<i>B</i> ^∗-Algebras and their Representation

Abstract

Let A be a complete lmc-∗-algebra with unit whose topology is given by a family P of submultiplicative pseudonorms p such that p(x∗ x) = p(x)2, x ∈ A (i.e. A is a b∗-algebra). We study the general properties of A (§2) and especially its representation in the commutative case. The description of the form of submultiplicative pseudonorms p such that p(x∗x) = p(x)2, x ∈ A (Lemma 3.2) enables us to obtain some structure theorems (§4). Thus we give an abstract characterisation of the algebra of all continuous complex functions on a locally compact Hausdorff space, endowed with the compact-open topology [Th. 4.1, Th. 4.2].