International Scholarly Research Network ISRN Algebra Volume 2012, Article ID 197468, 9 pages doi:10.5402/2012/197468 Research Article Tensor Products of Noncommutative Lp-Spaces Somlak Utudee Centre of Excellence in Mathematics, CHE, Si Ayutthaya RD, Bangkok 10400, Thailand Correspondence should be addressed to Somlak Utudee, [email protected] Received 27 January 2012; Accepted 1 March 2012 Academic Editor: F. Kittaneh Copyright q 2012 Somlak Utudee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the notion of tensor product of noncommutative Lp spaces associated with finite von Neumann algebras and define the notion of tensor product of Haagerup noncommutative Lp spaces associated with σ-finite von Neumann algebras. 1. Introduction and Preliminaries The main goal of this paper is explanation of the notion of tensor products of noncommu- tative Lp-spaces associated with von Neumann algebras. The notion of tensor products of noncommutative probability spaces was considered by Xu in 1. We will generalized that notations to the cases of noncommutative Lp-spaces associated with von Neumann algebras. In this section, we also give some necessary preliminaries on noncommutative Lp- spaces associated with von Neumann algebras and tensor product of von Neumann algebras. 1.1. Noncommutative Lp-Spaces Associated with Semifinite von Neumann Algebras We denote by M an infinite-dimensional von Neumann algebra acting on a separable Hilbert space H. Let us define a trace on M, the set of all positive elements of M. Definition 1.1. Let M be a von Neumann algebra. i A trace on M is a function τ : M → 0, ∞ satisfying the following. a τx λyτxλτy for any x, y ∈M and any λ ∈ R . ∗ ∗ b τxx τx x for any x ∈M tracial property. 2 ISRN Algebra ii A trace τ is faithful if τx0 implies x 0. iii A trace τ is normal if supιτ xι τ supιxι for any bounded increasing net xι in M . iv A trace τ is semifinite if for any nonzero x ∈M there exists a nonzero y ∈M such that y ≤ x and τy < ∞. v A trace τ is finite if τ1 < ∞. In this case, we will often assume that it is normalized. Recall that a von Neumann algebra M is called semifinite if any nonzero central projection contains a nonzero finite projection. The following theorem will always used in our construction and can be found in many references see, e.g., 2–4. Theorem 1.2. A von Neumann algebra M is semifinite von Neumann algebra if and only if there exists a faithful normal semifinite trace. Proof. Let M be a von Neumann algebra and τ a faithful normal semifinite trace. For any nonzero central projection p ∈M, there exist x ∈M, 0 / x ≤ p such that τx < ∞. Then, there exists a nonzero projection e ∈Mand a positive number ε such that xe ex ≥ εe.Thus, e is a finite projection. Hence, M is semifinite. Conversely, let M be a semifinite von Neumann algebra. We can assume that M is { } a uniform von Neumann algebra, that is, there exists a family ei i∈I of equivalent finite mutually orthogonal projections such that i∈I ei 1. For each ei, the von Neumann algebra M ei ei is finite and it then possesses a finite normal trace τi. Define a mapping by ∗ ∈M τ x τi vi xvi ,x , 1.1 i∈I ∈M ∗ ∗ where vi is a partial isometry such that vi vi ei vivi . Then, τ is a semifinite normal traces on M . Since the set of all semifinite normal traces on M , obtained in this manner, is sufficient. Then, M possesses a faithful normal semifinite trace. Let M be a von Neumann algebra equipped with a faithful normal semifinite trace τ. For 0 <p<∞,let | |p 1/p | | ∗ 1/2 x p τ x , where x x x . 1.2 The noncommutative Lp-space LpM,τ associated with M,τ is defined as the Banach space M · ∞M M completion of , p .WesetL ,τ equipped with the norm x ∞ x , the operator norm. Note that the usual commutative Lp-space is also in the family of noncommutative Lp-space see, e.g., 1, 5. Elements of the noncommutative Lp-space LpM,τ may be identified with unbounded operators. Definition 1.3. Let M be a von Neumann algebra equipped with a faithful normal semifinite trace τ. i A linear operator x :domx →His called affiliated with M if xu ux for all unitary u in the commutant M of M. ISRN Algebra 3 ii A closed densely defined operator x,affiliated with M, is called τ-measurable if for every ε>0 there exists an orthogonal projection p ∈Msuch that pH⊆domx and τ1 − p <ε. For 0 <p<∞, we have ∼ LpM,τ x | x is τ-measurable,τ|x|p < ∞ . 