Gelfand theory of reduced group $L^{p}$-operator algebra

Abstract

Let G be a locally compact Abelian group. The reduced group $L^{p}$-operator algebra $F^{p}_{\lambda }(G)$ is the completion of $L^{1}(G)$ with respect of to the norm induced by the left regular representation of $L^{1}(G)$ on $L^{p}(G)$. In this article, we show that the maximal ideal space of $F^{p}_{\lambda }(G)$ is homeomorphic to ${\widehat{G}}$. The Gelfand transformation $\Gamma _{p}:F^{p}_{\lambda }(G)\rightarrow C_{0}({\widehat{G}})$ is injective and $\Gamma _{p}$ is surjective if and only if G is finite for $p\in [1,+\infty )\setminus \{2\}$. As an application, we show that $F_{\lambda }^{p}(G)$ is a regular Banach algebra and the K-theory groups of $F^{p}_{\lambda }(G)$ do not depend on $p\in [1,+\infty )$.