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ISRN Algebra
VolumeΒ 2012Β (2012), Article IDΒ 197468, 9 pages
http://dx.doi.org/10.5402/2012/197468
Research Article

Tensor Products of Noncommutative 𝐿𝑝-Spaces

Centre of Excellence in Mathematics, CHE, Si Ayutthaya RD, Bangkok 10400, Thailand

Received 27 January 2012; Accepted 1 March 2012

Academic Editor: F.Β Kittaneh

Copyright Β© 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.

Abstract

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 β„³ an infinite-dimensional von Neumann algebra acting on a separable Hilbert space β„‹. Let us define a trace on β„³+, the set of all positive elements of β„³.

Definition 1.1. Let β„³ be a von Neumann algebra.(i)A trace on β„³+ is a function πœβˆΆβ„³+β†’[0,∞] satisfying the following.(a)𝜏(π‘₯+πœ†π‘¦)=𝜏(π‘₯)+πœ†πœ(𝑦) for any π‘₯,π‘¦βˆˆβ„³+ and any πœ†βˆˆβ„+.(b)𝜏(π‘₯π‘₯βˆ—)=𝜏(π‘₯βˆ—π‘₯) for any π‘₯βˆˆβ„³+ (tracial property).(ii)A trace 𝜏 is faithful if 𝜏(π‘₯)=0 implies π‘₯=0.(iii)A trace 𝜏 is normal if supπœ„πœ(π‘₯πœ„)=𝜏(supπœ„π‘₯πœ„) for any bounded increasing net (π‘₯πœ„) in β„³+.(iv)A trace 𝜏 is semifinite if for any nonzero π‘₯βˆˆβ„³+ there exists a nonzero π‘¦βˆˆβ„³+ such that 𝑦≀π‘₯ 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 β„³ 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 β„³ is semifinite von Neumann algebra if and only if there exists a faithful normal semifinite trace.

Proof. Let β„³ be a von Neumann algebra and 𝜏 a faithful normal semifinite trace. For any nonzero central projection π‘βˆˆβ„³, there exist π‘₯βˆˆβ„³+,0β‰ π‘₯≀𝑝 such that 𝜏(π‘₯)<∞. Then, there exists a nonzero projection π‘’βˆˆβ„³ and a positive number πœ€ such that π‘₯𝑒=𝑒π‘₯β‰₯πœ€π‘’. Thus, 𝑒 is a finite projection. Hence, β„³ is semifinite.
Conversely, let β„³ be a semifinite von Neumann algebra. We can assume that β„³ is a uniform von Neumann algebra, that is, there exists a family {𝑒𝑖}π‘–βˆˆπΌ of equivalent finite mutually orthogonal projections such that βˆ‘π‘–βˆˆπΌπ‘’π‘–=1. For each 𝑒𝑖, the von Neumann algebra 𝑒𝑖ℳ𝑒𝑖 is finite and it then possesses a finite normal trace πœπ‘–. Define a mapping by ξ“πœ(π‘₯)=π‘–βˆˆπΌπœπ‘–ξ€·π‘£βˆ—π‘–π‘₯𝑣𝑖,π‘₯βˆˆβ„³+,(1.1) where π‘£π‘–βˆˆβ„³ is a partial isometry such that π‘£βˆ—π‘–π‘£π‘–=𝑒𝑖=π‘£π‘–π‘£βˆ—π‘–. Then, 𝜏 is a semifinite normal traces on β„³+. Since the set of all semifinite normal traces on β„³+, obtained in this manner, is sufficient. Then, β„³ possesses a faithful normal semifinite trace.

Let β„³ be a von Neumann algebra equipped with a faithful normal semifinite trace 𝜏. For 0<𝑝<∞, letβ€–π‘₯‖𝑝=ξ€Ίπœ(|π‘₯|𝑝)ξ€»1/𝑝,whereξ€·π‘₯|π‘₯|=βˆ—π‘₯ξ€Έ1/2.(1.2) The noncommutative 𝐿𝑝-space 𝐿𝑝(β„³,𝜏) associated with (β„³,𝜏) is defined as the Banach space completion of (β„³,‖⋅‖𝑝). We set 𝐿∞(β„³,𝜏)=β„³ equipped with the norm β€–π‘₯β€–βˆž=β€–π‘₯β€–, 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(β„³,𝜏) may be identified with unbounded operators.

