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 τ x λy τ x λτ y for any x, y ∈M and any λ ∈ R . ∗ ∗ b τ xx τ x x for any x ∈M tracial property . 2 ISRN Algebra
ii A trace τ is faithful if τ x 0 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 τ y < ∞. v A trace τ is finite if τ 1 < ∞. 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 τ x < ∞. 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 The noncommutative Lp-space Lp M,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 Lp M,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 :dom x →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⊆dom x and τ 1 − p <ε. For 0 ∗ Note that L2 M,τ is a Hilbert space with respect to the scalar product x, y τ y x . If τ is a normal faithful finite trace, then it is normalized, that is, τ 1 1. 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,Aut M 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 L2 G, 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 ,fCc G, ,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 L2 G, 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, 2 R 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 f t σ−t x f t ,λs f t f t 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 L2 R, H affiliated with N is denoted by N. For each 0 We have ∞ 1 L M,ϕ M,LM,ϕ M∗. 1.14 For 0 ≤ ∞ p M · p M For 1 p< , L ,ϕ is a Banach space equipped with a norm p. For 0 It is well known that Lp M,ϕ is independent of ϕ up to isometric isomorphism preserving the order and modular structure of Lp M,ϕ see 6–8 . Sometimes, we denote Lp M,ϕ simply by Lp M . 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 x ξ ⊗ y η ∀ξ ∈H,η∈K. 1.16 Let M⊂B H , N⊂B K be two von Neumann algebras. The algebraic tensor product M⊗Nof M and N, n M⊗N ⊗ | ∈M ∈N xk yk xk ,yk ,n 1, 2,... , 1.17 k 1 is a ∗-subalgebra of operators on H⊗K. The von Neumann algebra generated by H⊗K in B H⊗K is denoted by M⊗N and it is called the tensor product of von Neumann algebras M and N. Since the map Mx −→ x⊗1 ∈M⊗N 1.18 is a ∗-isomorphism, we can view M as a von Neumann subalgebra of M⊗N. Similarly, we can also view N as a von Neumann subalgebra of M⊗N. By the Tomita commutation theorem, M and N commute and together generate M⊗N. Example 1.5. Let T be the unit circle equipped with the normalized Lebesque measure dm ∞ ∞ and M,τ a finite von Neumann algebra. Let L L T , dm ⊗ M,τ be consisting of all functions f such that τ xf z zn dm z , ∀x ∈ L1 M,τ ,n∈ Z,n>0. 1.19 ∞ ∞ Then, H T, M is a finite subdiagonal algebra of L T , dm ⊗ M,τ see 5 . 2. Tensor Products of Noncommutative Lp-Spaces Associated with von Neumann Algebras We first consider the simple case: finite von Neumann algebras. 6 ISRN Algebra 2.1. Tensor Products of Noncommutative Lp-Spaces Associated with Normal Faithful Finite von Neumann Algebras Theorem 2.1. Let M and N be finite von Neumann algebras equipped with normal faithful normalized traces τ1 and τ2, respectively. Then, there exists a normal faithful trace on the tensor product von Neumann algebra M⊗N such that ⊗ ∈M ∈N τ x y τ1 x τ2 y ,x ,y . 2.1 M N Proof. Since τ1 and τ2 are normal faithful normalized traces, we can view and as H 2 M K 2 N von Neumann algebras acting on L ,τ1 and L ,τ2 , respectively, by left M multiplication. Then, τ1 and τ2 are the vector states associated to the identities 1M of and 1N of N, respectively. That is, ∈M ∈N τ1 x x 1M , 1M ,τ2 y y 1N , 1N ,x ,y . 