Abstract We shall consider measure algebras associated to locally compact groups, bounded operators between them and properties of the underlying measures. take into account second dual provided with Arens products together tools Gélfand theory.

Let G be a locally compact group, let L1(G) be its group algebra, let M(G) be its usual measure algebra, let L1(G)∗∗ be the second dual of L1(G) with an Arens product, and let LUC(G)∗ be the conjugate of the space LUC(G) of bounded, left uniformly continuous, complex-valued functions on G with an Arens-type product. We find all the finite-dimensional left ideals of these algebras. We deduce tha...

Robin Harte Dedicated to Goldie Hawn, on her birthday Abstract We attempt to deconstruct the Arens-Royden Theorem. Suppose A is a Banach algebra (by default complex, with identity 1): we shall write 0.1 A−1 = {a ∈ A : 1 ∈ Aa∩aA} for the open subgroup of invertible elements, and A−1 0 for the connected component of the identity in A −1: it turns out ([8], [11] Theorem 7.11.4) that 0.2 A−1 0 = Ex...