tailieunhanh - Báo cáo toán học: "Perturbations of C*-algebras and K-theory "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nhiễu loạn của C *- đại số và lý thuyết K-. | 1. OPERATOR THEORY 12 1984 89 - 99 Copyright by INCREST 1984 PERTURBATIONS OF C -ALGEBRAS AND K-THEORY MAHMOOD KHOSHKAM INTRODUCTION We show that if A and B are c -subalgebras of a c -algebra c and A c B for a suitably small positive number y then with a mild assumption on A there exists a natural homomorphism Ĩ from K 4 into K 5 . If d A B y T is an isomorphism of K J onto K. B . To each closed two-sided ideal J in B there corresponds a unique closed two sided ideal I in A such that I G J. Using Ĩ we connect the six-term exact sequence of K-theory associated with ữ - I A - AỊI ữ to that associated with 0 - J - B - B J - 0 and show that the resulting diagram is commutative. We also show that if A and B are unital c -subalgebras of a c -algebra c and d A B is sufficiently small then the groups of unitaries of A and B are homotopically equivalent. These results are part of the author s Ph. D. dissertation done at Dalhousie University under the supervision of Professor John Phillips. I am indebted to George Skandalis for many helpful suggestions. The author would also wish to thank the referee for his comments. 1. PRELIMINARIES . All the tensor-products considered in this paper are the minimal tensor-product of c -algebras. . For c -subalgebras A B of a c algebra c and 0 y si 1 we use Christensen s definition 2 to say that A is y-contained in B 4 c B if for every a e A there exists an element b 6 B such that a z y a . If A c B and B c A then d A B 2y where d A B the distance between A and B is as defined in 5 . . If A is a c -algebra we denote by A the c -algebra obtained from 4 by-adjunction of an identity. We also denote by Aỵ the unit ball of A. 90 MAHMOOD KHOSHKAM . If A B are c -subalgebras of a c -algebra c and A is nuclear then A X D c B g D for any nuclear c -algebra D 2 Theorem . We also note that D A a. D B implies D A c D g B . To see this consider the exact sequence 0- Z g g C D - 0. Then j x X g 1 xeZ is a cross section for p. Let X