Gelfand theory of reduced group \(L^p\)-operator algebra
Let \(G\) be a locally compact abelian group and \(\lambda_p\) the left regular representation of \(G\) on \(L^p(G)\) (\(p\geq 1\)). The representation induces a contractive representation of \(L^1(G)\) on \(L^p(G)\). The closure of \(\lambda_p(L^1(G))\) in the operator norm topology of \(B(L^p(G))\) is a Banach algebra, denoted by \(F_\lambda^p(G)\). \textit{E.~Gardella} and \textit{H.~Thiel} have shown in [J. Fourier Anal. Appl. 21, No.~6, 1310--1343 (2015; Zbl 1334.22007)] that the maximal ideal space of \(F_\lambda^p(G)\) is isomorphic to the dual group \(\hat{G}\) of \(G\). In the paper under review, the authors provide another proof to the result.
scientific article; zbMATH DE number 7450626