1.3 ∗ Note that L2M,τ is a Hilbert space with respect to the scalar product x, y τy x. If τ is a normal faithful finite trace, then it is normalized, that is, τ11. In this case, M,τ is called a noncommutative probability space. 1.2. Noncommutative Lp-Spaces Associated with Arbitrary von Neumann Algebras In this subsection, we will recall the definitions of cross product see 2 and Haagerup noncommutative Lp-spaces. For details of the following results in Haagerup noncommutative Lp-spaces, we refer to 1, 5. Let M be a von Neumann algebra on a Hilbert space H,AutM the group of all ∗- automorphism of M, G a locally compact group equipped with its left Haar measure dg and −→ ∈ M G g πg Aut 1.4 a homomorphism of group, such that for any x ∈M, the mapping −→ ∈M G g πg x 1.5 M H is continuous for the weak operator topology in .LetCc G, be the space of all norm continuous functions defined on G and taking values in H which have compact supports. We endow it with the inner product: f1,f2 f1 g ,f2 g dg, 1.6 G and we denote by L2G, H the Hilbert space obtained by completion. ∈M ∈B 2 H For any x , the operator λx L G, is defined by the relations: −1 ∈ H ∈ λx f g πg x f g ,fCc G, ,g G, 1.7 ∈ ∈B 2 H whereas for any g G one defines the unitary operator ug L G, by the relations −1 ∈ H ∈ ug f g f g g ,fCcG, ,g G. 1.8 B 2 H ∈M ∈ The von Neumann algebra generated in L G, by the operators λx,x and ug ,g M M G, is called the cross-product of by the action π of G anditisdenotedby π G or simply by M G. 4 ISRN Algebra Remark 1.4. If M is a von Neumann algebra on a separable Hilbert space H and G is a separable abelian locally compact group acting by ∗-automorphisms of M, then the group G of the character of G acts by ∗-automorphisms of M G. M. Takesaki has proved that ∼ M G G M⊗B L2G, H . 1.9 ∼ In particular, if M is properly infinite, then M G G M. Let M be a von Neumann algebra on a Hilbert space H with a faithful normal semifinite weight ϕ. Let us recall the noncommutative Lp-space associated with M,ϕ constructed by Haagerup see, e.g., 1, 5. ϕ ∈ R R M Let σt σt ,t denote the one parameter modular automorphism group of on { ϕ} ∗ M associated with ϕ. The group σt is the only group of -automorphisms of , with respect N M× R to ϕ which satisfies the KMS-conditions. We consider the cross-product σ ,thatis, 2R H ∈M a von Neumann algebra acting on L , , generated by the operators πx,x ,andthe ∈ R operators λs,s , defined by − ∈ 2 R H ∈ R πx ft σ−txft,λs ft ft s for any f L , ,t . 1.10 It is well known that cross product N is semifinite see 5. By Theorem 10.29 of 2, there { } M exists a strong operator continuous group ut t∈R of unitary operators in such that ϕ ∗ ∈ R σt x utxut ,t . 1.11 Let τ be its unique faithful normal semifinite trace satisfying ◦ −t ∀ ∈ R τ σt e τ, t , 1.12 The ∗-algebra of all τ-measurable operators on L2R, H affiliated with N is denoted by N. For each 0 <p≤∞, we define the Haagerup noncommutative Lp-spaces by p M ∈ N| −t/p ∀ ∈ R L ,ϕ x σtx e x, t . 1.13 We have ∞ 1 L M,ϕ M,LM,ϕ M∗. 1.14 For 0 <p<∞, x ∈ LpM,ϕ if and only if |xp|∈L1M,ϕ, we then define | |p1/p ∈ p M x p x 1 ,xL ,ϕ . 1.15 ≤ ∞ pM · pM For 1 p< , L ,ϕ is a Banach space equipped with a norm p. For 0 <p<1, L ,ϕ · is a quasi-Banach space equipped with a p-norm p. ISRN Algebra 5 It is well known that LpM,ϕ is independent of ϕ up to isometric isomorphism preserving the order and modular structure of LpM,ϕsee 6–8. Sometimes, we denote LpM,ϕ simply by LpM. 1.3. Tensor Products of von Neumann Algebras Let H⊗K be the Hilbert space tensor product of H and K. For x ∈Mand y ∈N, the tensor product x⊗y is the bounded linear operator on H⊗K uniquely determined by x⊗y ξ ⊗ η xξ ⊗ y η ∀ξ ∈H,η∈K.