P-commutative Banach *-algebras, involutions, and extensions of pure positive functionals

Abstract

In this dissertation a class of *-algebras is defined which generalizes the class of commutative *-algebras. This is accomplished as follows. Let A be a complex *-algebra. If f is a positive functional on A, let I_f= {x in A: f(x*x) = 0} be the corresponding left ideal of A. Set P = intersection I_f, where the intersection is over all positive functionals on A. Then A is called P-commutative if xy - yx is in P for all x, y in A. Examples are given of noncommutative *-algebras which are P-commutative and many results are obtained for P-commutative Banach *-algebras which extend results known for commutative Banach *-algebras. Among them are the following: If A^2 = A, then every positive functional on A is continuous. If A has an approximate identity, then a nonzero positive functional on A is a pure state if and only if it is multiplicative. If A is symmetric, then the spectral radius in A is a continuous algebra semi-norm. Two theorems, not concerned with P-commutative *-algebras, are proven in Chapter III. The first says that, in a Banach *-algebra which is spanned by a finite number of its maximal commutative *-subalgebras, local continuity of the involution implies continuity of the involution. The second theorem is an extension theorem for pure positive functionals.