tailieunhanh - Báo cáo toán học: "Lifting projections from quotient C*-algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nâng dự báo từ thương C *- đại số. | Copyright by INCREST 1985 J. OPERATOR THEORY 10 1983 21 30 LIFTING PROJECTIONS FROM QUOTIENT C -ALGEBRAS MAN-DUEN CHOI 1. INTRODUCTION This paper deals with various aspects of the following lifting problem let be a closed two-sided ideal of a c -algebra 91 and let n 91 - 9IẠ be the quotient map for each projection PeM P must there exist a projection Q e 9Í satisfying IĨ Q P - The concrete results formulated here may serve to a certain extent as the C -algebraic realization of some K-theory implication. However no working knowledge of K-theory is needed for the reading of this paper. Notably there is an appropriate K-theory approach to an abstract counterpart of the lifting problem. Utilizing such fact Larry Brown has been successful to offer an affirmative answer for the lifting problem under the assumption that the ideal J is an AF c -algebra. Moreover Brown s result in conjunction with George Elliott s observation 3 Corollary p. 5 leads further to a remarkable structure theorem The extensions of AF c -algebras by AF ideals are AF. Readers are referred to the expository lecture notes by Edward Effros 2 Sections 8 9 for all details. While we may greatly admire the fascinating achievements attributed to C -algebraic K-theory techniques in recent research we may also become fully aware of the appearance of certain mysterious phenomena or indirect constructions associated with the sophisticated machinery. For the sake of better understanting it is often helpful to have explicit constructions and simple examples to show the reality of K-theory miracles. This paper motivated by K-theory methods is written for the exposition of certain plain facts and concrete solutions towards the lifting problem. Along with other outcomes a complete direct proof yielding in particular Brown s result is presented. This paper is organized as follows. 2 is intended for preliminaries. 3 consists of the main results establishing an affirmative answer for the lifting problem provided .