Multi-norms and modules over group algebras

TL;DR

This paper explores the conditions under which the Banach space L^p(G) becomes an injective module over the group algebra L^1(G), introducing (p,q)-amenability as a key concept linking group properties to module injectivity.

Contribution

It introduces (p,q)-amenability conditions and characterizes injectivity of L^p(G) modules in terms of these conditions for general and discrete groups.

Findings

Injectivity of L^p(G) implies (p,p)-amenability of G.

L^p(G) is injective iff G is (p,p)-amenable for discrete groups.

Established a link between group amenability and module injectivity.

Abstract

Let G be a locally compact group, and let 1 < p < \infty. In this paper we investigate the injectivity of the left L^1(G)-module L^p(G). We define a family of amenability type conditions called (p,q)-amenability, for any 1 <= p <= q. For a general locally compact group G we show if L^p(G) is injective, then G must be (p,p)-amenable. For a discrete group G we prove that l^p(G) is injective if and only if G is (p,p)-amenable.