tailieunhanh - Báo cáo toán học: "Transformation group C*-algebras with continuous trace. II "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Chuyển đổi nhóm C *- đại số với dấu vết liên tục. II | J. OPERATOR THEORY 11 1984 109-124 Copyright by INCREST 1984 TRANSFORMATION GROUP C -ALGEBRAS WITH CONTINUOUS TRACE. II PAUL s. MUHLY and DANA p. WILLIAMS 1. INTRODUCTION In this note G will be a locally compact group acting continuously on a locally compact Hausdorff space ÍÌ. The translate of an X e Í2 by an s e G will be denoted by s-x. The pair G Í2 will be referred to as a transformation group and the familiar transformation group c -algebra associated with G 12 3 will be denoted by C G ÍĨ . Of course these objects are special cases of C -dynamical systems and C -crossed products respectively. The question of when C G Í2 has continuous trace has attracted considerable attention lately and in a sense has been fairly well answered. Nevertheless a number of related and more delicate questions remain. Our objective in this note is to contribute to one of these. In 8 Green found necessary and sufficient conditions for C G Í2 to have continuous trace under the assumption that G is freely acting . under the assumption that s-x X implies J e. To be precise 8 Theorem 17 asserts that if G is freely acting and if G ÍĨ is second countable then C G Í2 has continuous trace if and only if compact subsets of Í2 are wandering in the following sense for each compact set K c Í2 the set í6 G sKCi K 0 is pre-compact in G. Green actually proved more than this. He showed that when G is freely acting and when compact subsets of Í2 are wandering then C G Í2 is isomorphic to the c -algebra determined by a continuous field of Hilbert spaces over the spectrum of C G Í2 . To say the same thing differently Green showed that when G is freely acting and when compact subsets of Í2 are wandering then C G Í2 has continuous trace and furthermore the Dixmier-Douady invariant of C G i2 vanishes cf. 2 . 110 PAUL s. MUHLY and DANA p. WILLIAMS In 14 the second author determined when C G Q has continuous trace in a variety of cases without assuming that G is freely acting. His results are .