# Tensor Products of NoncommutativeLp-Spaces

Authors: Somlak Utudee
Journal: ISRN Algebra
 (2012)

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## Abstract

We consider the notion of tensor product of noncommutativeLp 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.


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## Body

## 1. Introduction and Preliminaries

The main goal of this paper is explanation of the notion of tensor products of noncommutativeLp-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 noncommutativeLp-spaces associated with von Neumann algebras and tensor product of von Neumann algebras.


### 1.1. NoncommutativeLp-Spaces Associated with Semifinite von Neumann Algebras

We denote byℳ 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.
Letℳ be a von Neumann algebra.(i)
A trace onℳ+ 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).(ii)
A traceτ is faithful if τ(x)=0 implies x=0.(iii)
A traceτ is normal if supιτ(xι)=τ(supιxι) for any bounded increasing net (xι) in ℳ+.(iv)
A traceτ is semifinite if for any nonzero x∈ℳ+ there exists a nonzero y∈ℳ+ such that y≤x and τ(y)<∞.(v)
A traceτ is finite if τ(1)<∞. In this case, we will often assume that it is normalized.Recall that a von Neumann algebraℳ 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 algebraℳ is semifinite von Neumann algebra if and only if there exists a faithful normal semifinite trace.Proof.
Letℳ 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∈ℳ and a positive number ɛ such that xe=ex≥ɛe. Thus, e is a finite projection. Hence, ℳ is semifinite.
Conversely, letℳ 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∈Iei=1. For each ei, the von Neumann algebra eiℳei is finite and it then possesses a finite normal trace τi. Define a mapping by
(1.1)τ(x)=∑i∈Iτi(vi*xvi),x∈M+,
where vi∈ℳ 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.Letℳ be a von Neumann algebra equipped with a faithful normal semifinite trace τ. For 0<p<∞, let(1.2)‖x‖p=[τ(|x|p)]1/p,where|x|=(x*x)1/2.
The noncommutativeLp-spaceLp(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 noncommutativeLp-space Lp(ℳ,τ) may be identified with unbounded operators.Definition 1.3.
Letℳ be a von Neumann algebra equipped with a faithful normal semifinite trace τ.(i)
A linear operatorx:dom(x)→ℋ is called affiliated with ℳ if xu=ux for all unitary u in the commutant ℳ′ of ℳ.(ii)
A closed densely defined operatorx, affiliated with ℳ, is called τ-measurable if for every ɛ>0 there exists an orthogonal projection p∈ℳ such that pℋ⊆dom(x) and τ(1-p)<ɛ.For0<p<∞, we have(1.3)Lp(M,τ)≅{x∣xisτ-measurable,τ(|x|p)<∞}.
Note that L2(ℳ,τ) is a Hilbert space with respect to the scalar product 〈x,y〉=τ(y*x).Ifτ 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. NoncommutativeLp-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].Letℳ 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(1.4)G∋g⟼πg∈Aut(M)
a homomorphism of group, such that for any x∈ℳ, the mapping(1.5)G∋g⟼πg(x)∈M
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:(1.6)〈f1,f2〉=∫G〈f1(g),f2(g)〉dg,
and we denote by L2(G,ℋ) the Hilbert space obtained by completion.For anyx∈ℳ, the operator λx∈ℬ(L2(G,ℋ)) is defined by the relations:(1.7)(λx(f))(g)=πg-1(x)(f(g)),f∈Cc(G,H),g∈G,
whereas for any g∈G one defines the unitary operator ug∈ℬ(L2(G,ℋ)) by the relations(1.8)(ug(f))(g′)=f(g-1g′),f∈Cc(G,H),g′∈G.
The von Neumann algebra generated in ℬ(L2(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 ℳ⋊πG or simply by ℳ⋊G.Remark 1.4.
Ifℳ 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 Ĝ of the character of G acts by *-automorphisms of ℳ⋊G. M. Takesaki has proved that
(1.9)(M⋊G)⋊Ĝ≅M⊗¯B(L2(G,H)).
In particular, if ℳ is properly infinite, then (M⋊G)⋊Ĝ≅ℳ.Letℳ 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]).Letσt=σtφ,t∈ℝ denote the one parameter modular automorphism group of ℝ on ℳ associated with φ. The group {σtφ} is the only group of *-automorphisms of ℳ, with respect to φ which satisfies the KMS-conditions. We consider the cross-product 𝒩=ℳ×σℝ, 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(1.10)πx(f(t))=σ-t(x)f(t),λs(f(t))=f(t-s)foranyf∈L2(R,H),t∈R.
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(1.11)σ̂tφ(x)=utxut*,t∈R.Letτ be its (unique) faithful normal semifinite trace satisfying(1.12)τ∘σ̂t=e-tτ,∀t∈R,
The *-algebra of all τ-measurable operators on L2(ℝ,ℋ) affiliated with 𝒩 is denoted by 𝒩̃. For each 0<p≤∞, we define the Haagerup noncommutativeLp-spaces by(1.13)Lp(M,φ)={x∈Ñ∣σ̂t(x)=e-t/px,∀t∈R}.
We have(1.14)L∞(M,φ)=M,L1(M,φ)=M*.For0<p<∞, x∈Lp(ℳ,φ) if and only if |xp|∈L1(ℳ,φ), we then define(1.15)‖x‖p=‖|x|p‖11/p,x∈Lp(M,φ).
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.It is well known thatLp(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