Definition 1.3. Let β„³ be a von Neumann algebra equipped with a faithful normal semifinite trace 𝜏.(i)A linear operator π‘₯∢dom(π‘₯)β†’β„‹ is called affiliated with β„³ if π‘₯𝑒=𝑒π‘₯ for all unitary 𝑒 in the commutant β„³β€² of β„³.(ii)A closed densely defined operator π‘₯, affiliated with β„³, is called 𝜏-measurable if for every πœ€>0 there exists an orthogonal projection π‘βˆˆβ„³ such that π‘β„‹βŠ†dom(π‘₯) and 𝜏(1βˆ’π‘)<πœ€.

For 0<𝑝<∞, we have𝐿𝑝(β„³,𝜏)β‰…{π‘₯∣π‘₯is𝜏-measurable,𝜏(|π‘₯|𝑝)<∞}.(1.3) Note that 𝐿2(β„³,𝜏) is a Hilbert space with respect to the scalar product ⟨π‘₯,π‘¦βŸ©=𝜏(π‘¦βˆ—π‘₯).

If 𝜏 is a normal faithful finite trace, then it is normalized, that is, 𝜏(1)=1. In this case, (β„³,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 β„³ be a von Neumann algebra on a Hilbert space β„‹, Aut(β„³) the group of all βˆ—-automorphism of β„³, 𝐺 a locally compact group equipped with its left Haar measure 𝑑𝑔 andπΊβˆ‹π‘”βŸΌπœ‹π‘”βˆˆAut(β„³)(1.4) a homomorphism of group, such that for any π‘₯βˆˆβ„³, the mappingπΊβˆ‹π‘”βŸΌπœ‹π‘”(π‘₯)βˆˆβ„³(1.5) is continuous for the weak operator topology in β„³. Let 𝐢𝑐(𝐺,β„‹) be the space of all norm continuous functions defined on 𝐺 and taking values in β„‹ which have compact supports. We endow it with the inner product:βŸ¨π‘“1,𝑓2ξ€œβŸ©=πΊβŸ¨π‘“1(𝑔),𝑓2(𝑔)βŸ©π‘‘π‘”,(1.6) and we denote by 𝐿2(𝐺,β„‹) the Hilbert space obtained by completion.

For any π‘₯βˆˆβ„³, the operator πœ†π‘₯βˆˆβ„¬(𝐿2(𝐺,β„‹)) is defined by the relations:ξ€·πœ†π‘₯ξ€Έ(𝑓)(𝑔)=πœ‹π‘”βˆ’1(π‘₯)(𝑓(𝑔)),π‘“βˆˆπΆπ‘(𝐺,β„‹),π‘”βˆˆπΊ,(1.7) whereas for any π‘”βˆˆπΊ one defines the unitary operator π‘’π‘”βˆˆβ„¬(𝐿2(𝐺,β„‹)) by the relations𝑒𝑔𝑔(𝑓)ξ€Έξ€·ξ…žξ€Έξ€·π‘”=π‘“βˆ’1π‘”ξ…žξ€Έ,π‘“βˆˆπΆπ‘(𝐺,β„‹),π‘”β€²βˆˆπΊ.(1.8) The von Neumann algebra generated in ℬ(𝐿2(𝐺,β„‹)) by the operators πœ†π‘₯,π‘₯βˆˆβ„³ and 𝑒𝑔,π‘”βˆˆπΊ, is called the cross-product of β„³ by the action πœ‹ of 𝐺 and it is denoted by β„³β‹Šπœ‹πΊ or simply by β„³β‹ŠπΊ.

