Tensor Products of Noncommutative 𝐿^𝑝-Spaces

Abstract

We consider the notion of tensor product of noncommutative 𝐿^𝑝 spaces associated with finite von Neumann algebras and define the notion of tensor product of Haagerup noncommutative 𝐿^𝑝 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 𝐿^𝑝-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 𝐿^𝑝-spaces associated with von Neumann algebras.

In this section, we also give some necessary preliminaries on noncommutative 𝐿^𝑝-spaces associated with von Neumann algebras and tensor product of von Neumann algebras.

1.1. Noncommutative 𝐿^𝑝-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).(ii)A trace 𝜏 is faithful if 𝜏(𝑥)=0 implies 𝑥=0.(iii)A trace 𝜏 is normal if sup_𝜄𝜏(𝑥_𝜄)=𝜏(sup_𝜄𝑥_𝜄) for any bounded increasing net (𝑥_𝜄) 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∈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 {𝑒_𝑖}_𝑖∈𝐼 of equivalent finite mutually orthogonal projections such that ∑_𝑖∈𝐼𝑒_𝑖=1. For each 𝑒_𝑖, the von Neumann algebra 𝑒_𝑖ℳ𝑒_𝑖 is finite and it then possesses a finite normal trace 𝜏_𝑖. Define a mapping by 𝜏(𝑥)=_𝑖∈𝐼𝜏_𝑖𝑣^∗𝑖𝑥𝑣_𝑖,𝑥∈ℳ⁺,(1.1) where 𝑣_𝑖∈ℳ is a partial isometry such that 𝑣^∗𝑖𝑣_𝑖=𝑒_𝑖=𝑣_𝑖𝑣^∗𝑖. 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‖𝑥‖_𝑝=𝜏(|𝑥|^𝑝)^1/𝑝,where|𝑥|=𝑥^∗𝑥^1/2.(1.2) The noncommutative 𝐿^𝑝-space 𝐿^𝑝(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) 𝐿^𝑝-space is also in the family of noncommutative 𝐿^𝑝-space (see, e.g., [1, 5]).

Elements of the noncommutative 𝐿^𝑝-space 𝐿^𝑝(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)→ℋ is called affiliated with ℳ if 𝑥𝑢=𝑢𝑥 for all unitary 𝑢 in the commutant ℳ′ of ℳ.(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⊆dom(𝑥) and 𝜏(1−𝑝)<𝜀.

For 0<𝑝<∞, we have𝐿^𝑝(ℳ,𝜏)≅{𝑥∣𝑥is𝜏-measurable,𝜏(|𝑥|^𝑝)<∞}.(1.3) 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 𝐿^𝑝-Spaces Associated with Arbitrary von Neumann Algebras

In this subsection, we will recall the definitions of cross product (see [2]) and Haagerup noncommutative 𝐿^𝑝-spaces. For details of the following results in Haagerup noncommutative 𝐿^𝑝-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 and𝐺∋𝑔⟼𝜋_𝑔∈Aut(ℳ)(1.4) a homomorphism of group, such that for any 𝑥∈ℳ, the mapping𝐺∋𝑔⟼𝜋_𝑔(𝑥)∈ℳ(1.5) is continuous for the weak operator topology in ℳ. Let 𝐶_𝑐(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⟩=_𝐺⟨𝑓₁(𝑔),𝑓₂(𝑔)⟩𝑑𝑔,(1.6) and we denote by 𝐿²(𝐺,ℋ) the Hilbert space obtained by completion.

