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Journal of Operator Theory

Volume 86, Issue 2, Fall 2021  pp. 355-394.

$C^*$-Algebraic spectral sets, twisted groupoids and operators

Authors:  Marius Mantoiu
Author institution: Departmento de Matematicas, Facultad de Ciencias, Universidad de Chile, Santiago, Chile

Summary: 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 $\varepsilon$-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.

Keywords: spectrum, groupoid, $C^*$-algebra, numerical range, pseudodifferential operator, cocycle, decomposition principle

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