Get Actions of Discrete Amenable Groups on von Neumann Algebras PDF

By Adrian Ocneanu

ISBN-10: 3540156631

ISBN-13: 9783540156635

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Complete II -I12~0 on Let M ~. ~ e M,+ in the topology Proof. sequential be faithful and Then y e M L°. given by the seminorm The above topology being metrizable, completeness. Let w IIXn+ 1 -XnlI % + (Xn) n c (M~) h (M ~) h i8 x ÷ llxIIt~+ IlxyIIt~o . it is enough to prove be a sequence such that ~ w 2 -n II(Xn+ 1 -Xn)YWll ~ < (Xn)~" (YW)w be r e p r e s e n t i n g sequences for x n and y , with all h For each n we can modify x n"~ for ~ outside a n e i g h b o r h o o d xn @ M . Let 32 of e such that IlX~n+l- x~nlI~ + holds for all n and v.

3 was an almost e m b e d d i n g of this A F - a l g e b r a into the U H F - a l g e b r a ~, "ergodic" m o t i v a t e d by the fact that it is much easier to r e c o n s t r u c t U H F - a l g e b r a s inside a given W * - a l g e b r a than AF-algebras. _n+l The c o r o l l a r y shows that on n+l Zg n ~-- £g × i i K. _n n we have Li id , and so we obtamn at the limit a r e p r e s e n t a t i o n of • . in the weak closure of W. If -~ /_i K. × 3 i IInl = 1 ! for all n, then ~ G is an U H F - a l g e b r a and taking all m u l t i p l i c i t i e s If the p r o p o r t i o n of Kn lL n 1,3• in Isnl to be 1 we are done.

3 was an almost e m b e d d i n g of this A F - a l g e b r a into the U H F - a l g e b r a ~, "ergodic" m o t i v a t e d by the fact that it is much easier to r e c o n s t r u c t U H F - a l g e b r a s inside a given W * - a l g e b r a than AF-algebras. _n+l The c o r o l l a r y shows that on n+l Zg n ~-- £g × i i K. _n n we have Li id , and so we obtamn at the limit a r e p r e s e n t a t i o n of • . in the weak closure of W. If -~ /_i K. × 3 i IInl = 1 ! for all n, then ~ G is an U H F - a l g e b r a and taking all m u l t i p l i c i t i e s If the p r o p o r t i o n of Kn lL n 1,3• in Isnl to be 1 we are done.

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


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