Remark 1.4. If β„³ is a von Neumann algebra on a separable Hilbert space β„‹ and 𝐺 is a separable abelian locally compact group acting by βˆ—-automorphisms of β„³, then the group 𝐺 of the character of 𝐺 acts by βˆ—-automorphisms of β„³β‹ŠπΊ. M. Takesaki has proved that (π‘€β‹ŠπΊ)β‹ŠπΊβ‰…β„³ξ€·πΏβŠ—β„¬2ξ€Έ.(𝐺,β„‹)(1.9) In particular, if β„³ is properly infinite, then (π‘€β‹ŠπΊ)β‹ŠπΊβ‰…β„³.

Let β„³ be a von Neumann algebra on a Hilbert space β„‹ with a faithful normal semifinite weight πœ‘. Let us recall the noncommutative 𝐿𝑝-space associated with (β„³,πœ‘) constructed by Haagerup (see, e.g., [1, 5]).

Let πœŽπ‘‘=πœŽπœ‘π‘‘,π‘‘βˆˆβ„ denote the one parameter modular automorphism group of ℝ on β„³ associated with πœ‘. The group {πœŽπœ‘π‘‘} is the only group of βˆ—-automorphisms of β„³, with respect to πœ‘ which satisfies the KMS-conditions. We consider the cross-product 𝒩=β„³Γ—πœŽβ„, that is, a von Neumann algebra acting on 𝐿2(ℝ,β„‹), generated by the operators πœ‹π‘₯,π‘₯βˆˆβ„³, and the operators πœ†π‘ ,π‘ βˆˆβ„, defined byπœ‹π‘₯(𝑓(𝑑))=πœŽβˆ’π‘‘(π‘₯)𝑓(𝑑),πœ†π‘ (𝑓(𝑑))=𝑓(π‘‘βˆ’π‘ )foranyπ‘“βˆˆπΏ2(ℝ,β„‹),π‘‘βˆˆβ„.(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 {𝑒𝑑}π‘‘βˆˆβ„ of unitary operators in β„³ such thatξπœŽπœ‘π‘‘(π‘₯)=𝑒𝑑π‘₯π‘’βˆ—π‘‘,π‘‘βˆˆβ„.(1.11)

Let 𝜏 be its (unique) faithful normal semifinite trace satisfyingπœβˆ˜ξπœŽπ‘‘=π‘’βˆ’π‘‘πœ,βˆ€π‘‘βˆˆβ„,(1.12) The βˆ—-algebra of all 𝜏-measurable operators on 𝐿2(ℝ,β„‹) affiliated with 𝒩 is denoted by 𝒩. For each 0<π‘β‰€βˆž, we define the Haagerup noncommutative 𝐿𝑝-spaces by𝐿𝑝(β„³,πœ‘)=π‘₯βˆˆπ’©βˆ£ξπœŽπ‘‘(π‘₯)=π‘’βˆ’π‘‘/𝑝.π‘₯,βˆ€π‘‘βˆˆβ„(1.13) We have𝐿∞(β„³,πœ‘)=β„³,𝐿1(β„³,πœ‘)=β„³βˆ—.(1.14)

For 0<𝑝<∞, π‘₯βˆˆπΏπ‘(β„³,πœ‘) if and only if |π‘₯𝑝|∈𝐿1(β„³,πœ‘), we then defineβ€–π‘₯‖𝑝=β€–|π‘₯|𝑝‖11/𝑝,π‘₯βˆˆπΏπ‘(β„³,πœ‘).(1.15) For 1≀𝑝<∞, 𝐿𝑝(β„³,πœ‘) is a Banach space equipped with a norm ‖⋅‖𝑝. For 0<𝑝<1, 𝐿𝑝(β„³,πœ‘) is a quasi-Banach space equipped with a 𝑝-norm ‖⋅‖𝑝.

It is well known that 𝐿𝑝(β„³,πœ‘) is independent of πœ‘ up to isometric isomorphism preserving the order and modular structure of 𝐿𝑝(β„³,πœ‘) (see [6–8]). Sometimes, we denote 𝐿𝑝(β„³,πœ‘) simply by 𝐿𝑝(β„³).

