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Báo cáo toán học: "Hilbert C*-modules: theorems of Stinespring and Voiculescu "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hilbert C module *-: định lý của Stinespring và Voiculescu. | J. OPERATOR THEORY 4 1980 133-150 Copyright by INCREST 1980 HILBERT C -MODULES THEOREMS OF STINESPRING AND VO1CULESCU G. G. KASPAROV There are two well known technical results ill the theory of extensions of c -algebras. Stinespring s theorem 11 describes the structure of completely positive maps of a c -algebra A into the full operator algebra Voiculescu s theorem 12 establishes the existence of the identity element in the semigroup of extensions of the type 0 - X - SỀ A ồ X being the algebra of compact operators in .X X . The theory of general extensions 0 - X announced in 13 is based on a suitable generalization of these two theorems with i X replaced by the multiplier algebra BỴ In the present paper we give this generalization Theorems 3 and 6 . Besides this the paper also contains two theorems about Hilbert c -modules which are of independent interest Theorems 1 and 2 . In particular Theorem 2 asserts that every countably generated Hilbert B--module is a direct summand in the canonical Hilbert space over B. 1 NOTATION 1. In what follows all the algebras are c -algebras the homomorphisms are -ho-momorphisms and the ideals are closed and two-sided. All the results are valid for the next three categories of algebras complex algebras real algebras and real algebras. A complex algebra is called real if it is equipped with an antilinear involution bb satisfying two conditions bib 2 b b b b . The homomorphisms of such algebras must preserve the real involution. Moreover we suppose that a fixed compact second countable group G acts as a group of automorphisms on all algebras. All homomorphisms are supposed to be equivariant. In the real case the group and the action must be real i.e. there exists an involution g - g on G such that g b g b . Some explanations are necessary here. c -algebras i.e. algebras of operators are characterized in the complex case by a usual condition V.v. In. the real case however this condition ÍS not sufficient. It must be replaced by I M 2 .