A Quantitative Version of the Non-Abelian Idempotent Theorem

Suppose that G is a finite group and f is a complex-valued function on G. f induces a (left) convolution operator from L2(G) to L2(G) by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${g \mapsto f \ast g}$$\end{document} where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \ast g(z) := \mathbb{E}_{xy=z}f(x)g(y)\,\, {\rm for\,\,all} \, z \in G.$$\end{document} This operator is a linear map L2(G) → L2(G) between two finite dimensional Hilbert spaces, and so it has well-defined singular values; we write || f ||A(G) for their sum.