1.3. Tensor Products of von Neumann Algebras

Let β„‹βŠ—π’¦ be the Hilbert space tensor product of β„‹ and 𝒦. For π‘₯βˆˆβ„³ and π‘¦βˆˆπ’©, the tensor product π‘₯βŠ—π‘¦ is the bounded linear operator on β„‹βŠ—π’¦ uniquely determined byξ‚€π‘₯ξ‚βŠ—π‘¦(πœ‰βŠ—πœ‚)=π‘₯(πœ‰)βŠ—π‘¦(πœ‚)βˆ€πœ‰βˆˆβ„‹,πœ‚βˆˆπ’¦.(1.16)

Let β„³βŠ‚β„¬(β„‹),π’©βŠ‚β„¬(𝒦) be two von Neumann algebras. The algebraic tensor product β„³βŠ—π’© of β„³ and 𝒩,ξƒ―β„³βŠ—π’©=π‘›ξ“π‘˜=1π‘₯π‘˜βŠ—π‘¦π‘˜βˆ£π‘₯π‘˜βˆˆβ„³,π‘¦π‘˜ξƒ°,βˆˆπ’©,𝑛=1,2,…(1.17) is a βˆ—-subalgebra of operators on β„‹βŠ—π’¦. The von Neumann algebra generated by β„‹βŠ—π’¦ in ℬ(β„‹βŠ—π’¦) is denoted by β„³βŠ—π’© and it is called the tensor product of von Neumann algebras β„³ and 𝒩. Since the mapβ„³βˆ‹π‘₯⟼π‘₯βŠ—1βˆˆβ„³βŠ—π’©(1.18) is a βˆ—-isomorphism, we can view β„³ as a von Neumann subalgebra of β„³βŠ—π’©. Similarly, we can also view 𝒩 as a von Neumann subalgebra of β„³βŠ—π’©. By the Tomita commutation theorem, β„³ and 𝒩 commute and together generate β„³βŠ—π’©.

Example 1.5. Let 𝕋 be the unit circle equipped with the normalized Lebesque measure dm and (β„³,𝜏) a finite von Neumann algebra. Let (𝐿∞(𝐿∞(𝕋),dm)βŠ—(β„³,𝜏) be consisting of all functions 𝑓 such that ξ€œπœ(π‘₯𝑓(𝑧))𝑧𝑛dm(𝑧),βˆ€π‘₯∈𝐿1(β„³,𝜏),π‘›βˆˆβ„€,𝑛>0.(1.19) Then, 𝐻∞(𝕋,β„³) is a finite subdiagonal algebra of (𝐿∞(𝕋),dm)βŠ—(β„³,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.

2.1. Tensor Products of Noncommutative 𝐿𝑝-Spaces Associated with Normal Faithful Finite von Neumann Algebras

Theorem 2.1. Let β„³ 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 β„³βŠ—π’© such that 𝜏(π‘₯βŠ—π‘¦)=𝜏1(π‘₯)𝜏2(𝑦),π‘₯βˆˆβ„³,π‘¦βˆˆπ’©.(2.1)

Proof. Since 𝜏1 and 𝜏2 are normal faithful normalized traces, we can view β„³ and 𝒩 as von Neumann algebras acting on β„‹=𝐿2(β„³,𝜏1) and 𝒦=𝐿2(𝒩,𝜏2), respectively, by left multiplication. Then, 𝜏1 and 𝜏2 are the vector states associated to the identities 1β„³ of β„³ and 1𝒩 of 𝒩, respectively. That is, 𝜏1π‘₯ξ€·1(π‘₯)=β„³ξ€Έ,1ℳ,𝜏2𝑦1(𝑦)=𝒩,1𝒩,π‘₯βˆˆβ„³,π‘¦βˆˆπ’©.(2.2) Let 𝜏 be the vector state associated to 1β„³βŠ—1𝒩 on β„³βŠ—π’©. Then, 𝜏 is uniquely determined by 𝜏(π‘₯βŠ—π‘¦)=𝜏1(π‘₯)𝜏2(𝑦) for all π‘₯βˆˆβ„³,π‘¦βˆˆπ’©. Therefore, 𝜏 is tracial and faithful.

