TAILIEUCHUNG - Báo cáo toán học: "The linear span of projections in simple C*-algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Khoảng tuyến tính dự báo trong đơn giản C *- đại số. | Copyright by INCREST 1980 J. OPERATOR THEORY 4 1980 289-296 THE LINEAR SPAN OF PROJECTIONS IN SIMPLE C -ALGEBRAS GERT K. PEDERSEN Let A be a c -algebra and consider the following chain of conditions AF A is approximately finite-dimensional. FS sa has a dense set of elements with finite spectrum. CP The convex hull of projections is dense in tfie unit ball of A . LP The linear span of projections is dense in A. AP The algebra generated by projections is dense in A. BP A has non-trivial projections. Clearly each of the conditions implies every other below it. The question under discussion is whether any lower condition will imply a higher one if A is assumed to be simple. We show that BP implies AP an unpublished result by the authors of 1 and that BP implies LP in a number of interesting cases among which are Cuntz s algebras 7 and Rieffel s irrational rotation c -algebras 11 12 . Nevertheless 3P does not in general imply LP just as LP does not imply CP . It is conceivable that CP implies FS for any simple c -algebra but it is false that FS implies AF . Finally it should be mentioned that not all simple c -algebras satisfy BP see 3 and 4 For background and terminology we refer to 9 A preliminary version of this paper was substantially enriched by contributions from B. Blackadar and G. A. Elliott. It is a pleasure to thank them both for their generous help. POSITIVE RESULTS Let D be a closed -invariant subspace of a c -algebra A. Following 1 we say that A derives D if ad da ỄŨ for every a in Ạ and d in D. We say that D is unitarily invariant if uDu D for every unitary u in Ầ. The equation exp ừ ỉ d exp itlì d it hd dh . t2 290 GERT PEDERSEN valid for every real t and every h in 4sa shows that if D is unitarily invariant then A derives D cf. 1 . Since the projections in A is a unitarily invariant set we immediately obtain the following result. Lemma 1. Given a c -algebra A let L P and A P respectively denote the closed linear span and the c -algebra generated by

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