On the Tensor Products of Maximal Abelian JW-Algebras

Abstract

It is well known in the work of Kadison and Ringrose (1983)that if 𝐴 and 𝐵 are maximal abelian von Neumann subalgebras of von Neumann algebras 𝑀 and 𝑁, respectively, then 𝐴⊗𝐵 is a maximal abelian von Neumann subalgebra of 𝑀⊗𝑁. It is then natural to ask whether a similar result holds in the context of 𝐽𝑊-algebras and the 𝐽𝑊-tensor product. Guided to some extent by the close relationship between a 𝐽𝑊-algebra M and its universal enveloping von Neumann algebra 𝑊^∗(𝑀), we seek in this article to investigate the answer to this question.

1. Introduction

A 𝐽𝐶-algebra   𝐴 is a norm (uniformly) closed Jordan subalgebra of the Jordan algebra 𝐵(𝐻)_𝑠.𝑎 of all bounded self adjoint operators on a Hilbert space 𝐻. The Jordan product is given by 𝑎∘𝑏=(𝑎𝑏+𝑏𝑎)/2. A subspace 𝐼 of a 𝐽𝐶-algebra 𝐴 is called a Jordan ideal if 𝑎∘𝑏∈𝐼 for every 𝑎∈𝐴 and every 𝑏∈𝐼. A 𝐽𝐶-algebra is said to be simple if it has no nontrivial norm closed Jordan ideals. A 𝐽𝑊-algebra 𝑀⊆𝐵(𝐻)_𝑠.𝑎 is a weakly closed 𝐽𝐶-algebra. If 𝑀 is a 𝐽𝐶-algebra (resp., 𝐽𝑊-algebra), let 𝐶^∗(𝑀) (resp., 𝑊^∗(𝑀)) be the universal enveloping 𝐶^∗-algebra (resp., von Neumann algebra) of 𝑀, and let 𝜃_𝑀 (resp., Φ_𝑀) be the canonical involutive *-antiautomorphism of 𝐶^∗(𝑀) (resp., 𝑊^∗(𝑀)). Usually we will regard 𝑀 as a generating Jordan subalgebra of 𝐶^∗(𝑀)) and 𝑊^∗(𝑀) so that 𝜃_𝑀 and Φ_𝑀 fix each point of 𝑀. The real 𝐶^∗-algebra 𝑅^∗(𝑀)={𝑥∈𝐶^∗(𝑀)∶𝜃_𝑀(𝑥)=𝑥^∗} satisfies 𝑅^∗(𝑀)∩𝑖𝑅^∗(𝑀)=0,𝐶^∗(𝑀)=𝑅^∗(𝑀)⊕𝑖𝑅^∗(𝑀), (1.1) and the real von Neumann algebra 𝑅𝑊^∗(𝑀)={𝑥∈𝑊^∗(𝑀)∶Φ_𝑀(𝑥)=𝑥^∗} satisfies 𝑅𝑊^∗(𝑀)∩𝑖𝑅𝑊^∗(𝑀)=0,𝑊^∗(𝑀)=𝑅𝑊^∗(𝑀)⊕𝑖𝑅𝑊^∗(𝑀). (1.2) The reader is refered to [1–5] for a detailed account of the theory of 𝐽𝐶-algebras and 𝐽𝑊-algebras. The relevant background on the theory of 𝐶^∗-algebras and von Neumann algebras can be found in [6–8].

A projection 𝑒 of a 𝐽𝑊-algebra 𝑀 is said to be  abelian if 𝑒𝑀𝑒 is associative, and it is called minimal if it is nonzero and contains no other nonzero projections of 𝑀, or equivalently, 𝑒 is minimal if and only if 𝑒𝑀𝑒=ℝ𝑒. A 𝐽𝑊-factor is a 𝐽𝑊-algebra with trivial centre; a Type 𝐼𝐽𝑊-factor is a 𝐽𝑊-factor which contains a minimal projection. A 𝐽𝑊-algebra is said to be of Type  𝐼_n if there is a family of abelian projections (𝑒_𝛼)_𝛼∈𝐽 such that the central support 𝑐_M(𝑒_𝛼) of 𝑒_𝛼 in 𝑀 equals the unit 1_M of M, ∑_𝛼∈𝐽𝑒_𝛼=1_M and card 𝐽=𝑛 (see [1, Section  5.3]).   A spin factor 𝑉=𝐻⊕ℝ1_𝑉 is a real Jordan algebra with identity 1_𝑉, where 𝐻 is a real Hilbert space of dimension at least two. The Jordan product on 𝑉 is defined by (𝑎+𝜆1_𝑉)∘(𝑏+𝜇1_𝑉)=(𝜇𝑎+𝜆𝑏)+(⟨𝑎,𝑏⟩+𝜆𝜇)1_𝑉,𝑎,𝑏∈𝑉,𝜆,𝜇∈ℝ, (1.3) and the norm on 𝑉 is given by ‖𝑎+𝜆1_𝑉‖=⟨𝑎,𝑎⟩^1/2+|𝜆|. (1.4) A spin factor 𝑉 is universally reversible when dim𝑉=3 or 4, nonreversible when dim𝑉≠3,4 or 6, and it can be either reversible or nonreversible when dim𝑉=6. A spin factor is a simple reflexive 𝐽𝑊-algebra and constitutes the Type 𝐼₂𝐽𝑊-factor (see [2, Section  6.1]).