For any 𝑥∈ℳ, the operator 𝜆_𝑥∈ℬ(𝐿²(𝐺,ℋ)) is defined by the relations:𝜆_𝑥(𝑓)(𝑔)=𝜋^𝑔−1(𝑥)(𝑓(𝑔)),𝑓∈𝐶_𝑐(𝐺,ℋ),𝑔∈𝐺,(1.7) whereas for any 𝑔∈𝐺 one defines the unitary operator 𝑢_𝑔∈ℬ(𝐿²(𝐺,ℋ)) by the relations𝑢_𝑔(𝑓)𝑔^=𝑓𝑔⁻¹𝑔^,𝑓∈𝐶_𝑐(𝐺,ℋ),𝑔′∈𝐺.(1.8) The von Neumann algebra generated in ℬ(𝐿²(𝐺,ℋ)) by the operators 𝜆_𝑥,𝑥∈ℳ and 𝑢_𝑔,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. 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 𝐿^𝑝-space associated with (ℳ,𝜑) constructed by Haagerup (see, e.g., [1, 5]).

Let 𝜎_𝑡=𝜎^𝜑𝑡,𝑡∈ℝ denote the one parameter modular automorphism group of ℝ on ℳ associated with 𝜑. The group {𝜎^𝜑𝑡} 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 𝜋_𝑥,𝑥∈ℳ, and the operators 𝜆_𝑠,𝑠∈ℝ, defined by𝜋_𝑥(𝑓(𝑡))=𝜎_−𝑡(𝑥)𝑓(𝑡),𝜆_𝑠(𝑓(𝑡))=𝑓(𝑡−𝑠)forany𝑓∈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 {𝑢_𝑡}_𝑡∈ℝ of unitary operators in ℳ such that𝜎^𝜑𝑡(𝑥)=𝑢_𝑡𝑥𝑢^∗𝑡,𝑡∈ℝ.(1.11)

Let 𝜏 be its (unique) faithful normal semifinite trace satisfying𝜏∘𝜎_𝑡=𝑒^−𝑡𝜏,∀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 𝐿^𝑝-spaces by𝐿^𝑝(ℳ,𝜑)=𝑥∈𝒩∣𝜎_𝑡(𝑥)=𝑒^−𝑡/𝑝𝑥,∀𝑡∈ℝ.(1.13) We have𝐿^∞(ℳ,𝜑)=ℳ,𝐿¹(ℳ,𝜑)=ℳ_∗.(1.14)

For 0<𝑝<∞, 𝑥∈𝐿^𝑝(ℳ,𝜑) if and only if |𝑥^𝑝|∈𝐿¹(ℳ,𝜑), we then define‖𝑥‖_𝑝=‖|𝑥|^𝑝‖^11/𝑝,𝑥∈𝐿^𝑝(ℳ,𝜑).(1.15) For 1≤𝑝<∞, 𝐿^𝑝(ℳ,𝜑) is a Banach space equipped with a norm ‖⋅‖_𝑝. For 0<𝑝<1, 𝐿^𝑝(ℳ,𝜑) is a quasi-Banach space equipped with a 𝑝-norm ‖⋅‖_𝑝.

It is well known that 𝐿^𝑝(ℳ,𝜑) is independent of 𝜑 up to isometric isomorphism preserving the order and modular structure of 𝐿^𝑝(ℳ,𝜑) (see [6–8]). Sometimes, we denote 𝐿^𝑝(ℳ,𝜑) simply by 𝐿^𝑝(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)=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⊗𝒩 of ℳ and 𝒩,ℳ⊗𝒩=^𝑛_𝑘=1𝑥_𝑘⊗𝑦_𝑘∣𝑥_𝑘∈ℳ,𝑦_𝑘∈𝒩,𝑛=1,2,…,(1.17) 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 mapℳ∋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(𝑥𝑓(𝑧))𝑧^𝑛dm(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 𝐿^𝑝-Spaces Associated with von Neumann Algebras

