TAILIEUCHUNG - Báo cáo toán học: "A non-commutative Gelfand-Naimark theorem "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một không giao hoán định lý Gelfand-Naimark. | J. OPERATOR THEORY 8 1982 361-389 Copyright by INCREST 1982 A NON-COMMUTATIVE GELFAND-NAIM ARK THEOREM PAWEL KRUSZYNSKI and STANISLAW L. WORONOWICZ INTRODUCTION One of the most fundamental facts in the theory of c -algebras was discovered in early forties by Gelfand and Naimark. It says that any commutative c -algebra with unit is isomorphic to the algebra of all continuous functions on a compact topological space. Moreover any homomorphism of commutative c -algebras is related to a continuous map of the underlying compact spaces. Briefly speaking the category of commutative c -algebras with unit is dual to the category of compact topological spaces. With this result any fact dealing with compact topological spaces has a counterpart in the commutative c -algebras theory. In many cases these counterparts possess natural generalizations for noncommutative c -algebras and may be proved. It shows that even in the noncommutative case consideration of c -algebras as algebras of continuous functions is of a great inspiratory value. This point of view provides us with interesting conjectures and sometimes suggest a way of a proof. To formalize this point of view one may introduce the notion of compact pseudospaces 6 . Briefly a compact pseudospace is an object of the category dual to the category of c -algebras with unit. Then any c -algebra can be viewed as the algebra of all continuous functions on the corresponding pseudospace. One should stress however that introducing the category of pseudospaces we do not create any new theory we introduce only a new language to the c -algebras theory. In the present paper we try to develop more constructive approach to the problem. We consider so called domains and operator functions defined on them. To explain better our ideas let US recall two possible formulations of the basic Gelfand--Naimark result. In general the Gelfand-Naimark theorem deals with abstract compact topological spaces and continuous complex valued functions. .

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