2.2 Let τ be the vector state associated to 1M ⊗ 1N on M⊗N. Then, τ is uniquely determined by ⊗ ∈M ∈N τ x y τ1 x τ2 y for all x ,y . Therefore, τ is tracial and faithful. ⊗ τ is called the tensor product trace of τ1 and τ2, and we denote it by τ1 τ2. Then, we p p M⊗N ⊗ can define the noncommutative L -spaces L ,τ1 τ2 and called it the noncommutative p M N L -tensor product of ,τ1 and ,τ2 . Example 2.2. Let us consider two cases see 1, 5 . ∞ 1 Let Ω,P be a probability space. We can represent L Ω as a von Neumann algebra on H L2 Ω by multiplication and the integral against P is a normal ∞ faithful normalized trace on L Ω .Let M,τ be a noncommutative probability ∞ space. Then, Lp L Ω ⊗M, ⊗τ is isometric to Lp Ω,Lp M , the usual Lp-space of p-integrable functions from Ω to Lp M . 2 Let B l2 be equipped with the usual trace Tr and let M,τ be a noncommutative probability space. Then, the element of Lp B l2 ⊗M, Tr ⊗τ , the noncommutative Lp-tensor product of B l2 , Tr and M,τ can be identified with an infinite matrix with entries in Lp M,τ . 2.2. Infinite Tensor Products of Noncommutative Lp-Spaces Associated with Finite von Neumann Algebras ∈ N M ⊗ M For n ,let n be a von Neumann algebras. The infinite algebraic tensor product n≥1 n of M ⊗ ∈M n is the set of all finite linear combinations of elementary tensors n≥1xn, where xn n and all but finitely many xn are 1, that is, m M k | k ∈M ∈ N n xn xn n and all but finitely many xn are 1,m . 2.3 n≥1 k 1 n≥1 First, let us consider infinite tensor products of noncommutative Lp-spaces associated with finite factors. ISRN Algebra 7 ∈ N M For n ,let n be a finite factor equipped with a unique normal faithful normalized ⊗ M trace τn. We have the product state τ on n≥1 n, defined by ∈M τ xn τn xn ,xn n. 2.4 n≥1 n≥1 ⊗ M The infinite von Neumann tensor product n≥1 n is the weak-closure of the image of the ⊗ M 2 ⊗ M representation of n≥1 n by the left multiplication on the Hilbert space L n≥1 n .Itis a finite factor with the trace τ is the extension of τ, which is the unique normalized trace. τ is ⊗ called the infinite tensor product trace of τn anddenotedby n≥1τn see 7 . Then, we can define p p ⊗ M ⊗ the noncommutative L -spaces L n≥1 n, n≥1τn and called it the infinite noncommutative p M L -tensor product of n,τn . Next, let us consider the infinite tensor products of noncommutative Lp-Spaces associated with normal faithful finite von Neumann algebras. Theorem 2.3. M Let m m∈N be a sequence of finite von Neumann algebras equipped with normal A ∪ M ⊗M ⊗···⊗M H A faithful normalized traces τm.Let m≥1 1 2 m .Let be the completion of with respect to the inner product m ⊗···⊗ ⊗···⊗ ∗ x1 xm,y1 ym τk ykxk . 2.5 k 1 Let π : A−→B H be defined by π x Λ a Λ xa ,x∈A,a∈A, where Λ : A −→ H is the inclusion. 2.6 ∗ Let N be the weak -closure of π A in B H . Then, there exists a normal state ν on N such that m ⊗···⊗ ∈M ∈ N ν x1 xm τk xk ,xk k,m . 2.7 k 1 H 2 M M H Proof. Let m L m and consider m as a von Neumann algebra on m by left multiplication. Let N M ⊗M ⊗···⊗M m 1 2 m, ⊗ ⊗···⊗ 2.8 νm τ1 τ2 τm. N N We view m as a von Neumann subalgebra of m 1 via the inclusion: ⊗···⊗ −→ ⊗···⊗ ⊗ x1 xm x1 xm 1Mm 1 . 2.9 | A ∗ Since τm 1 1 1, νm 1 Nm νm.Notethat is a unital -algebra and the traces νm induce a A faithful normal state νo on . Since νo is faithful, the representation π is faithful. Therefore, A N B H N ,andall m, can be viewed as subalgebras of .Letν the restriction to of the vector 8 ISRN Algebra Λ | state given by 1 . Then, ν is tracial and faithful. The trace ν Mm τm and ν is the unique normal state on N such that m ⊗···⊗ ∈M ∈ N ν x1 xm τk xk ,xk k,m . 2.10 k 1 N p M ,ν is called the infinite tensor products of noncommutative L -spaces of m,τm see 1 . C × Example 2.4. Let M2 be the full algebra of 2 2 matrices. Murray and von Neumann proved that the infinite tensor product ⊗ C WOT n≥1M2 , 2.11 C produced with respect to the unique normalized trace tr2 on M2 , is the unique AFD II1- factor see, e.g., 7 . 