Fourier transform and its analogues are the cornerstone of classical harmonic analysis. In the absence of Fourier transform for non-Abelian groups, the new theory of non-commutative harmonic analysis was developed. A major trend in non-commutative harmonic analysis is to investigate function algebras related to Fourier analysis (and representation theory) of non-Abelian groups. The Fourier algebra, which is associated with the regular representation of the ambient group, is a fundamental example of such function algebras.
In this talk, we investigate Banach algebraic behavior, in particular spectral theory, of the Fourier algebra and its weighted versions for various classes of locally compact (Lie) groups, and show that these function algebras encode the properties of the underlying groups in various ways.

This talk is based on joint work with Lee, Ludwig, Spronk, and Turowska.