Abstract
A large part of the theory of Banach *-algebras is developed and generalized to continuous inverse *-algebras (i.e. complex locally convex unital *-algebras with open unit group and continuous inversion) which are (Mackey) complete. If the involution is continuous, the closed unit ball with respect to the greatest C*-semi-norm is the closed convex hull of the unitary elements. (This is originally due to Palmer.) For hermitian continuous inverse *-algebras, we generalize characterizations due to Raĭikov, Pták, and Palmer, we prove the Shirali–Ford Theorem, and we show that closed subalgebras are equispectrally embedded.
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