𝜏 is called the tensor product trace of 𝜏1 and 𝜏2, and we denote it by 𝜏1βŠ—πœ2. Then, we can define the noncommutative 𝐿𝑝-spaces 𝐿𝑝(β„³βŠ—π’©,𝜏1βŠ—πœ2) and called it the noncommutative 𝐿𝑝-tensor product of (β„³,𝜏1) and (𝒩,𝜏2).

Example 2.2. Let us consider two cases (see [1, 5]).(1)Let (Ξ©,𝑃) be a probability space. We can represent 𝐿∞(Ξ©) as a von Neumann algebra on β„‹=𝐿2(Ξ©) by multiplication and the integral against 𝑃 is a normal faithful normalized trace on 𝐿∞(Ξ©). Let (β„³,𝜏) be a noncommutative probability space. Then, 𝐿𝑝(𝐿∞(Ξ©)βˆ«βŠ—β„³,βŠ—πœ) is isometric to 𝐿𝑝(Ξ©,𝐿𝑝(β„³)), the usual 𝐿𝑝-space of 𝑝-integrable functions from Ξ© to 𝐿𝑝(β„³).(2)Let ℬ(𝑙2) be equipped with the usual trace Tr and let (β„³,𝜏) be a noncommutative probability space. Then, the element of 𝐿𝑝(ℬ(𝑙2)βŠ—β„³,TrβŠ—πœ), the noncommutative 𝐿𝑝-tensor product of (ℬ(𝑙2),Tr) and (β„³,𝜏) can be identified with an infinite matrix with entries in 𝐿𝑝(β„³,𝜏).

2.2. Infinite Tensor Products of Noncommutative 𝐿𝑝-Spaces Associated with Finite von Neumann Algebras

For π‘›βˆˆβ„•, let ℳ𝑛 be a von Neumann algebras. The infinite algebraic tensor product βŠ—π‘›β‰₯1ℳ𝑛 of ℳ𝑛 is the set of all finite linear combinations of elementary tensors βŠ—π‘›β‰₯1π‘₯𝑛, where π‘₯π‘›βˆˆβ„³π‘› and all but finitely many π‘₯𝑛 are 1, that is,𝑛β‰₯1ℳ𝑛=ξƒ―π‘šξ“π‘˜=1𝑛β‰₯1π‘₯𝑛(π‘˜)ξƒͺ∣π‘₯𝑛(π‘˜)βˆˆβ„³π‘›andallbutfinitelymanyπ‘₯𝑛areξƒ°.1,π‘šβˆˆβ„•(2.3)

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

For π‘›βˆˆβ„•, let ℳ𝑛 be a finite factor equipped with a unique normal faithful normalized trace πœπ‘›. We have the product state 𝜏 on βŠ—π‘›β‰₯1ℳ𝑛, defined byπœξƒ©ξ·π‘›β‰₯1π‘₯𝑛ξƒͺ=𝑛β‰₯1πœπ‘›ξ€·π‘₯𝑛,π‘₯π‘›βˆˆβ„³π‘›.(2.4) The infinite von Neumann tensor product βŠ—π‘›β‰₯1ℳ𝑛 is the weak-closure of the image of the representation of βŠ—π‘›β‰₯1ℳ𝑛 by the left multiplication on the Hilbert space 𝐿2(βŠ—π‘›β‰₯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 πœπ‘› and denoted by βŠ—π‘›β‰₯1πœπ‘› (see [7]). Then, we can define the noncommutative 𝐿𝑝-spaces 𝐿𝑝(βŠ—π‘›β‰₯1ℳ𝑛,βŠ—π‘›β‰₯1πœπ‘›) and called it the infinite noncommutative 𝐿𝑝-tensor product of (ℳ𝑛,πœπ‘›).

