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Volume 2012 (2012), Article ID 197468, 9 pages
Tensor Products of Noncommutative -Spaces
Centre of Excellence in Mathematics, CHE, Si Ayutthaya RD, Bangkok 10400, Thailand
Received 27 January 2012; Accepted 1 March 2012
Academic Editor: F. Kittaneh
Copyright © 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 spaces associated with finite von Neumann algebras and define the notion of tensor product of Haagerup noncommutative spaces associated with -finite von Neumann algebras.
1. Introduction and Preliminaries
The main goal of this paper is explanation of the notion of tensor products of noncommutative -spaces associated with von Neumann algebras. The notion of tensor products of noncommutative probability spaces was considered by Xu in . We will generalized that notations to the cases of noncommutative -spaces associated with von Neumann algebras.
In this section, we also give some necessary preliminaries on noncommutative -spaces associated with von Neumann algebras and tensor product of von Neumann algebras.
1.1. Noncommutative -Spaces Associated with Semifinite von Neumann Algebras
We denote by an infinite-dimensional von Neumann algebra acting on a separable Hilbert space . Let us define a trace on , the set of all positive elements of .
Definition 1.1. Let be a von Neumann algebra.(i)A trace on is a function satisfying the following.(a) for any and any .(b) for any (tracial property).(ii)A trace is faithful if implies .(iii)A trace is normal if for any bounded increasing net in .(iv)A trace is semifinite if for any nonzero there exists a nonzero such that and .(v)A trace is finite if . In this case, we will often assume that it is normalized.
Recall that a von Neumann algebra 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 is semifinite von Neumann algebra if and only if there exists a faithful normal semifinite trace.
Proof. Let be a von Neumann algebra and a faithful normal semifinite trace. For any nonzero central projection , there exist such that . Then, there exists a nonzero projection and a positive number such that . Thus, is a finite projection. Hence, is semifinite.
Conversely, let be a semifinite von Neumann algebra. We can assume that is a uniform von Neumann algebra, that is, there exists a family of equivalent finite mutually orthogonal projections such that . For each , the von Neumann algebra is finite and it then possesses a finite normal trace . Define a mapping by where is a partial isometry such that . Then, is a semifinite normal traces on . Since the set of all semifinite normal traces on , obtained in this manner, is sufficient. Then, possesses a faithful normal semifinite trace.
Let be a von Neumann algebra equipped with a faithful normal semifinite trace . For , let The noncommutative -space associated with is defined as the Banach space completion of . We set equipped with the norm , the operator norm. Note that the usual (commutative) -space is also in the family of noncommutative -space (see, e.g., [1, 5]).
Elements of the noncommutative -space may be identified with unbounded operators.
Definition 1.3. Let be a von Neumann algebra equipped with a faithful normal semifinite trace .(i)A linear operator is called affiliated with if for all unitary in the commutant of .(ii)A closed densely defined operator , affiliated with , is called -measurable if for every there exists an orthogonal projection such that and .
For , we have Note that is a Hilbert space with respect to the scalar product .
If is a normal faithful finite trace, then it is normalized, that is, . In this case, is called a noncommutative probability space.
1.2. Noncommutative -Spaces Associated with Arbitrary von Neumann Algebras
In this subsection, we will recall the definitions of cross product (see ) and Haagerup noncommutative -spaces. For details of the following results in Haagerup noncommutative -spaces, we refer to [1, 5].
Let be a von Neumann algebra on a Hilbert space , the group of all -automorphism of , a locally compact group equipped with its left Haar measure and a homomorphism of group, such that for any , the mapping is continuous for the weak operator topology in . Let be the space of all norm continuous functions defined on and taking values in which have compact supports. We endow it with the inner product: and we denote by the Hilbert space obtained by completion.
For any , the operator is defined by the relations: whereas for any one defines the unitary operator by the relations The von Neumann algebra generated in by the operators and , is called the cross-product of by the action of and it is denoted by or simply by .
Remark 1.4. If is a von Neumann algebra on a separable Hilbert space and is a separable abelian locally compact group acting by -automorphisms of , then the group of the character of acts by -automorphisms of . M. Takesaki has proved that In particular, if is properly infinite, then .
Let denote the one parameter modular automorphism group of on associated with . The group is the only group of -automorphisms of , with respect to which satisfies the KMS-conditions. We consider the cross-product , that is, a von Neumann algebra acting on , generated by the operators , and the operators , defined by It is well known that cross product is semifinite (see ). By Theorem 10.29 of , there exists a strong operator continuous group of unitary operators in such that
Let be its (unique) faithful normal semifinite trace satisfying The -algebra of all -measurable operators on affiliated with is denoted by . For each , we define the Haagerup noncommutative -spaces by We have
For , if and only if , we then define For , is a Banach space equipped with a norm . For , is a quasi-Banach space equipped with a -norm .
1.3. Tensor Products of von Neumann Algebras
Let be the Hilbert space tensor product of and . For and , the tensor product is the bounded linear operator on uniquely determined by
Let be two von Neumann algebras. The algebraic tensor product of and , is a -subalgebra of operators on . The von Neumann algebra generated by in is denoted by and it is called the tensor product of von Neumann algebras and . Since the map is a -isomorphism, we can view as a von Neumann subalgebra of . Similarly, we can also view as a von Neumann subalgebra of . By the Tomita commutation theorem, and commute and together generate .
