Linear mappings of Banach algebras and Banach algebras of vector-valued functions
Fields, Jerry Wayne
Fields, Jerry Wayne
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1972
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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.
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Banach algebras
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Dissertation
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iv, 95 leaves, bound
Department
Mathematics