Banach-Orlicz Algebras on a Locally Compact Group

Abstract

Let G be a locally compact group with a fixed left Haar measure λ. Given an N-function φ, we consider the Orlicz space \({L^{\varphi}(G)}\) under the convolution multiplication and establish that, for amenable groups under mild conditions on φ, it is a convolution algebra if and only if G is compact. Also we prove that for a locally compact group G, the convolution algebra \({L^{\varphi}(G)}\) has a bounded approximate identity if and only if G is discrete.