tailieunhanh - Báo cáo toán học: " The Embedding of Haagerup Lop Spaces"

Mục đích của bài viết này là để cho một minh chứng cho một định lý do S. Goldstein: Nếu có một sự phóng chiếu σ-yếu liên tục trung thành định mức một từ một đại số von Neumann M vào von Neumann N subalgebra, sau đó Lp (N) có thể được canonically embeded vào Lp (M). | Vietnam Journal of Mathematics 34 3 2006 353-356 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2006 The Embedding of Haagerup Lp Spaces Phan Viet Thu Faculty of Math. Mech. and Inform. Hanoi University of Science 334 Nguyen Trai Thanh Xuan Hanoi Vietnam 2 Received April 18 2006 Abstract. The aim of this paper is to give a proof for a theorem due to S. Goldstein that If there is a ơ- weakly continuous faithful projection of norm one from a von Neumann algebra M onto its von Neumann subalgebra N then Lp N can be canonically embeded into Lp M . Here Lp A 6 denotes the Haagerup Lp space over the von Neumann algebra A. 2000 Mathematics Subject Classification 46L52 81R15. Keywords von Neumann algebras Haagerup spaces conditional expection for von Neumann algebras. Let M be a von Neumann algebra acting in a Hilbert space H and Ộ a normal faithful semifinite weight on M. Let ơị t R denote the modular automorphism group on M associated with . The crossed product M M xiCTt R is a von Neumann algebra acting on H L2 R H generated by nM a e t ơtt a e t Am s e t e t - s e e H t e R. 1 Theorem. Let N be a von Neumann subalgebra of M. Suppose that N is semifinite and ơị N ơ N for each t e R. Then N the crossed product of N is canonically embeded into M and for each p e 1 ro the space Lp N can be canonically embeded into Lp M so that for any k e Lp N WIN Wk M where . N and . M denote the norms of Lp N and Lp M respectively. 354 Phan Viet Thu Proof. The condition ơ f n ơf N means that Vb G N Ớị b ơ N b G N . ơị leaves N invariant Together with the condition that N is semifinite it implies by a theorem of Takesaki 5 that there is a ơ-weakly continuous projection E of norm one of M onto N such that ý -0 N o E. It is not hard to show that E o ơ ơ o E see for example 4 Proposition . Let N N Si N R it is a von Neumann algebra acting on L2 R H H generated by operators nN b b G N and An s s G R defined by n b e t ơ N b e t A s e t e t - s e G H t G R. 2 Sine ơ N b ơ t N b for b G N 1