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

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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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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, 2006 Abstract. The aim of this paper is to give a proof for a theorem due to S. Goldstein that: If there is a σ- weakly continuous faithful projection of norm one from a von Neumann 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 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 ψ a normal ψ faithful semiﬁnite weight on M . Let {σt }t∈R denote the modular automorphism group on M associated with ψ. The crossed product M = M σt R is a von Neumann 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 = σt is canonically embeded into M and for each p ∈ [1, ∞] the space Lp (N ) can be canonically 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 p
354 Phan Viet Thu ψ |N ψ |N ψ ψ Proof. The condition σt |N = σt means that ∀b ∈ N , σt (b) = σt (b) ∈ N , φ i.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 continuous projection E of norm one of M onto N such that ψ = (ψ|N ) ◦ E . It is not hard to 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 , t generated 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 ψ Sine 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 ∈ M and λM (s), s ∈ R. Then M is the σ-weak closure of M0 and any element x0 ∈ M0 can 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 =1 for some {sk }m ⊂ R; {bk}m ⊂ N . The σ-weak closure of N0 is N. Then we have 1 1 N0 ⊂ 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 is σ-weakly continuous on M; for all s ∈ R we have θs (N) ⊂ N. The continuity of θs in measure implies also
The Embedding of Haagerup Lp Spaces 355 θs (N) ⊂ N and M N θs |N = θs , ∀s ∈ R, M N where θs and θs denote the dual action θs of R on M and on N respectively. By deﬁnition of Lp (N ) and Lp (M ) and the above results, it follows that s Lp (N ) = {k ∈ N|∀s ∈ R : θs k = e− p k} N (5) s −p M p = {k ∈ N ⊂ M|∀s ∈ R : θs k =e k } ⊂ L (M ). Then we have Lp (N ) ⊂ Lp (M ). It remains now to show that M N for any k ∈ Lp (N ) ⊂ Lp (M ). k =k p p It suﬃces to demonstrate it for the case p = 1. Note that L1 (M ) M∗ ; L1 (N ) N∗ and for any φ ∈ N∗ ; φ ◦ E ∈ M∗ . In [1, 2] the author has ˆ ∧ ∧ proved that E can be extended canonically to E : M+ → N+ ; E : M → N 1 1 and E1 : L (M ) → L (N ), given by h(φ) → hφ◦E . It is extended also to Ep : Lp (M ) → Lp (N ); and for any φ ∈ N∗ φ = ( φ ◦ E ) − |N . (N ) Let us calculate the norm of hN = hM E . Note that ||hN ||1 = ||φ||(N ) and φ φ◦ φ (M ) ||hM E ||1 = ||φ ◦ E ||(M ) for any φ ∈ N∗ . We have φ◦ (N ) (M ) φ = sup |φ(b)| ≥ sup |(φ ◦ E )(a)| = φ ◦ E b∈N, b ≤1 a∈M, a ≤1 (N ) ≥ sup |(φ ◦ E )(b)| = sup |φ(b)| = φ . (6) b∈N, b ≤1 b∈N, b ≤1 (N ) (M ) (N ) (M ) ; i.e. hN = hM E This implies φ = φ◦E , which shows that φ 1 φ◦ 1 (N ) (M ) 1 1 for any k ∈ L (N ) ⊂ L (M ), one has k = k . It is now obvious that, 1 1 for each p ∈ [1, ∞], (N ) (M ) ∀k ∈ Lp (N ) ⊂ Lp (M ). k =k , p p References 1. S. Goldstein, Conditional expectations and Stochastic integrals in non commu- tative Lp -spaces, Math. Proc. Camb. Phil. Soc. 110 (1991) 365–383. 2. S. Goldstein, Norm convergence of martingales in Lp -spaces over von Neumann algebras, Revue Roumaine de Math. Pures et Appl. 32 (1987) 531–541. 3. R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Alge- bras, Vol. I, 1983; Vol. II, 1986. Academic Press, New York – London. 4. C. E. Lance, Martingale convergence in von Neumann algebras, Math. Proc. Camb. Phil. Soc. 84 (1978) 47–56.
356 Phan Viet Thu 5. M. Takesaki, Conditional expectations in von Neumann algebras, J. Funct. Anal. 9 (1972) 306–321. 6. M. Terp, Lp -spaces Associated with von Neumann Algebras, Notes Kφbenhavns Universitet, Matematisk Institut, N0 . 3, 1981.