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

2.1. Tensor Products of Noncommutative 𝐿^𝑝-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(𝑥)𝜏₂(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 1_ℳ of ℳ and 1_𝒩 of 𝒩, respectively. That is, 𝜏₁(𝑥)=𝑥1_ℳ,1_ℳ,𝜏₂(𝑦)=𝑦1_𝒩,1_𝒩,𝑥∈ℳ,𝑦∈𝒩.(2.2) Let 𝜏 be the vector state associated to 1_ℳ⊗1_𝒩 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 𝐿^𝑝-spaces 𝐿^𝑝(ℳ⊗𝒩,𝜏₁⊗𝜏₂) and called it the noncommutative 𝐿^𝑝-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, 𝐿^𝑝(𝐿^∞(Ω)⊗ℳ,∫⊗𝜏) is isometric to 𝐿^𝑝(Ω,𝐿^𝑝(ℳ)), the usual 𝐿^𝑝-space of 𝑝-integrable functions from Ω to 𝐿^𝑝(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 𝐿^𝑝(ℬ(𝑙²)⊗ℳ,Tr⊗𝜏), the noncommutative 𝐿^𝑝-tensor product of (ℬ(𝑙²),Tr) and (ℳ,𝜏) can be identified with an infinite matrix with entries in 𝐿^𝑝(ℳ,𝜏).

2.2. Infinite Tensor Products of Noncommutative 𝐿^𝑝-Spaces Associated with Finite von Neumann Algebras

For 𝑛∈ℕ, let ℳ_𝑛 be a von Neumann algebras. The infinite algebraic tensor product ⊗_𝑛≥1ℳ_𝑛 of ℳ_𝑛 is the set of all finite linear combinations of elementary tensors ⊗_𝑛≥1𝑥_𝑛, where 𝑥_𝑛∈ℳ_𝑛 and all but finitely many 𝑥_𝑛 are 1, that is,_𝑛≥1ℳ_𝑛=^𝑚_𝑘=1_𝑛≥1𝑥^𝑛(𝑘)∣𝑥^𝑛(𝑘)∈ℳ_𝑛andallbutfinitelymany𝑥_𝑛are1,𝑚∈ℕ.(2.3)

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

For 𝑛∈ℕ, let ℳ_𝑛 be a finite factor equipped with a unique normal faithful normalized trace 𝜏_𝑛. We have the product state 𝜏 on ⊗_𝑛≥1ℳ_𝑛, defined by𝜏_𝑛≥1𝑥_𝑛=_𝑛≥1𝜏_𝑛𝑥_𝑛,𝑥_𝑛∈ℳ_𝑛.(2.4) The infinite von Neumann tensor product ⊗_𝑛≥1ℳ_𝑛 is the weak-closure of the image of the representation of ⊗_𝑛≥1ℳ_𝑛 by the left multiplication on the Hilbert space 𝐿²(⊗_𝑛≥1ℳ_𝑛). 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 𝜏_𝑛 and denoted by ⊗_𝑛≥1𝜏_𝑛 (see [7]). Then, we can define the noncommutative 𝐿^𝑝-spaces 𝐿^𝑝(⊗_𝑛≥1ℳ_𝑛,⊗_𝑛≥1𝜏_𝑛) and called it the infinite noncommutative 𝐿^𝑝-tensor product of (ℳ_𝑛,𝜏_𝑛).

Next, let us consider the infinite tensor products of noncommutative 𝐿^𝑝-Spaces associated with normal faithful finite von Neumann algebras.

Theorem 2.3. Let (ℳ_𝑚)_𝑚∈ℕ be a sequence of finite von Neumann algebras equipped with normal faithful normalized traces 𝜏_𝑚. Let 𝒜=∪_𝑚≥1(ℳ₁⊗ℳ₂⊗⋯⊗ℳ_𝑚). Let ℋ be the completion of 𝒜 with respect to the inner product 𝑥¹⊗⋯⊗𝑥_𝑚,𝑦₁⊗⋯⊗𝑦_𝑚=^𝑚_𝑘=1𝜏_𝑘𝑦^∗𝑘𝑥_𝑘.(2.5) Let 𝜋∶𝒜→ℬ(ℋ) be defined by 𝜋(𝑥)Λ(𝑎)=Λ(𝑥𝑎),𝑥∈𝒜,𝑎∈𝒜,whereΛ∶𝐴⟶ℋ𝑖𝑠𝑡ℎ𝑒𝑖𝑛𝑐𝑙𝑢𝑠𝑖𝑜𝑛.(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 𝜈𝑥₁⊗⋯⊗𝑥_𝑚=^𝑚_𝑘=1𝜏_𝑘𝑥_𝑘,𝑥_𝑘∈ℳ_𝑘,𝑚∈ℕ.(2.7)