Letℋ⊗¯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(1.16)(x⊗¯y)(ξ⊗η)=x(ξ)⊗y(η)∀ξ∈H,η∈K.Letℳ⊂ℬ(ℋ),𝒩⊂ℬ(𝒦) be two von Neumann algebras. The algebraic tensor productℳ⊗𝒩 of ℳ and 𝒩,(1.17)M⊗N={∑k=1nxk⊗yk∣xk∈M,yk∈N,n=1,2,…},
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 algebrasℳ and 𝒩. Since the map(1.18)M∋x⟼x⊗¯1∈M⊗¯N
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.
Let𝕋 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
(1.19)∫τ(xf(z))z¯ndm(z),∀x∈L1(M,τ),n∈Z,n>0.
Then, H∞(𝕋,ℳ) is a finite subdiagonal algebra of (L∞(𝕋),dm)⊗¯(ℳ,τ) (see [5]).


## 1.1. NoncommutativeLp-Spaces Associated with Semifinite von Neumann Algebras

We denote byℳ 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.
Letℳ be a von Neumann algebra.(i)
A trace onℳ+ 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).(ii)
A traceτ is faithful if τ(x)=0 implies x=0.(iii)
A traceτ is normal if supιτ(xι)=τ(supιxι) for any bounded increasing net (xι) in ℳ+.(iv)
A traceτ is semifinite if for any nonzero x∈ℳ+ there exists a nonzero y∈ℳ+ such that y≤x and τ(y)<∞.(v)
A traceτ is finite if τ(1)<∞. In this case, we will often assume that it is normalized.Recall that a von Neumann algebraℳ 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 algebraℳ is semifinite von Neumann algebra if and only if there exists a faithful normal semifinite trace.Proof.
Letℳ 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∈ℳ and a positive number ɛ such that xe=ex≥ɛe. Thus, e is a finite projection. Hence, ℳ is semifinite.
Conversely, letℳ 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∈Iei=1. For each ei, the von Neumann algebra eiℳei is finite and it then possesses a finite normal trace τi. Define a mapping by
(1.1)τ(x)=∑i∈Iτi(vi*xvi),x∈M+,
where vi∈ℳ 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.Letℳ be a von Neumann algebra equipped with a faithful normal semifinite trace τ. For 0<p<∞, let(1.2)‖x‖p=[τ(|x|p)]1/p,where|x|=(x*x)1/2.
The noncommutativeLp-spaceLp(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 noncommutativeLp-space Lp(ℳ,τ) may be identified with unbounded operators.Definition 1.3.
Letℳ be a von Neumann algebra equipped with a faithful normal semifinite trace τ.(i)
A linear operatorx:dom(x)→ℋ is called affiliated with ℳ if xu=ux for all unitary u in the commutant ℳ′ of ℳ.(ii)
A closed densely defined operatorx, affiliated with ℳ, is called τ-measurable if for every ɛ>0 there exists an orthogonal projection p∈ℳ such that pℋ⊆dom(x) and τ(1-p)<ɛ.For0<p<∞, we have(1.3)Lp(M,τ)≅{x∣xisτ-measurable,τ(|x|p)<∞}.
Note that L2(ℳ,τ) is a Hilbert space with respect to the scalar product 〈x,y〉=τ(y*x).Ifτ 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. NoncommutativeLp-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].Letℳ 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(1.4)G∋g⟼πg∈Aut(M)
