Banach ∗-algebras, B∗-seminorms, and positive functionals

This paper concerns Banach ∗-algebras which are nonunital or have bounded approximate identity. A necessary and sufficient condition is given for a B∗-seminorm to be regular. A one-to-one correspondence between the carrier space of a complex Banach ∗-algebra and the set of all regular B∗-seminorms is established. Interaction between α-bounded functionals and regular B∗-seminorms is examined. We prove that the carrier space of a Banach ∗-algebra is identical with the set of all extreme points of positive linear functionals. The main approach is via an ordering of the algebra, the positive cone being the closure of the set of certain elements in the set of self-adjoint (hermitian) elements of the algebra (but not in the algebra) where the involution on the algebra is not continuous. A characterization of the set of centralizers for algebra in terms of positive cones where the involution is not continuous is given. This result improves some previous results on this topic. In the context of approximate identity of norm less than or equal to the real number 1, we prove that a cone is closed with respect to a special product if and only if the algebra is commutative modulo its radical. An application of the Shirali–Ford theorem is also discussed. In particular, the equality between the two positive cones is established if the algebra is symmetric.