Proof. Let ℋ_𝑚=𝐿²(ℳ_𝑚) and consider ℳ_𝑚 as a von Neumann algebra on ℋ_𝑚 by left multiplication. Let 𝒩𝜈_𝑚𝑚=ℳ₁⊗ℳ₂⊗⋯⊗ℳ_𝑚,=𝜏₁⊗𝜏₂⊗⋯⊗𝜏_𝑚.(2.8) We view 𝒩_𝑚 as a von Neumann subalgebra of 𝒩_𝑚+1 via the inclusion: 𝑥₁⊗⋯⊗𝑥_𝑚⟼𝑥₁⊗⋯⊗𝑥_𝑚⊗1_ℳ𝑚+1.(2.9) Since 𝜏_𝑚+1(1)=1, 𝜈_𝑚+1|_𝒩𝑚=𝜈_𝑚. Note that 𝒜 is a unital ∗-algebra and the traces 𝜈_𝑚 induce a faithful normal state 𝜈_𝑜 on 𝒜. Since 𝜈_𝑜 is faithful, the representation 𝜋 is faithful. Therefore, 𝒜, and all 𝒩_𝑚, 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=𝜏_𝑚 and 𝜈 is the unique normal state on 𝒩 such that 𝜈𝑥₁⊗⋯⊗𝑥_𝑚=^𝑚_𝑘=1𝜏_𝑘𝑥_𝑘,𝑥_𝑘∈ℳ_𝑘,𝑚∈ℕ.(2.10)

(𝒩,𝜈) is called the infinite tensor products of noncommutative 𝐿^𝑝-spaces of (ℳ_𝑚,𝜏_𝑚) (see [1]).

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≥1𝑀₂(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 𝐿^𝑝-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 𝐿^𝑝-spaces to the corresponding ones on those associated with finite von Neumann algebras (see, e.g., [6, 8]).

For each 𝑘∈{1,2}, let ℳ_𝑘 be a 𝜎-finite von Neumann algebra. Let 𝐿^𝑝(ℳ_𝑘) be the Haagerup noncommutative 𝐿^𝑝-spaces. By the reduction theorem, there exist a Banach space (𝑋_𝑝)_𝑘 (a quasi Banach space if 𝑝<1), a sequence (ℛ_𝑘,𝑚)_𝑚∈ℕ of finite von Neumann algebras, each equipped with a faithful normal finite trace 𝜏_𝑘,𝑚, and for each 𝑚∈ℕ an isometric embedding 𝐽_𝑘,𝑚∶𝐿^𝑝(ℛ_𝑘,𝑚,𝜏_𝑘,𝑚)→(𝑋_𝑝)_𝑘 such that(1)𝐽_𝑘,𝑚1(𝐿^𝑝(ℛ_𝑘,𝑚1,𝜏_𝑘,𝑚1))⊂𝐽_𝑘,𝑚2(𝐿^𝑝(ℛ_𝑘,𝑚2,𝜏_𝑘,𝑚2)) for all 𝑚₁,𝑚₂∈ℕ such that 𝑚₁≤𝑚₂;(2)⋃_𝑚∈ℕ𝐽_𝑘,𝑚(𝐿^𝑝(ℛ_𝑘,𝑚,𝜏_𝑘,𝑚)) is dense in (𝑋_𝑝)_𝑘;(3)𝐿^𝑝(ℳ_𝑘) is isometric to a subspace (𝑌_𝑝)_𝑘 of (𝑋_𝑝)_𝑘;(4)(𝑌_𝑝)_𝑘 and all 𝐽_𝑘(𝐿^𝑝(ℛ_𝑘,𝑚,𝜏_𝑘,𝑚)),𝑚∈ℕ are 1-complemented in (𝑋_𝑝)_𝑘 for 1≤𝑝<∞.

