C*-Algebraic Spectral Sets, Twisted Groupoids and Operators

TL;DR

This paper develops a framework using twisted groupoid $C^*$-algebras to analyze spectral properties of operators, including pseudo-differential, Toeplitz-like, and band-dominated operators, emphasizing the role of orbit structures.

Contribution

It introduces a novel approach linking twisted groupoid $C^*$-algebras with spectral analysis of various classes of operators, including a Decomposition Principle for spectral quantities.

Findings

Spectral, norm, and pseudospectral properties depend on orbit closure structures.

Established a Decomposition Principle connecting spectral properties of operators to their restrictions.

Applied the framework to pseudo-differential, Toeplitz-like, and band-dominated operators.

Abstract

We treat spectral problems by twisted groupoid methods. To Hausdorff locally compact groupoids endowed with a continuous $2$-cocycle one associates the reduced twisted groupoid $C^*$-algebra. Elements (or multipliers) of this algebra admit natural Hilbert space representations. We show the relevance of the orbit closure structure of the unit space of the groupoid in dealing with spectra, norms, numerical ranges and $\epsilon$-pseudospectra of the resulting operators. As an example, we treat a class of pseudo-differential operators introduced recently, associated to group actions. We also prove a Decomposition Principle for bounded operators connected to groupoids, showing that several relevant spectral quantities of these operators coincide with those of certain non-invariant restrictions. This is applied to Toeplitz-like operators with variable coefficients and to band dominated operators on discrete metric spaces.