generated, polynomial growth Type R Lie groups.
• To define and verify the temperedness condition on the action of G on M that ensures spectral invariance.
• To provide a framework for spectral invariance in noncommutative harmonic analysis and operator algebras.

Proposed method

• The paper constructs the smooth crossed product S(G ⋉ M) as a dense subalgebra of the C*-crossed product C*(G ⋉ M).
• It assumes G is a compactly generated, polynomial growth, Type R Lie group.
• The action of G on the space M of Schwartz functions S(M) is required to be tempered in a specific sense.
• Spectral invariance is established via analysis of the growth properties of the group and the action's regularity.
• The proof relies on techniques from functional analysis and operator algebras, particularly in the context of noncommutative geometry.
• The framework generalizes known spectral invariance results to a broader class of dynamical systems.

Experimental results

Research questions

• RQ1 Under what conditions is the smooth crossed product S(G ⋉ M) spectral invariant in the C*-crossed product C*(G ⋉ M)?
• RQ2 How does the polynomial growth and Type R structure of a Lie group G affect spectral invariance in crossed products?
• RQ3 What is the precise meaning and role of the 'tempered action' condition on the G-action on M?
• RQ4 Can spectral invariance be extended to dynamical systems with non-abelian, non-unimodular Lie group symmetries?
• RQ5 What is the relationship between smoothness of the crossed product and spectral properties in C*-algebras?

Key findings

• The smooth crossed product S(G ⋉ M) is dense in the C*-crossed product C*(G ⋉ M).
• Under the tempered action condition, S(G ⋉ M) is spectral invariant in C*(G ⋉ M).
• The result holds for compactly generated, polynomial growth, Type R Lie groups.
• The spectral invariance is established via the interplay between group growth and regularity of the action.
• The framework provides a new class of spectral invariant subalgebras in C*-dynamical systems.
• The result generalizes prior spectral invariance theorems to a broader class of non-abelian group actions.