Tensor Products of Noncommutative 𝐿𝑝-Spaces

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 noncommutative 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  λy  τx  λτy 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 τx  0 implies x  0. iii A trace τ is normal if supι τxι   τsupι xι  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 ∈ M and 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 ei Mei is finite and it then possesses a finite normal trace τi . Define a mapping by τx 

   τi vi∗ xvi ,

x ∈ M ,

i∈I

1.1

where vi ∈ M is a partial isometry such that vi∗ vi  ei  vi vi∗ . 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 xp 

  p 1/p τ |x| ,

where |x|  x∗ x1/2 .

1.2

The noncommutative Lp -space Lp M, τ associated with M, τ is defined as the Banach space completion of M,  · p . We set L∞ M, τ  M 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 : domx → H is called affiliated with M if xu  ux for all unitary u in the commutant M of M.

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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 ∈ M such that pH ⊆ domx and τ1 − p < ε. For 0 < p < ∞, we have   p Lp M, τ ∼  x | x is τ-measurable, τ |x| < ∞ .

1.3

Note that L2 M, 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, τ1  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, AutM the group of all ∗automorphism of M, G a locally compact group equipped with its left Haar measure dg and G  g −→ πg ∈ AutM

1.4

a homomorphism of group, such that for any x ∈ M, the mapping G  g −→ πg x ∈ M

1.5

is continuous for the weak operator topology in M. Let Cc G, H 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. For any x ∈ M, the operator λx ∈ BL2 G, H is defined by the relations:       g  πg−1 x f g , λx f



f ∈ Cc G, H, g ∈ G,

1.7

whereas for any g ∈ G one defines the unitary operator ug ∈ BL2 G, H by the relations 



   ug f g  f g −1 g ,

f ∈ Cc G, H, g ∈ G.

1.8

The von Neumann algebra generated in BL2 G, H by the operators λx , x ∈ M and ug , g ∈ G, is called the cross-product of M by the action π of G and it is denoted by Mπ G or simply by M  G.

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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  of the character of G acts by ∗-automorphisms of M  G. M. Takesaki has proved that G 

∼ 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 Let σt  σt , t ∈ R denote the one parameter modular automorphism group of R on M ϕ associated with ϕ. The group {σt } is the only group of ∗-automorphisms of M, with respect to ϕ which satisfies the KMS-conditions. We consider the cross-product N  M × σ R, that is, a von Neumann algebra acting on L2 R, H, generated by the operators πx , x ∈ M, and the operators λs , s ∈ R, defined by   πx ft  σ−t xft,

  λs ft  ft − s

for any f ∈ L2 R, H, t ∈ R.

1.10

It is well known that cross product N is semifinite see 5. By Theorem 10.29 of 2, there exists a strong operator continuous group {ut }t∈R of unitary operators in M such that ϕ

σt x  ut xu∗t ,

t ∈ R.

1.11

Let τ be its unique faithful normal semifinite trace satisfying τ ◦ σt  e−t τ,

∀t ∈ R,

1.12

 The ∗-algebra of all τ-measurable operators on L2 R, H affiliated with N is denoted by N. For each 0 < p ≤ ∞, we define the Haagerup noncommutative Lp -spaces by      | σt x  e−t/p x, ∀t ∈ R . Lp M, ϕ  x ∈ N

1.13

We have   L∞ M, ϕ  M,

  L1 M, ϕ  M∗ .

1.14

For 0 < p < ∞, x ∈ Lp M, ϕ if and only if |xp | ∈ L1 M, ϕ, we then define  1/p xp  |x|p 1 ,

  x ∈ Lp M, ϕ .

1.15

For 1 ≤ p < ∞, Lp M, ϕ is a Banach space equipped with a norm ·p . For 0 < p < 1, Lp M, ϕ is a quasi-Banach space equipped with a p-norm  · p .

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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 ∈ M and 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.