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

Theorem 2.3. Let (β„³π‘š)π‘šβˆˆβ„• be a sequence of finite von Neumann algebras equipped with normal faithful normalized traces πœπ‘š. Let π’œ=βˆͺπ‘šβ‰₯1(β„³1βŠ—β„³2βŠ—β‹―βŠ—β„³π‘š). Let β„‹ be the completion of π’œ with respect to the inner product π‘₯1βŠ—β‹―βŠ—π‘₯π‘š,𝑦1βŠ—β‹―βŠ—π‘¦π‘šξ‚­=π‘šξ‘π‘˜=1πœπ‘˜ξ€·π‘¦βˆ—π‘˜π‘₯π‘˜ξ€Έ.(2.5) Let πœ‹βˆΆπ’œβ†’β„¬(β„‹) be defined by πœ‹(π‘₯)Ξ›(π‘Ž)=Ξ›(π‘₯π‘Ž),π‘₯βˆˆπ’œ,π‘Žβˆˆπ’œ,whereΞ›βˆΆπ΄βŸΆβ„‹π‘–π‘ π‘‘β„Žπ‘’π‘–π‘›π‘π‘™π‘’π‘ π‘–π‘œπ‘›.(2.6) Let 𝒩 be the weakβˆ—-closure of πœ‹(𝐴) in ℬ(β„‹). Then, there exists a normal state 𝜈 on 𝒩 such that πœˆξ‚€π‘₯1βŠ—β‹―βŠ—π‘₯π‘šξ‚=π‘šξ‘π‘˜=1πœπ‘˜ξ€·π‘₯π‘˜ξ€Έ,π‘₯π‘˜βˆˆβ„³π‘˜,π‘šβˆˆβ„•.(2.7)

Proof. Let β„‹π‘š=𝐿2(β„³π‘š) and consider β„³π‘š as a von Neumann algebra on β„‹π‘š by left multiplication. Let π’©π‘š=β„³1βŠ—β„³2βŠ—β‹―βŠ—β„³π‘š,πœˆπ‘š=𝜏1βŠ—πœ2βŠ—β‹―βŠ—πœπ‘š.(2.8) We view π’©π‘š as a von Neumann subalgebra of π’©π‘š+1 via the inclusion: π‘₯1βŠ—β‹―βŠ—π‘₯π‘šβŸΌπ‘₯1βŠ—β‹―βŠ—π‘₯π‘šβŠ—1β„³π‘š+1.(2.9) Since πœπ‘š+1(1)=1, πœˆπ‘š+1|π’©π‘š=πœˆπ‘š. Note that π’œ is a unital βˆ—-algebra and the traces πœˆπ‘š induce a faithful normal state πœˆπ‘œ on π’œ. Since πœˆπ‘œ is faithful, the representation πœ‹ is faithful. Therefore, π’œ, and all π’©π‘š, can be viewed as subalgebras of ℬ(β„‹). Let 𝜈 the restriction to 𝒩 of the vector state given by Ξ›(1). Then, 𝜈 is tracial and faithful. The trace 𝜈|β„³π‘š=πœπ‘š and 𝜈 is the unique normal state on 𝒩 such that πœˆξ‚€π‘₯1βŠ—β‹―βŠ—π‘₯π‘šξ‚=π‘šξ‘π‘˜=1πœπ‘˜ξ€·π‘₯π‘˜ξ€Έ,π‘₯π‘˜βˆˆβ„³π‘˜,π‘šβˆˆβ„•.(2.10)

(𝒩,𝜈) is called the infinite tensor products of noncommutative 𝐿𝑝-spaces of (β„³π‘š,πœπ‘š) (see [1]).

