TL;DR
This paper explores advanced topics in non-commutative harmonic analysis, focusing on Fourier transforms on Lie groups, operational calculus, and spectral separation problems, highlighting solvable open problems in the field.
Contribution
It presents a discussion of open problems in non-commutative harmonic analysis related to Fourier transforms and spectral theory, emphasizing their solvability.
Findings
Identification of open problems in non-commutative harmonic analysis
Proposed approaches to the Gelfand--Gindikin problem
Insights into operational calculus on Lie groups
Abstract
We discuss two topics related to Fourier transforms on Lie groups and on homogeneous spaces: the operational calculus and the Gelfand--Gindikin problem (program) about separation of non-uniform spectra. Our purpose is to indicate some non-solved problems of non-commutative harmonic analysis that definitely are solvable. This is a sketch of my talks on VI School "Geometry and Physics", Bialowieza, Poland, June 2017.
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