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).
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Báo cáo toán học: " The Embedding of Haagerup Lop Spaces" Vietnam Journal of Mathematics 34:3 (2006) 353–356 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 ‹ 9$ 67 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 Received April 18, 2006Abstract. The aim of this paper is to give a proof for a theorem due to S. Goldsteinthat: If there is a σ- weakly continuous faithful projection of norm one from a vonNeumann algebra M onto its von Neumann subalgebra N , then Lp (N ) can be canon-ically embeded into Lp (M ). Here Lp (A) [6] denotes the Haagerup Lp space over thevon Neumann algebra A.2000 Mathematics Subject Classification: 46L52, 81R15.Keywords: von Neumann algebras, Haagerup spaces, conditional expection for vonNeumann algebras.Let M be a von Neumann algebra acting in a Hilbert space H and ψ a normal ψfaithful semifinite weight on M . Let {σt }t∈R denote the modular automorphismgroup on M associated with ψ. The crossed product M = M σt R is a vonNeumann algebra acting on H = L2 (R, H ) generated by ψ (πM (a)ξ )(t) = σ−t (a)ξ (t), ξ ∈ H, t ∈ R. (λM (s)ξ )(t) = ξ (t − s) (1)Theorem. Let N be a von Neumann subalgebra of M . Suppose that ψ|N is ψ |N ψ for each t ∈ R. Then N, the crossed product of N ,semifinite and σt |N = σtis canonically embeded into M and for each p ∈ [1, ∞] the space Lp (N ) can becanonically embeded into Lp (M ), so that for any k ∈ Lp (N ) N M k =k p, p N M denote the norms of Lp (N ) and Lp (M ) respectively.where . and . p p354 Phan Viet Thu ψ |N ψ |N ψ ψProof. The condition σt |N = σt means that ∀b ∈ N , σt (b) = σt (b) ∈ N , φi.e. σt leaves N invariant; Together with the condition that ψ|N is semifinite,it implies, by a theorem of Takesaki [5], that there is a σ-weakly continuousprojection E of norm one of M onto N such that ψ = (ψ|N ) ◦ E . It is not hardto show that E ◦ σψ = σψ ◦ E (see for example, [4, Proposition 3.2]). Let N = N σψ|N R, it is a von Neumann algebra acting on L2 (R, H ) = H , tgenerated by operators πN (b), b ∈ N and λN (s), s ∈ R; defined by ψ |N (π(b)ξ (t) = σ−t (b)ξ (t)), (λ(s)ξ (t) = ξ (t − s)) ξ ∈ H, t ∈ R. (2) ψ |N ψSine σ−t (b) = (σ−t|N )(b) for b ∈ N ; (1) and (2) imply π M |N = π N , (3) λM = λN ,and M, N act on the same Hilbert space H. Let M0 be the * algebra generated algebraically by operators πM (a), a ∈ Mand λM (s), s ∈ R. Then M is the σ-weak closure of M0 and any element x0 ∈ M0can be represented as n λM (sk )πM (ak ) for some {sk }n ⊂ R; {ak}n ⊂ M. x0 = 1 1 k =1 We define N0 in the same way. Thus ∀y0 ∈ N0 , m m λM (sk )(πM |N )(bk ) ∈ M0 y0 = λN (sk )πN (bk ) = k =1 k =1for some {sk }m ⊂ R; {bk}m ⊂ N . The σ-weak closure of N0 is N. Then we have 1 1N0 ⊂ M0 and their σ-weak closures verify N ⊂ M. It is clear that ∀x ∈ N ⊂M; ||x||(N ) = ||x||(M ).Consider now the dual action θs of R in M, characterized by θs (πM (a)) = πM (a), ∀a ∈ M, θs (λM (t)) = e−istλM (t), ∀t, s ∈ R. (4)By (3), we have θs (πN (a)) = πN (a), ∀a ∈ N, θs (λN (t)) = e−istλN (t), ∀t, s ∈ R.Thus θs (y0 ) ∈ N0 for y0 ∈ N0, ∀s ∈ R. So that θs (N0 ) ⊂ N0 ⊂ N. Since θs isσ-weakly continuous on M; for all s ∈ R we have ...