Example 2.4. Let 𝑀2(β„‚) be the full algebra of 2Γ—2 matrices. Murray and von Neumann proved that the infinite tensor product ξ€·βŠ—π‘›β‰₯1𝑀2ξ€Έ(β„‚)WOT,(2.11) produced with respect to the unique normalized trace tr2 on 𝑀2(β„‚), 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 π‘˜βˆˆ{1,2}, let β„³π‘˜ be a 𝜎-finite von Neumann algebra. Let 𝐿𝑝(β„³π‘˜) be the Haagerup noncommutative 𝐿𝑝-spaces. By the reduction theorem, there exist a Banach space (𝑋𝑝)π‘˜ (a quasi Banach space if 𝑝<1), a sequence (β„›π‘˜,π‘š)π‘šβˆˆβ„• of finite von Neumann algebras, each equipped with a faithful normal finite trace πœπ‘˜,π‘š, and for each π‘šβˆˆβ„• an isometric embedding π½π‘˜,π‘šβˆΆπΏπ‘(β„›π‘˜,π‘š,πœπ‘˜,π‘š)β†’(𝑋𝑝)π‘˜ such that(1)π½π‘˜,π‘š1(𝐿𝑝(β„›π‘˜,π‘š1,πœπ‘˜,π‘š1))βŠ‚π½π‘˜,π‘š2(𝐿𝑝(β„›π‘˜,π‘š2,πœπ‘˜,π‘š2)) for all π‘š1,π‘š2βˆˆβ„• such that π‘š1β‰€π‘š2;(2)β‹ƒπ‘šβˆˆβ„•π½π‘˜,π‘š(𝐿𝑝(β„›π‘˜,π‘š,πœπ‘˜,π‘š)) is dense in (𝑋𝑝)π‘˜;(3)𝐿𝑝(β„³π‘˜) is isometric to a subspace (π‘Œπ‘)π‘˜ of (𝑋𝑝)π‘˜;(4)(π‘Œπ‘)π‘˜ and all π½π‘˜(𝐿𝑝(β„›π‘˜,π‘š,πœπ‘˜,π‘š)),π‘šβˆˆβ„• are 1-complemented in (𝑋𝑝)π‘˜ for 1≀𝑝<∞.

Here, 𝐿𝑝(β„›π‘˜,π‘š,πœπ‘˜,π‘š) is the tracial noncommutative 𝐿𝑝-space associated with (β„›π‘˜,π‘š,πœπ‘˜,π‘š).

Thus, we have a sequence (β„›π‘˜,π‘š,πœπ‘˜,π‘š) of finite von Neumann algebras. We then have the noncommutative 𝐿𝑝-tensor product (β„›π‘š,πœπ‘š)∢=(β„›1,π‘šβŠ—β„›2,π‘š,𝜏1,π‘šβŠ—πœ2,π‘š). Applying the construction in Section 2.2, we will be able to construct the infinite tensor products of noncommutative 𝐿𝑝-spaces of (β„›π‘š,πœπ‘š). Hence, we have the tensor products of noncommutative 𝐿𝑝-spaces of 𝐿𝑝(β„³1) and 𝐿𝑝(β„³2).

With this setting, if {β„³π‘˜}π‘˜βˆˆβ„• 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.

Let β„³ be an (arbitrary) von Neumann algebra. Then, β„³ admits the following direct sum decomposition:ℳ=π‘—βˆˆπ½π’©π‘—ξ€·πΎβŠ—π΅π‘—ξ€Έ,(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 β„³ be a general von Neumann algebra and 0<𝑝<∞. Let 𝐿𝑝(β„³) be the Haagerup noncommutative 𝐿𝑝-space associated with β„³. Then, there exist a Banach space 𝑋𝑝 (a quasi Banach space if 𝑝<1), a family (ℛ𝑖)π‘–βˆˆπΌ of finite von Neumann algebras, each equipped with a normal faithful finite trace πœπ‘–, and, for each π‘–βˆˆπΌ, an isometric embedding π½π‘–βˆΆπΏπ‘(ℛ𝑖,πœπ‘–)→𝑋𝑝 such that(1)𝐽𝑖(𝐿𝑝(ℛ𝑖,πœπ‘–))βŠ‚π½π‘—(𝐿𝑝(ℛ𝑗,πœπ‘—)) for all 𝑖,π‘—βˆˆπΌ such that 𝑖≀𝑗;(2)β‹ƒπ‘–βˆˆπΌπ½π‘–(𝐿𝑝(ℛ𝑖,πœπ‘–)) is dense in 𝑋𝑝;(3)𝐿𝑝(β„³) is isometric to a subspace π‘Œπ‘ of 𝑋𝑝;(4)π‘Œπ‘ and all 𝐽𝑖(𝐿𝑝(ℛ𝑖,πœπ‘–)),π‘–βˆˆπΌ are 1-complemented in 𝑋𝑝 for 1≀𝑝<∞.

Here, 𝐿𝑝(ℛ𝑖,πœπ‘–) is the tracial noncommutative 𝐿𝑝-space associated with (ℛ𝑖,πœπ‘–).

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.

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