a homomorphism of group, such that for any x∈ℳ, the mapping(1.5)G∋g⟼πg(x)∈M
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:(1.6)〈f1,f2〉=∫G〈f1(g),f2(g)〉dg,
and we denote by L2(G,ℋ) the Hilbert space obtained by completion.For anyx∈ℳ, the operator λx∈ℬ(L2(G,ℋ)) is defined by the relations:(1.7)(λx(f))(g)=πg-1(x)(f(g)),f∈Cc(G,H),g∈G,
whereas for any g∈G one defines the unitary operator ug∈ℬ(L2(G,ℋ)) by the relations(1.8)(ug(f))(g′)=f(g-1g′),f∈Cc(G,H),g′∈G.
The von Neumann algebra generated in ℬ(L2(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 ℳ⋊πG or simply by ℳ⋊G.Remark 1.4.
Ifℳ 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 Ĝ of the character of G acts by *-automorphisms of ℳ⋊G. M. Takesaki has proved that
(1.9)(M⋊G)⋊Ĝ≅M⊗¯B(L2(G,H)).
In particular, if ℳ is properly infinite, then (M⋊G)⋊Ĝ≅ℳ.Letℳ 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]).Letσt=σtφ,t∈ℝ denote the one parameter modular automorphism group of ℝ on ℳ associated with φ. The group {σtφ} is the only group of *-automorphisms of ℳ, with respect to φ which satisfies the KMS-conditions. We consider the cross-product 𝒩=ℳ×σℝ, 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(1.10)πx(f(t))=σ-t(x)f(t),λs(f(t))=f(t-s)foranyf∈L2(R,H),t∈R.
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(1.11)σ̂tφ(x)=utxut*,t∈R.Letτ be its (unique) faithful normal semifinite trace satisfying(1.12)τ∘σ̂t=e-tτ,∀t∈R,
The *-algebra of all τ-measurable operators on L2(ℝ,ℋ) affiliated with 𝒩 is denoted by 𝒩̃. For each 0<p≤∞, we define the Haagerup noncommutativeLp-spaces by(1.13)Lp(M,φ)={x∈Ñ∣σ̂t(x)=e-t/px,∀t∈R}.
We have(1.14)L∞(M,φ)=M,L1(M,φ)=M*.For0<p<∞, x∈Lp(ℳ,φ) if and only if |xp|∈L1(ℳ,φ), we then define(1.15)‖x‖p=‖|x|p‖11/p,x∈Lp(M,φ).
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.It is well known thatLp(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

Letℋ⊗¯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(1.16)(x⊗¯y)(ξ⊗η)=x(ξ)⊗y(η)∀ξ∈H,η∈K.Letℳ⊂ℬ(ℋ),𝒩⊂ℬ(𝒦) be two von Neumann algebras. The algebraic tensor productℳ⊗𝒩 of ℳ and 𝒩,(1.17)M⊗N={∑k=1nxk⊗yk∣xk∈M,yk∈N,n=1,2,…},
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 algebrasℳ and 𝒩. Since the map(1.18)M∋x⟼x⊗¯1∈M⊗¯N
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.
Let𝕋 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
(1.19)∫τ(xf(z))z¯ndm(z),∀x∈L1(M,τ),n∈Z,n>0.
Then, H∞(𝕋,ℳ) is a finite subdiagonal algebra of (L∞(𝕋),dm)⊗¯(ℳ,τ) (see [5]).


## 2. Tensor Products of NoncommutativeLp-Spaces Associated with von Neumann Algebras

We first consider the simple case: finite von Neumann algebras.


