Linear mappings of Banach algebras and Banach algebras of vector-valued functions

Abstract

In this dissertation, an effort is made to give sufficient conditions that self-adjoint linear mappings of C*-algebras be C*-homomorphisms. Sufficient conditions are given so that a positive linear functional on a Banach *-algebra is multiplicative. Positive linear mappings of C*-algebras are characterized in terms of the numerical range. If A is a C* -algebra and x in A, the set V(x) = {f(x): f is a pure state on A} is studied. Also, a detailed examination is made of algebras of continuous functions vanishing at infinity on a locally compact Hausdorff space and having values in an arbitrary Banach algebra. These algebras are denoted by C_0 (X, A). If A is a commutative Banach algebra with an approximate identity, then the Shilov boundary of C_0 (X, A) is computed. If A is a C*-algebra, the pure states on C_0 (X, A) are characterized in terms of the pure states on A. If X is compact, it is shown that the pure states on C(X,A) with the weak*- topology is homeomorphic with a topological product.