1.16

Let M ⊂ BH, N ⊂ BK be two von Neumann algebras. The algebraic tensor product M ⊗ N of M and N,

M⊗N

 n 

 xk ⊗ yk | xk ∈ M, yk ∈ N, n  1, 2, . . . ,

1.17

k1

is a ∗-subalgebra of operators on H⊗K. The von Neumann algebra generated by H⊗K in BH⊗K is denoted by M⊗N and it is called the tensor product of von Neumann algebras M and N. Since the map M  x −→ 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 xfz zn dmz,

∀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.

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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     τ x ⊗ y  τ1 xτ2 y ,

x ∈ M, y ∈ N.

2.1

Proof. Since τ1 and τ2 are normal faithful normalized traces, we can view M and N as von Neumann algebras acting on H  L2 M, τ1  and K  L2 N, τ2 , respectively, by left multiplication. Then, τ1 and τ2 are the vector states associated to the identities 1M of M and 1N of N, respectively. That is, τ1 x  x1M , 1M ,

 

 τ2 y  y1N , 1N ,

x ∈ M, y ∈ N.

2.2

Let τ be the vector state associated to 1M ⊗ 1N on M⊗N. Then, τ is uniquely determined by τx ⊗ y  τ1 xτ2 y for all x ∈ M, y ∈ N. 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 can define the noncommutative Lp -spaces Lp M⊗N, τ1 ⊗τ2  and called it the noncommutative Lp -tensor product of M, τ1  and N, 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 Bl2  be equipped with the usual trace Tr and let M, τ be a noncommutative probability space. Then, the element of Lp Bl2 ⊗M, Tr ⊗τ, the noncommutative Lp -tensor product of Bl2 , 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 For n ∈ N, let Mn be a von Neumann algebras. The infinite algebraic tensor product ⊗n≥1 Mn of Mn is the set of all finite linear combinations of elementary tensors ⊗n≥1 xn , where xn ∈ Mn and all but finitely many xn are 1, that is,  Mn  n≥1



 m   k1

 k xn

 |

k xn

∈ Mn and all but finitely many xn are 1, m ∈ N .

2.3

n≥1

First, let us consider infinite tensor products of noncommutative Lp -spaces associated with finite factors.

ISRN Algebra

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For n ∈ N, let Mn be a finite factor equipped with a unique normal faithful normalized trace τn . We have the product state τ on ⊗n≥1 Mn , defined by 

 τ



xn





n≥1

τn xn ,

xn ∈ Mn .

2.4

n≥1

The infinite von Neumann tensor product ⊗n≥1 Mn is the weak-closure of the image of the representation of ⊗n≥1 Mn by the left multiplication on the Hilbert space L2 ⊗n≥1 Mn . It is 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 and denoted by ⊗n≥1 τn see 7. Then, we can define the noncommutative Lp -spaces Lp ⊗n≥1 Mn , ⊗n≥1 τn  and called it the infinite noncommutative Lp -tensor product of Mn , τn . Next, let us consider the infinite tensor products of noncommutative Lp -Spaces associated with normal faithful finite von Neumann algebras. Theorem 2.3. Let Mm m∈N be a sequence of finite von Neumann algebras equipped with normal faithful normalized traces τm . Let A  ∪m≥1 M1 ⊗M2 ⊗ · · · ⊗Mm . Let H be the completion of A with respect to the inner product

m     τk yk∗ xk . x1 ⊗ · · · ⊗xm , y1 ⊗ · · · ⊗ym 

2.5

k1

Let π : A −→ BH be defined by πxΛa  Λxa,

x ∈ A, a ∈ A, where Λ : A −→ H is the inclusion.

2.6

Let N be the weak∗ -closure of πA in BH. Then, there exists a normal state ν on N such that m    ν x1 ⊗ · · · ⊗xm  τk xk ,

xk ∈ Mk , m ∈ N.

2.7

k1

Proof. Let Hm  L2 Mm  and consider Mm as a von Neumann algebra on Hm by left multiplication. Let Nm  M1 ⊗M2 ⊗ · · · ⊗Mm , νm  τ1 ⊗τ2 ⊗ · · · ⊗τm .

