The predual of the space of convolutors on a locally compact group

Abstract

Let Cvp(G) be the space of convolution operators on the Lebesgue space LP(G), for an arbitrary locally compact group G. We describe Cvp(G) as a dual space, whose predual, is a Banach algebra of functions on G, under pointwise operations, with maximal ideal space G. This involves a variation of Herz's definition of AP(G); the benefit of this new definition is that all of Cvp(G) is obtained as the dual in the nonamenable setting. We also discuss further developments of this idea.