A generalized Fourier inversion theorem

TL;DR

This paper extends the Fourier inversion theorem to operator-valued transforms on non-Abelian groups, broadening the scope of harmonic analysis and improving upon previous Abelian-specific results.

Contribution

It introduces a generalized Fourier inversion theorem for operator-valued transforms on non-Abelian groups, enhancing prior work by Ruy Exel.

Findings

Generalized Fourier inversion theorem for non-Abelian groups

Operator-valued Fourier transforms defined for integrable elements

Improved upon previous Abelian group results

Abstract

In this work we define operator-valued Fourier transforms for suitable integrable elements with respect to the Plancherel weight of a (not necessarily Abelian) locally compact group. Our main result is a generalized version of the Fourier inversion Theorem for strictly-unconditionally integrable Fourier transforms. Our results generalize and improve those previously obtained by Ruy Exel in the case of Abelian groups.