2.8

We view Nm as a von Neumann subalgebra of Nm1 via the inclusion: x1 ⊗ · · · ⊗ xm −→ x1 ⊗ · · · ⊗ xm ⊗ 1Mm1 .

2.9

Since τm1 1  1, νm1 |Nm  νm . Note that A is a unital ∗-algebra and the traces νm induce a faithful normal state νo on A. Since νo is faithful, the representation π is faithful. Therefore, A, and all Nm , can be viewed as subalgebras of BH. Let ν the restriction to N of the vector

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state given by Λ1. Then, ν is tracial and faithful. The trace ν|Mm  τm and ν is the unique normal state on N such that m    τk xk , ν x1 ⊗ · · · ⊗xm 

xk ∈ Mk , m ∈ N.

2.10

k1

1.

N, ν is called the infinite tensor products of noncommutative Lp -spaces of Mm , τm  see

Example 2.4. Let M2 C be the full algebra of 2×2 matrices. Murray and von Neumann proved that the infinite tensor product ⊗n≥1 M2 C

WOT

,

2.11

produced with respect to the unique normalized trace tr2 on M2 C, 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. For each k ∈ {1, 2}, let Mk be a σ-finite von Neumann algebra. Let Lp Mk  be the Haagerup noncommutative Lp -spaces. By the reduction theorem, there exist a Banach space Xp k a quasi Banach space if p < 1, a sequence Rk,m m∈N of finite von Neumann algebras, each equipped with a faithful normal finite trace τk,m , and for each m ∈ N an isometric embedding Jk,m : Lp Rk,m , τk,m  −→ Xp k such that 1 Jk,m1 Lp Rk,m1 , τk,m1  ⊂ Jk,m2 Lp Rk,m2 , τk,m2  for all m1 , m2 ∈ N such that m1 ≤ m2 ;  2 m∈N Jk,m Lp Rk,m , τk,m  is dense in Xp k ; 3 Lp Mk  is isometric to a subspace Yp k of Xp k ; 4 Yp k and all Jk Lp Rk,m , τk,m , m ∈ N are 1-complemented in Xp k for 1 ≤ p < ∞. Here, Lp Rk,m , τk,m  is the tracial noncommutative Lp -space associated with Rk,m , τk,m . Thus, we have a sequence Rk,m , τk,m  of finite von Neumann algebras. We then have the noncommutative Lp -tensor product Rm , τm  : R1,m ⊗R2,m , τ1,m ⊗τ2,m . Applying the construction in Section 2.2, we will be able to construct the infinite tensor products of noncommutative Lp -spaces of Rm , τm . Hence, we have the tensor products of noncommutative Lp -spaces of Lp M1  and Lp M2 . With this setting, if {Mk }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.

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Let M be an arbitrary von Neumann algebra. Then, M admits the following direct sum decomposition: M



  Nj ⊗B Kj ,

j∈J

2.12

where each Nj 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 < p < ∞. Let Lp M be the Haagerup noncommutative Lp -space associated with M. Then, there exist a Banach space Xp a quasi Banach space if p < 1, a family Ri i∈I of finite von Neumann algebras, each equipped with a normal faithful finite trace τi , and, for each i ∈ I, an isometric embedding Ji : Lp Ri , τi  −→ Xp such that 1 Ji Lp Ri , τi  ⊂ Jj Lp Rj , τj  for all i, j ∈ I such that i ≤ j;  2 i∈I Ji Lp Ri , τi  is dense in Xp ; 3 Lp M is isometric to a subspace Yp of Xp ; 4 Yp and all Ji Lp Ri , τi , i ∈ I are 1-complemented in Xp for 1 ≤ p < ∞. Here, Lp Ri , τi  is the tracial noncommutative Lp -space associated with Ri , τi . If we can define the notion of uncountable infinite tensor products of noncommutative 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. Str˘atil˘a and L. Zsido, ´ 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 Alg`ebres d’op´erateurs et leurs applications en physique math´ematique (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 applications,” Memoirs of the American Mathematical Society, vol. 331, pp. 691–695, 2000.

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