2.3. Tensor Products of Noncommutative Lp-Spaces Associated with σ-Finite von Neumann Algebras In the case of tensor products of σ-finite von Neumann algebras, we will apply the reduction theorem. This theorem was proved by Haaagerup in 1979 and can be used to reduce the problems on general noncommutative Lp-spaces to the corresponding ones on those associated with finite von Neumann algebras see, e.g., 6, 8 . ∈{ } M p M For each k 1, 2 ,let k be a σ-finite von Neumann algebra. Let L k be the Haagerup noncommutative Lp-spaces. By the reduction theorem, there exist a Banach space R Xp k a quasi Banach space if p<1 , a sequence k,m m∈N of finite von Neumann algebras, ∈ N each equipped with a faithful normal finite trace τk,m, and for each m an isometric p R −→ embedding Jk,m : L k,m,τk,m Xp k such that p R ⊂ p R ∈ N ≤ 1 Jk,m1 L k,m1 ,τk,m1 Jk,m2 L k,m2 ,τk,m2 for all m1,m2 such that m1 m2; p R 2 m∈N Jk,m L k,m,τk,m is dense in Xp k; p M 3 L k is isometric to a subspace Yp k of Xp k; p R ∈ N ≤ ∞ 4 Yp k and all Jk L k,m,τk,m ,m are 1-complemented in Xp k for 1 p< . p R p R Here, L k,m,τk,m is the tracial noncommutative L -space associated with k,m,τk,m . R Thus, we have a sequence k,m,τk,m of finite von Neumann algebras. We then have p R R ⊗R ⊗ the noncommutative L -tensor product m,τm : 1,m 2,m,τ1,m τ2,m . Applying the construction in Section 2.2, we will be able to construct the infinite tensor products of non- p R commutative L -spaces of m,τm . Hence, we have the tensor products of noncommutative p p M p M L -spaces of L 1 and L 2 . {M } With this setting, if k k∈N be a sequence of σ-finite von Neumann algebra, we will also be able to construct the infinite tensor product of noncommutative Lp-spaces associated with σ-finite von Neumann algebras. ISRN Algebra 9 Let M be an arbitrary von Neumann algebra. Then, M admits the following direct sum decomposition: M N ⊗ j B Kj , 2.12 j∈J N where each j is an σ-finite von Neumann algebra. Using the reduction theorem in general case, the approximation theorem can be extended to the general case as follows. Let M be a general von Neumann algebra and 0 1 J Lp R ,τ ⊂ J Lp R ,τ for all i, j ∈ I such that i ≤ j; i i i j j j p R 2 i∈I Ji L i,τi is dense in Xp; p M 3 L is isometric to a subspace Yp of Xp; p R ∈ ≤ ∞ 4 Yp and all Ji L i,τi ,i I are 1-complemented in Xp for 1 p< . p R p R Here, L i,τi is the tracial noncommutative L -space associated with i,τi . If we can define the notion of uncountable infinite tensor products of noncommu- tative Lp-spaces associated with finite von Neumann algebras, we should be able to define tensor products of Haagerup noncommutative Lp-spaces. Acknowledgment This research is supported by the Centre of Excellence in Mathematics, the commission on Higher Education, Thailand. References 1 Q. Xu, Operator Spaces and Noncommutative Lp, Lecture in the Summer School on Banach Spaces and Operator Spaces, Nankai University, China. 2 S. Stratil˘ aandL.Zsid˘ o,´ Lectures on von Neumann Algebras, Abacus Press, Tunbridge Well, Kent, UK, 1979. 3 G. K. Pedersen, “The trace in semi-finite von Neumann algebras,” Mathematica Scandinavica, vol. 37, no. 1, pp. 142–144, 1975. 4 M. Takesaki, Theory of Operator Algebras—I, Springer, Berlin, Germany, 2003. 5 G. Pisier and Q. Xu, “Non-commutative Lp-spaces,” in Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517, North-Holland, Amsterdam, The Netherlands, 2003. 6 U. Haagerup, “Lp-spaces associated with an arbitrary von Neumann algebra,” in Algebres` d’operateurs´ et leurs applications en physique mathematique´ (Proc. Colloq., Marseille, 1977), vol. 274 of CNRS International Colloquium, pp. 175–184, CNRS, Paris, France, 1979. 7 M. Terp, Lp Spaces Associated with von Neumann Algebras Notes, Mathematical Institute, Copenhagen University, 1981. 8 U. Haagerup, M. Junge, and Q. Xu, “A reduction method for noncommutative Lp-spaces and applica- tions,” Memoirs of the American Mathematical Society, vol. 331, pp. 691–695, 2000. 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