tailieunhanh - Báo cáo toán học: "SAW*-algebras and corona C*-algebras, contributions to non- commuttative topology "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: SAW *- đại số và C *- đại số, vầng hào quang đóng góp commuttative topo. | J. OPERATOR THEORY 15 1986 15-32 Copyright by INCREST 1986 S ll4 -ALGEBRAS AMD CORONA C -ALGEBRAS CONTRIBUTIONS TO NON-COMMUTATIVE TOPOLOGY GERT K. PEDERSEN INTRODUCTION One aspect of non-commutative topology is to view a general c -algebra as a non-commutative C0 Af . Each property concerning a locally compact Haus-dorff space Xcan in principle be formulated in terms of the function algebra C0 X and will then usually make sense and hopefully be true for any non-commutative c -algebra. Instead of this translation one may look directly for the objects that in a non-commutative c -algebra A replaces the open and closed sets from the case A C0 Z . It is generally agreed by Chuck Akemann and me that these objects are the open and the closed projections in the enveloping von Neumann algebra A. of A. The open projections are in a bijective correspondence with the hereditary C -suba1gebras of A of the form L n A for some closed left ideal L of A see 11 . Unfortunately their complements the closed projections only correspond to quotients when they are central in A so that the complementary open projection corresponds to a closed ideal of 7Í . In this paper the results are framed exclusively in terms of hereditary algebras and ideals. We shall be much concerned with c -algebras or c -subalgebras that are ơ-unital which by definition means that they contain a strictly positive element. As shown by Aarnes and Kadison A is 7-unital if and only if it contains a countable approximate unit e . Indeed any sequence e fn h will do provided that h is strictly positive and the functions fn increase pointwise to 1 on Sp A 11 -Furthermore the approximate unit e may be chosen quasi-central with respect to any fixed countable subset of the multiplier algebra M A of A 11 Clearly a c -algebra C0 Jf is ơ-unital precisely when X is T-compact. The ơ-unital algebras occur quite frequently in c -algebra theory now and a proper terminology is long overdue. In .