Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Derivations và nhóm hành động miễn phí trên C *- đại số. | Copyright by INCREST 198Ố J. OPERATOR THEORY 15 1986 377-410 DERIVATIONS AND FREE GROUP ACTIONS ON C -ALGEBRAS OLA BRATTELI and AKITAKA KISHIMOTO 1. MAIN RESULTS AND NOTATION Let G be a locally compact group which is abelian or compact with dual G and a an action of G on a c -algebra srf as a group of -automorphisms. Let j p U 3 I K G K compact denote the dense -subalgebra of G-finite elements where sda K is the spectral subspace of sđ corresponding to the compact set K 16 11 4 When G is a Lie group it is an open problem whether or not all derivations 5 defined on j p and mapping into j p in short ỗ e Der j p 3 p are pregenerators . Ô is closable and its closure generates a one-parameter group of -automorphisms see 16 . Throughout this paper we adopt the convention that derivations commute with the -operation . This is false if G is not Lie see 12 Example A systematic treatment of positive results in this direction is given in 4 we will here only mention the following theorems If G is compact abelian and Lie a is ergodic 3 is simple and ỏ e Derộs p J p then Ỗ has the decomposition 5 50 3 where 50 is the generator of a one-parameter subgroup of a G and Ỗ is inner so a posteriori 5 is a pregenerator 7 Theorem If G T the circle group ỉđ is separable and a fixes any closed two-sided ideal in sđ and ỗ e Der p j then 5 is a generator 25 Theorem and 26 Theorem . If G is compact not necessarily abelian and there exists a faithful G-covariant representation of with j n s4 Cl where sứ is the fixed point algebra for a and Ỗ 6 DerfiaZp J3 with ổ j a j p then Ỏ again has the decomposition ỗ ỗữ Ỗ with Ỗ bounded and thus 5 is a pregenerator 11 Theorem . If G is compact abelian Lie and there exists a faithful G-covariant representation on a Hilbert space such that the range projections E y j t y . are all equal to 1 and Ỏ e Der j p sýị then 5 is a pregenerator 11 Theorem . Our first main result is Theorem . Let G be a separable locally compact group .