### 2.1. Tensor Products of NoncommutativeLp-Spaces Associated with Normal Faithful Finite von Neumann Algebras

Theorem 2.1.
Letℳ 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
(2.1)τ(x⊗y)=τ1(x)τ2(y),x∈M,y∈N.Proof.
Sinceτ1 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,
(2.2)τ1(x)=〈x(1M),1M〉,τ2(y)=〈y(1N),1N〉,x∈M,y∈N.
Let τ be the vector state associated to 1ℳ⊗1𝒩 on ℳ⊗¯𝒩. Then, τ is uniquely determined by τ(x⊗y)=τ1(x)τ2(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 noncommutativeLp-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)
Letℬ(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 NoncommutativeLp-Spaces Associated with Finite von Neumann Algebras

Forn∈ℕ, let ℳn be a von Neumann algebras. The infinite algebraic tensor product⊗n≥1ℳn of ℳn is the set of all finite linear combinations of elementary tensors ⊗n≥1xn, where xn∈ℳn and all but finitely many xn are 1, that is,(2.3)⨂n≥1Mn={∑k=1m(⨂n≥1xn(k))∣xn(k)∈Mnandallbutfinitelymanyxnare1,m∈N}.First, let us consider infinite tensor products of noncommutativeLp-spaces associated with finite factors.Forn∈ℕ, let ℳn be a finite factor equipped with a unique normal faithful normalized trace τn. We have the product state τ on ⊗n≥1ℳn, defined by(2.4)τ(⨂n≥1xn)=∏n≥1τn(xn),xn∈Mn.
The infinite von Neumann tensor product⊗¯n≥1ℳn is the weak-closure of the image of the representation of ⊗n≥1ℳn by the left multiplication on the Hilbert space L2(⊗n≥1ℳn). 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≥1τn (see [7]). Then, we can define the noncommutative Lp-spaces Lp(⊗¯n≥1ℳn,⊗¯n≥1τn) and called it the infinite noncommutativeLp-tensor product of (ℳn,τn).Next, let us consider the infinite tensor products of noncommutativeLp-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
(2.5)〈x1⊗¯⋯⊗¯xm,y1⊗¯⋯⊗¯ym〉=∏k=1mτk(yk*xk).
Let π:𝒜→ℬ(ℋ) be defined by
(2.6)π(x)Λ(a)=Λ(xa),x∈A,a∈A,whereΛ:A⟶Histheinclusion.
Let 𝒩 be the weak*-closure of π(A) in ℬ(ℋ). Then, there exists a normal state ν on 𝒩 such that
(2.7)ν(x1⊗¯⋯⊗¯xm)=∏k=1mτk(xk),xk∈Mk,m∈N.Proof.
Letℋm=L2(ℳm) and consider ℳm as a von Neumann algebra on ℋm by left multiplication. Let
(2.8)Nm=M1⊗¯M2⊗¯⋯⊗¯Mm,νm=τ1⊗¯τ2⊗¯⋯⊗¯τm.
We view 𝒩m as a von Neumann subalgebra of 𝒩m+1 via the inclusion:
(2.9)x1⊗⋯⊗xm⟼x1⊗⋯⊗xm⊗1Mm+1.
Since τm+1(1)=1, νm+1|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 ℬ(ℋ). Let ν the restriction to 𝒩 of the vector state given by Λ(1). Then, ν is tracial and faithful. The trace ν|ℳm=τm and ν is the unique normal state on 𝒩 such that
(2.10)ν(x1⊗¯⋯⊗¯xm)=∏k=1mτk(xk),xk∈Mk,m∈N.(𝒩,ν) is called the infinite tensor products of noncommutativeLp-spaces of (ℳm,τm) (see [1]).Example 2.4.
LetM2(ℂ) be the full algebra of 2×2 matrices. Murray and von Neumann proved that the infinite tensor product
(2.11)(⊗n≥1M2(C))¯WOT,
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 NoncommutativeLp-Spaces Associated with σ-Finite von Neumann Algebras

In the case of tensor products ofσ-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 eachk∈{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∈ℕJk,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.Letℳ be an (arbitrary) von Neumann algebra. Then, ℳ admits the following direct sum decomposition:(2.12)M=⨂j∈JNj⊗¯B(Kj),
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.Letℳ 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∈IJi(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 noncommutativeLp-spaces associated with finite von Neumann algebras, we should be able to define tensor products of Haagerup noncommutative Lp-spaces.


