Introverted subspaces of the duals of measure algebras

Abstract

Let G be a locally compact group. In continuation of our studies on the first and second duals of measure algebras by the use of the theory of generalised functions, here we study the C * -subalgebra GL 0 (G) of GL(G) as an introverted subspace of M (G) * . In the case where G is non-compact we show that any topological left invariant mean on GL(G) lies in GL 0 (G) ⊥ . We then endow GL 0 (G) * with an Arens-type product which contains M (G) as a closed subalgebra and M a (G) as a closed ideal which is a solid set with respect to absolute continuity in GL 0 (G) * . Among other things, we prove that G is compact if and only if GL 0 (G) * has a non-zero left (weakly) completely continuous element.Mathematics Subject Classification (2010). 43A10, 43A15, 43A20, 47B07.