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.
DOI: http://dx.doi.org/10.7900/jot.2020may05.2272
Keywords: spectrum, groupoid, $C^*$-algebra, numerical range, pseudodifferential operator, cocycle, decomposition principle
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