## 2.1. Tensor Products of NoncommutativeLp-Spaces Associated with Normal Faithful Finite von Neumann Algebras

Theorem 2.1.
Letℳ 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
(2.1)τ(x⊗y)=τ1(x)τ2(y),x∈M,y∈N.Proof.
Sinceτ1 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,
(2.2)τ1(x)=〈x(1M),1M〉,τ2(y)=〈y(1N),1N〉,x∈M,y∈N.
Let τ be the vector state associated to 1ℳ⊗1𝒩 on ℳ⊗¯𝒩. Then, τ is uniquely determined by τ(x⊗y)=τ1(x)τ2(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 noncommutativeLp-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)
Letℬ(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 NoncommutativeLp-Spaces Associated with Finite von Neumann Algebras

Forn∈ℕ, let ℳn be a von Neumann algebras. The infinite algebraic tensor product⊗n≥1ℳn of ℳn is the set of all finite linear combinations of elementary tensors ⊗n≥1xn, where xn∈ℳn and all but finitely many xn are 1, that is,(2.3)⨂n≥1Mn={∑k=1m(⨂n≥1xn(k))∣xn(k)∈Mnandallbutfinitelymanyxnare1,m∈N}.First, let us consider infinite tensor products of noncommutativeLp-spaces associated with finite factors.Forn∈ℕ, let ℳn be a finite factor equipped with a unique normal faithful normalized trace τn. We have the product state τ on ⊗n≥1ℳn, defined by(2.4)τ(⨂n≥1xn)=∏n≥1τn(xn),xn∈Mn.
The infinite von Neumann tensor product⊗¯n≥1ℳn is the weak-closure of the image of the representation of ⊗n≥1ℳn by the left multiplication on the Hilbert space L2(⊗n≥1ℳn). 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≥1τn (see [7]). Then, we can define the noncommutative Lp-spaces Lp(⊗¯n≥1ℳn,⊗¯n≥1τn) and called it the infinite noncommutativeLp-tensor product of (ℳn,τn).Next, let us consider the infinite tensor products of noncommutativeLp-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
(2.5)〈x1⊗¯⋯⊗¯xm,y1⊗¯⋯⊗¯ym〉=∏k=1mτk(yk*xk).
Let π:𝒜→ℬ(ℋ) be defined by
(2.6)π(x)Λ(a)=Λ(xa),x∈A,a∈A,whereΛ:A⟶Histheinclusion.
Let 𝒩 be the weak*-closure of π(A) in ℬ(ℋ). Then, there exists a normal state ν on 𝒩 such that
(2.7)ν(x1⊗¯⋯⊗¯xm)=∏k=1mτk(xk),xk∈Mk,m∈N.Proof.
Letℋm=L2(ℳm) and consider ℳm as a von Neumann algebra on ℋm by left multiplication. Let
(2.8)Nm=M1⊗¯M2⊗¯⋯⊗¯Mm,νm=τ1⊗¯τ2⊗¯⋯⊗¯τm.
We view 𝒩m as a von Neumann subalgebra of 𝒩m+1 via the inclusion:
(2.9)x1⊗⋯⊗xm⟼x1⊗⋯⊗xm⊗1Mm+1.
Since τm+1(1)=1, νm+1|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 ℬ(ℋ). Let ν the restriction to 𝒩 of the vector state given by Λ(1). Then, ν is tracial and faithful. The trace ν|ℳm=τm and ν is the unique normal state on 𝒩 such that
(2.10)ν(x1⊗¯⋯⊗¯xm)=∏k=1mτk(xk),xk∈Mk,m∈N.(𝒩,ν) is called the infinite tensor products of noncommutativeLp-spaces of (ℳm,τm) (see [1]).Example 2.4.
LetM2(ℂ) be the full algebra of 2×2 matrices. Murray and von Neumann proved that the infinite tensor product
(2.11)(⊗n≥1M2(C))¯WOT,
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 NoncommutativeLp-Spaces Associated with σ-Finite von Neumann Algebras

In the case of tensor products ofσ-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 eachk∈{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∈ℕJk,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.Letℳ be an (arbitrary) von Neumann algebra. Then, ℳ admits the following direct sum decomposition:(2.12)M=⨂j∈JNj⊗¯B(Kj),
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.Letℳ 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∈IJi(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 noncommutativeLp-spaces associated with finite von Neumann algebras, we should be able to define tensor products of Haagerup noncommutative Lp-spaces.



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*Source: 197468-2012-05-14.xml*