Example 1.5. Let be the unit circle equipped with the normalized Lebesque measure dm and a finite von Neumann algebra. Let be consisting of all functions such that Then, is a finite subdiagonal algebra of (see ).
2. Tensor Products of Noncommutative -Spaces Associated with von Neumann Algebras
We first consider the simple case: finite von Neumann algebras.
2.1. Tensor Products of Noncommutative -Spaces Associated with Normal Faithful Finite von Neumann Algebras
Theorem 2.1. Let and be finite von Neumann algebras equipped with normal faithful normalized traces and , respectively. Then, there exists a normal faithful trace on the tensor product von Neumann algebra such that
Proof. Since and are normal faithful normalized traces, we can view and as von Neumann algebras acting on and , respectively, by left multiplication. Then, and are the vector states associated to the identities of and of , respectively. That is, Let be the vector state associated to on . Then, is uniquely determined by for all . Therefore, is tracial and faithful.
is called the tensor product trace of and , and we denote it by . Then, we can define the noncommutative -spaces and called it the noncommutative -tensor product of and .
Example 2.2. Let us consider two cases (see [1, 5]).(1)Let be a probability space. We can represent as a von Neumann algebra on by multiplication and the integral against is a normal faithful normalized trace on . Let be a noncommutative probability space. Then, is isometric to , the usual -space of -integrable functions from to .(2)Let be equipped with the usual trace Tr and let be a noncommutative probability space. Then, the element of , the noncommutative -tensor product of and can be identified with an infinite matrix with entries in .
2.2. Infinite Tensor Products of Noncommutative -Spaces Associated with Finite von Neumann Algebras
For , let be a von Neumann algebras. The infinite algebraic tensor product of is the set of all finite linear combinations of elementary tensors , where and all but finitely many are 1, that is,
First, let us consider infinite tensor products of noncommutative -spaces associated with finite factors.
For , let be a finite factor equipped with a unique normal faithful normalized trace . We have the product state on , defined by The infinite von Neumann tensor product is the weak-closure of the image of the representation of by the left multiplication on the Hilbert space . It is a finite factor with the trace is the extension of , which is the unique normalized trace. is called the infinite tensor product trace of and denoted by (see ). Then, we can define the noncommutative -spaces and called it the infinite noncommutative -tensor product of .
Next, let us consider the infinite tensor products of noncommutative -Spaces associated with normal faithful finite von Neumann algebras.
Theorem 2.3. Let be a sequence of finite von Neumann algebras equipped with normal faithful normalized traces . Let . Let be the completion of with respect to the inner product Let be defined by Let be the weak-closure of in . Then, there exists a normal state on such that
Proof. Let and consider as a von Neumann algebra on by left multiplication. Let We view as a von Neumann subalgebra of via the inclusion: Since , . Note that is a unital -algebra and the traces induce a faithful normal state on . Since is faithful, the representation is faithful. Therefore, , and all , can be viewed as subalgebras of . Let the restriction to of the vector state given by . Then, is tracial and faithful. The trace and is the unique normal state on such that
is called the infinite tensor products of noncommutative -spaces of (see ).
Example 2.4. Let be the full algebra of matrices. Murray and von Neumann proved that the infinite tensor product produced with respect to the unique normalized trace on , is the unique AFD II-factor (see, e.g., ).
2.3. Tensor Products of Noncommutative -Spaces Associated with -Finite von Neumann Algebras
In the case of tensor products of -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 -spaces to the corresponding ones on those associated with finite von Neumann algebras (see, e.g., [6, 8]).
For each , let be a -finite von Neumann algebra. Let be the Haagerup noncommutative -spaces. By the reduction theorem, there exist a Banach space (a quasi Banach space if ), a sequence of finite von Neumann algebras, each equipped with a faithful normal finite trace , and for each an isometric embedding such that(1) for all such that ;(2) is dense in ;(3) is isometric to a subspace of ;(4) and all are 1-complemented in for .
Here, is the tracial noncommutative -space associated with .
Thus, we have a sequence of finite von Neumann algebras. We then have the noncommutative -tensor product . Applying the construction in Section 2.2, we will be able to construct the infinite tensor products of noncommutative -spaces of . Hence, we have the tensor products of noncommutative -spaces of and .
With this setting, if be a sequence of -finite von Neumann algebra, we will also be able to construct the infinite tensor product of noncommutative -spaces associated with -finite von Neumann algebras.
Let be an (arbitrary) von Neumann algebra. Then, admits the following direct sum decomposition: where each is an -finite von Neumann algebra. Using the reduction theorem in general case, the approximation theorem can be extended to the general case as follows.
Let be a general von Neumann algebra and . Let be the Haagerup noncommutative -space associated with . Then, there exist a Banach space (a quasi Banach space if ), a family of finite von Neumann algebras, each equipped with a normal faithful finite trace , and, for each , an isometric embedding such that(1) for all such that ;(2) is dense in ;(3) is isometric to a subspace of ;(4) and all are 1-complemented in for .
Here, is the tracial noncommutative -space associated with .
If we can define the notion of (uncountable) infinite tensor products of noncommutative -spaces associated with finite von Neumann algebras, we should be able to define tensor products of Haagerup noncommutative -spaces.
This research is supported by the Centre of Excellence in Mathematics, the commission on Higher Education, Thailand.
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