TAILIEUCHUNG - Báo cáo toán học: "The Fubini theorem for exact C*-algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Định lý Fubini cho chính xác C *- đại số. | J. OPERATOR THEORY 10 1983 3 8 Copyright by INCREST 1983 THE FUBIN1 THEOREM FOR EXACT C -ALGEBRAS EBERHARD KIRCHBERG 1. INTRODUCTION Let A and B be c -algebras. By A B vve denote the minimal or spatial CMensor product of A and B 7 Further algebra subalgebra tensor product and ideal mean c -algebra c -subalgebra c -tensor product and closed ideal respectively. A B is a subalgebra of the spatial PF -tensor product X X B see . 6 . For every linear functional p e X there is a unique ultraweakly continuous linear map Lp from X B into B with LẶa b p a b for a e X be B see . 6 8 . Hence L p X B B if p 0 and p G X - Lip e A B B is an isometric linear map. Now let D be a subalgebra of B. Then D is a fp -subalgebra of B and 0 B D. The Fubini algebra F X D A B of X and D in X B is the set of all t 6 X B with Lyự G D for every state of X. From J9 n B D LẠA B B and from the commutation theorem for ff -algebras we get F X D A B A B n F X B X g B and _ F X D A B1- A ữ D . F X D A B is a subalgebra of X B containing X D. For fixed X the minimal tensor product is a functor B - V X B in the category of c -algebras. We call X exact if this functor is an exact functor 3 . According to 4 X is exact if and only if F X J A B X J for every algebra B and every ideal J of B. By 5 quotiens of subalgebras of nuclear algebras are exact. Now let H be a Hilbert space of countable dimension and let and denote the algebras of bounded and compact linear operators on H respectively. The kernel of the canonical epimorphism X X is F X A Hence 0 - X - X - X H - 0 is an exact sequence if and only if F X X 7 X In Section 2 we prove the following Fubini type theorem. 4 EBERHARD KIRCHBERG Theorem . Let A be a C -algebra. The following properties of A are equivalent i F A D A B -A D for every C -algebra B and every hereditary C - -subalgebra D of B . DBD c D . ii A is exact. iii The canonical sequence O A - A ỰỈ - A - 0 is exact. According to 4 subalgebras of exact algebras are exact c M M-.

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