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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một đơn giản unital projectionless C *- đại số. | Copyright by INCREST 1981 J. OPERATOR THEORY 5 1981 63-71 A SIMPLE UNITAL PROJECTIONLESS C -ALGEBRA BRUCE E. BLACKADAR 1. INTRODUCTION It has long been an open question whether there exists a simple projectionless c -algebra i.e. a c -algebra with no nontrivial projections 9 p. 18 . In 1 the author constructed a simple nonunital projectionless c -algebra. Using the same general method we now construct a unital projectionless simple c -algebra. The idea is to construct a projectionless unital c -algebra r with Prim r a circle and a unital twice-around map ijr.r - r. Then if A s r and P A - - A 1 is taken to be ý A lim A p is the desired algebra. As in 1 r will be a c -algebra of the form . 0 1 - B f V ơ 0 where B is an appropriate simple unital AF algebra and Ơ is an appropriate automorphism of B. The important property that Ơ must have is that ơ p is not equivalent to p for every nontrivial projection p e B this property insures that r is projectionless. B and Ơ must have two other technical properties in order that the twice-around embedding can be constructed. B and Ơ are constructed in Section 2. Actually what is constructed is the dimension group Kữ B and the automorphism O of K0 B . Then in Section 3 the construction of A is described. Section 4 is a description of some of the structure of A. It is clear from its definition that A is nuclear. Proposition 4.2 shows that A has a unique trace which of course must be faithful and finite since A is simple and unital . We also show in Section 4 that XoG4 z. Also in Section 4 some variations on the construction of A are discussed. It is shown in Theorems 4.10 and 4.12 that if X is any compact totally disconnected metric space and N is any countable torsion-free Abelian group then there is a simple unital projectionless c -algebra A I with X Ah s H C X Z N with the strict ordering from the first coordinate and with ET An X. Section 5 concludes with some open questions. 64 BRUCE E. BLACKADAR The research for this .
