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Báo cáo toán học: "A converse to Ocneanu's theorem "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một trò chuyện với định lý của Ocneanu. | Copyright by INCREST 1983 J. OPERATOR THEORY 10 1983 61 -63 A CONVERSE TO OCNEANU S THEOREM V. F. R. JONES Ocneanu s theorem 2 states grosso modo that for any countable amenable discrete group G there is only one action by outer automorphisms on the hyper-finite type Hi factor R up to outer conjugacy. Here we exhibit for any countable non-amenable discrete group G two outer actions of G on R which are not outer conjugate. The invariant we shall use is the fixed point algebra for the action on the algebra of central sequences. The proof will be based on the following elementary Hilbert space lemma obtained by K. Schmidt and the author. The setting is as follows. Let 9 be a separable Hilbert space with orthonormal basis e I i e I u 0 . If G is a countable group we form the infinite tensor product where each Ji g is a copy of Ji . geG taken along the vector e0 g. Let g Ug be the unitary representation corres-geG ponding to the Bernoulli shift i.e. ug ei l . free geG Lemma 1. The representation ug is a direct sum of the trivial representation and a subrepresentation of a direct sum of copies of the regular representation. Proof. An orthonormal basis for 3C is given by the collection F of all functions i G - Zu 0 such that i g 0 except on a finite subset of G. Moreover the action of G on ar comes from permutations of the basis gi i ilhg . If 0 denotes the space of orbits of F under the action of G then the representation g ug is Tig where Tig is the representation coming from right translation on z2 of the eeo orbit 0. Such a representation Tig is a subrepresentation of the regular representation whenever the stabilizer subgroup of a point i e 0 is finite. It thus suffices to show that the stabilizer of every point i e F is finite except when i g 0 V g e G. But if T g e G I i g 0 then T is finite and if h is in the stabilizer of i then Th T. This is only possible for finitely many h. Q.E.D. 62 V. R R. JONES Now let T be the trace on R r 1 1 and let US say that an .