Multi-norms and the injectivity of $L^p(G)$

TL;DR

This paper establishes that the injectivity of the Banach module $L^p(G)$ over $L^1(G)$ characterizes the amenability of the group $G$, using the novel concept of multi-norms.

Contribution

It introduces the theory of multi-normed spaces and applies it to characterize the injectivity of $L^p(G)$ modules in terms of group amenability.

Findings

Injectivity of $L^p(G)$$ as an $L^1(G)$-module iff $G$ is amenable.

Development of the theory of multi-normed spaces.

Application of multi-norms to Banach module theory.

Abstract

Let $G$ be a locally compact group, and take $p\in(1,\infty)$. We prove that the Banach left $L^1(G)$-module $L^p(G)$ is injective (if and) only if the group $G$ is amenable. Our proof uses the notion of multi-norms. We also develop the theory of multi-normed spaces.