# Actions of Discrete Amenable Groups on von Neumann Algebras by Adrian Ocneanu

Similar linear books

Lie Groups Beyond an Introduction

This publication takes the reader from the tip of introductory Lie staff idea to the brink of infinite-dimensional workforce representations. Merging algebra and research all through, the writer makes use of Lie-theoretic tips on how to enhance a stunning concept having vast functions in arithmetic and physics. The publication at the start stocks insights that utilize genuine matrices; it later is determined by such structural positive aspects as houses of root structures.

Lectures on Tensor Categories and Modular Functors

This ebook provides an exposition of the kin one of the following 3 subject matters: monoidal tensor different types (such as a class of representations of a quantum group), third-dimensional topological quantum box thought, and 2-dimensional modular functors (which clearly come up in 2-dimensional conformal box theory).

Proper Maps of Toposes

We advance the speculation of compactness of maps among toposes, including linked notions of separatedness. This concept is outfitted round types of 'propriety' for topos maps, brought right here in a parallel style. the 1st, giving what we easily name 'proper' maps, is a comparatively vulnerable as a result of Johnstone.

Additional info for Actions of Discrete Amenable Groups on von Neumann Algebras

Example text

Let us choose (n) ek = le(k0) Step C is proved. small, to make y = 0 in this procedure N ' N M~ k such that k, thus e0 = fq+ I, can be taken arbitrarily Let us describe N'A M e . ,q, gE K (i + IKI)-I Ifkl~ Ifk+ll#~< ( i - ( i + Step D. for and lek~g(ek) I~ < Ylekl ~ We have gE K . ,q ; sub C*-algebra ek A N, Let '" of ~ ( I~, M~) Z~(IN,M w) is kept globally term by term on yielded then ek ~(~,M~)). ,q, geE. 48 We also have for all Ad Um(e k) and thus e k E N' N M~. 4 or M~ Ad Um(~(ek)) = ~(ek Mm, we shall part by E= G apply bE = 1 ~ k=0 the Index in o r d e r of the p r o o f .....

To p r o v e * - s u b s e t of s c a l a r values; o u t e r and let Then q' < 1 - (q V ~(q) V B-I(q)) it we i n f e r Let takes < ¼ T(q). (q' V 8(q'))(qV 8(q)) = 0 dicted is, by T x E M e. ~(q~(q)) if not, same reasoning for 8 e Aut M such t h a t Indeed, then is a factor, [x[T = ~(Ix[) maximal q' ~ 0 M (~n)n be a t o t a l sequences ~E~ that B = (~)e there exists We r e m a r k fixed. for a By m e a n s e Aut M e a projection t h a t in the a l g e b r a Me we have T~(18(q)aV-a~qI2 ) = Te(] (8(q)-q)aV] 2) = Te(laVlZ)T((8(q)-q)) 2 > ~/~ Te(laVl2) H e n c e we can p i c k o u t of a r e p r e s e n t i n g element q~e M such that llqV I[ <~ 1 (q~)~ r e p r e s e n t s Ho~(q)a-aq[l and the c o n t r a d i c t i o n q an >i 1/21ja~ll~ , 1 ll[q ,Sk]ll# ~< U T h e n the s e q u e n c e for and II~(q~)a~ -a~q~)I1# 1 li[qv,~k]N <~ ~ sequence 2 i> thus o b t a i n e d k=l ' .....

Such that for any 05 = W - lim T (x n) n÷t0 if X n = X for all = ~(~(x)) sequences automorphisms leave it globally ~: e ÷ k = (Xn) n E d (i) part lemma. sequence. M ~, which there w E V n \ V n + 1 , and for (xk(V)) w. to the one of the p r e c e d i n g w e can p u t t o g e t h e r predual if by (a(Xn)) n of . 37 Remark. then From Proof. = u Cn n invariant Let a unital by ~, and M n • M n + l • M, sequence any (5) ~ • M, , Let Vn, be n~ 1 such be finite sets w i t h on M. n) Yp(n) (9) lla~(Xp(n))- for all The lemma and now In w h a t follows, i/n as in the such , x= union M,.