B ∗ ‐Algebras and their Representation

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].