Harmonic analysis and functional analysis are two closely related areas of mathematics, which evolved from tools developed to study differential equations arising in engineering and physics. Fourier algebras of locally compact groups sit at the interface between these two areas; they are normed algebras of functions that encode both the topological and the algebraic structure of the original groups.
In addition to being studied for intrinsic interest, they have found external uses within pure mathematics (approximation properties for operator algebras) and applied mathematics (higher-dimensional wavelet transforms).
This Research in Groups will investigate certain numerical invariants of Fourier algebras. These invariants, called amenability constants, have been much studied for other classes of Banach algebras, such as semigroup algebras [Dales–Lau–Strauss, Mem. Amer. Math. Soc. 2010, Ch. 10] or C*-algebras [Haagerup, Inv. Math. 1983], but much less has been done for Fourier algebras. Our project aims to improve the existing bounds on these invariants, by combining abstract tools from functional analysis with explicit formulas from noncommutative harmonic analysis. We will build on recent progress by the lead organizer (IMRN 2023), drawing on complementary expertise of the team members.