Here, 𝐿^𝑝(ℛ_𝑘,𝑚,𝜏_𝑘,𝑚) is the tracial noncommutative 𝐿^𝑝-space associated with (ℛ_𝑘,𝑚,𝜏_𝑘,𝑚).

Thus, we have a sequence (ℛ_𝑘,𝑚,𝜏_𝑘,𝑚) of finite von Neumann algebras. We then have the noncommutative 𝐿^𝑝-tensor product (ℛ_𝑚,𝜏_𝑚)∶=(ℛ_1,𝑚⊗ℛ_2,𝑚,𝜏_1,𝑚⊗𝜏_2,𝑚). Applying the construction in Section 2.2, we will be able to construct the infinite tensor products of noncommutative 𝐿^𝑝-spaces of (ℛ_𝑚,𝜏_𝑚). Hence, we have the tensor products of noncommutative 𝐿^𝑝-spaces of 𝐿^𝑝(ℳ₁) and 𝐿^𝑝(ℳ₂).

With this setting, if {ℳ_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 𝐿^𝑝-spaces associated with 𝜎-finite von Neumann algebras.

Let ℳ be an (arbitrary) von Neumann algebra. Then, ℳ admits the following direct sum decomposition:ℳ=_𝑗∈𝐽𝒩_𝑗⊗𝐵𝐾_𝑗,(2.12) where each 𝒩_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<𝑝<∞. Let 𝐿^𝑝(ℳ) be the Haagerup noncommutative 𝐿^𝑝-space associated with ℳ. Then, there exist a Banach space 𝑋_𝑝 (a quasi Banach space if 𝑝<1), a family (ℛ_𝑖)_𝑖∈𝐼 of finite von Neumann algebras, each equipped with a normal faithful finite trace 𝜏_𝑖, and, for each 𝑖∈𝐼, an isometric embedding 𝐽_𝑖∶𝐿^𝑝(ℛ_𝑖,𝜏_𝑖)→𝑋_𝑝 such that(1)𝐽_𝑖(𝐿^𝑝(ℛ_𝑖,𝜏_𝑖))⊂𝐽_𝑗(𝐿^𝑝(ℛ_𝑗,𝜏_𝑗)) for all 𝑖,𝑗∈𝐼 such that 𝑖≤𝑗;(2)⋃_𝑖∈𝐼𝐽_𝑖(𝐿^𝑝(ℛ_𝑖,𝜏_𝑖)) is dense in 𝑋_𝑝;(3)𝐿^𝑝(ℳ) is isometric to a subspace 𝑌_𝑝 of 𝑋_𝑝;(4)𝑌_𝑝 and all 𝐽_𝑖(𝐿^𝑝(ℛ_𝑖,𝜏_𝑖)),𝑖∈𝐼 are 1-complemented in 𝑋_𝑝 for 1≤𝑝<∞.

Here, 𝐿^𝑝(ℛ_𝑖,𝜏_𝑖) is the tracial noncommutative 𝐿^𝑝-space associated with (ℛ_𝑖,𝜏_𝑖).

If we can define the notion of (uncountable) infinite tensor products of noncommutative 𝐿^𝑝-spaces associated with finite von Neumann algebras, we should be able to define tensor products of Haagerup noncommutative 𝐿^𝑝-spaces.

Acknowledgment

This research is supported by the Centre of Excellence in Mathematics, the commission on Higher Education, Thailand.