Left inversion in a dual Banach algebra endowed with an Arens product and the Jacobson radical

Abstract

With some suitable Arens product, the dual space of the C$^{*}$ algebra of left uniformly continous complex valued functions on a locally compact group G admits a Banach algebra structure. The nature of the left invertible elements is important for its connection with the Jacobson radical. Our aim in this article is to determine conditions of left invertibility and some of their consequences.