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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nhận xét về phổ Connes C *- động học hệ thống. | J. OPERATOR THEORY 3 1980 143-Ỉ48 CT Copyright by INCREST 1980 REMARKS ON THE CONNES SPECTRUM FOR C -DYNAMICAL SYSTEMS GERT K. PEDERSEN In 2 2.3.17 A. Connes showed that for a IF -dynamical system 5DI G a where ỈƠÌ is a factor the Connes spectrum r a of the representation a is the intersection of all Arveson spectra Sp j5 where p ranges over the representations exterior equivalent to a. Regarding p as a perturbation of a by a unitary one-cocycle this result characterizes r a as the part of Sp a which is invariant under all perturbations. In the case where Gjripi is compact Connes shows in 2 2.3.13 the existence of a perturbation such that Sp 7 F a thus J has the minimal spectrum within its exterior equivalence class. Connes also solves the problem of finding for such a minimal p a state or weight satisfying the KMS condition with respect to p generalized traces on IIIA factors see 2 4.3 . We shall in this paper obtain a version of Connes results for C -dynamical systems. The main tool will be a theorem of L. G. Brown 1 on stably isomorphic c -algebras. The author is indebted to A. Connes and D. Olesen for valuable discussions during a stay in April 1978 at the Institut des Hautes Etudes Scientifiques France where this work was carried out. Consider a C -dynamical system A G a i.e. a c -algebra A a locally compact abelian group G and a pointwise continuous automorphic representation a G - - Aut of G on A. Recall from 5 3.3 that A is G-simple if it contains no nontrivial closed G-invariant ideals. Fix an infinite dimensional separable Hilber space H and let X denote the c -algebra of compact operators on H. Proposition. Let A G a be a C -dynamical system where A is separable and G-simple. For each non-zero G-invariant hereditary c -subalgebra B of A there is a system A X G P such that p is exterior equivalent to a X I on A AỐ and Sp 0 Sp a B . Proof. Since A is G-simple B is not contained in any proper closed ideal of A. Since moreover A is separable we conclude from
