Topology Counterparts in C*-Algebras
AuthorCorder, Daniel Surrell
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This paper describes a contravariant category equivalence between the category of unital commutative C*-algebras with unital *-homomorphisms and the category of compact Hausdorff spaces with continuous functions in order to characterize semiprojective C*-algebras. Results preliminary to establishment of the equivalence yield homeomorphisms between any compact Hausdorff space X, the space of maximal ideals on C(X) endowed with the hull-kernel topology, and the space of characters on C(X) under the weak* topology. The functional calculus herein constructed provides a link between normal elements of a C*-algebra and continuous functions on the spectra of the elements. The equivalences established, along with the functional calculus, provide a means to develop the C*-algebra theory of semiprojectivity by analogy to the topological concept of absolute neighborhood retracts on compact metrizable spaces; the analogy yields many examples of semiprojective C*-algebras. Semiprojectivity theory is an instance of extending well-established consequences from one mathematical context for use in another context via category equivalence and it additionally motivates an exploration of the extent to which results from one context can be developed analogously in the other beyond the limits of the equivalence.