A linear map 𝜑∶𝐴→𝐵 between 𝐽𝐶-algebras 𝐴 and 𝐵 is called a (Jordan) homomorphism if it preserves the Jordan product. A Jordan homomorphism which is one to one is called a Jordan isomorphism. A factor representation of a 𝐽𝐶-algebra 𝐴 is a (Jordan) homomorphism of 𝐴 onto a weakly dense subalgebra of a 𝐽𝑊-factor 𝑀. Type I factor representations are defined accordingly.

A 𝐽𝐶-algebra 𝐴 is said to be reversible if 𝑎₁𝑎₂⋯𝑎_𝑛+𝑎_𝑛𝑎_𝑛−1⋯𝑎₁∈𝐴 whenever 𝑎₁,𝑎₂,…,𝑎_𝑛∈𝐴 and is said to be universally reversible if 𝜋(𝐴) is reversible for every representation 𝜋 of 𝐴 [2, page 5]. The only universally reversible spin factors are 𝑉₂=𝑀₂(ℝ)_𝑠.𝑎 and 𝑉₃=𝑀₂(ℂ)_𝑠.𝑎 [2, Theorem  2.1]. A 𝐽𝐶-algebra 𝐴 is universally reversible if and only if it has no spin factor representations other than onto 𝑉₂ and 𝑉₃ [2, Theorems  2.2]. Every 𝐽𝑊-algebra without a direct summand of Type 𝐼₂ is universally reversible [1, 5.1.5, 5.3.5, 6.2.3].

Two elements 𝑎 and 𝑏 of a 𝐽𝐶-algebra 𝐴 are said to operator commute if 𝑇_𝑎𝑇_𝑏=𝑇_𝑏𝑇_𝑎, where 𝑇_𝑎∶𝐴→𝐴 is the multiplication operator defined by 𝑇_𝑎(𝑥)=𝑎∘𝑥, for all 𝑥∈𝐴. A 𝐽𝑊-algebra 𝑀 is called associative if all its elements operators commute. A 𝐽𝑊-subalgebra 𝐴 of a 𝐽𝑊-algebra 𝑀 is called maximal associative if it is not contained in any larger associative 𝐽𝑊-subalgebra of 𝑀. If 𝐴 is a 𝐽𝑊-subalgebra of a 𝐽𝑊-algebra 𝑀⊆𝐵(𝐻)_𝑠.𝑎 and 𝐴^′ is the set of all elements of 𝐵(𝐻)_𝑠.𝑎 which operator commutes with all elements of 𝐴, then 𝐴 is a maximal associative 𝐽𝑊-subalgebra of 𝑀 if and only if 𝐴=𝐴^′∩𝑀. Indeed, since 𝐴 is associative, 𝐴⊆𝐴^′∩𝑀 and 𝐴 together with any element of 𝐴^′∩𝑀 generates an associative 𝐽𝑊-subalgebra of 𝑀 which implies that 𝐴^′∩𝑀⊆𝐴 since 𝐴 is maximal abelian. In particular, if 𝐴⊆𝐵(𝐻)_𝑠.𝑎 is an associative 𝐽𝑊-algebra, then 𝐴 is maximal associative if and only if 𝐴=𝐴^′.

This article aims to study the relationship between the maximality of an associative 𝐽𝑊-subalgebra 𝐵 of a 𝐽𝑊-algebra 𝑀 and that of 𝑊^∗(𝐵) in 𝑊^∗(𝑀). We give a counterexample which rules out the establishing of a result in the theory of 𝐽𝑊-tensor products analog to that given in [6,Theorem  11.2.18] for von Neumann tensor products (cf. Example 2.2). Then we prove that a Jordan analog of Theorem  11.2.18 in [6] can be established in some particular cases.

Theorem 1.1 (see [9, Proposition  1]). Let 𝑀⊆𝐵(𝐻)_𝑠.𝑎 be a 𝐽𝑊-algebra, and let 𝑎,𝑏∈𝑀. Then the following are equivalent:(i)𝑎𝑏=𝑏𝑎;(ii)𝑇_𝑎𝑇_𝑏=𝑇_𝑏𝑇_𝑎;(iii)𝑎²∘𝑏=𝑎𝑏𝑎.That is, 𝑎 and 𝑏 operators commute if and only if they commute under ordinary operator multiplication.

Definition 1.2. Let  𝑀  and  𝑁  be a pair of 𝐽𝑊-algebras canonically embedded in their respective universal enveloping von Neumann algebras 𝑊^∗(M)  and 𝑊^∗(𝑁). Then the 𝐽𝑊-tensor product  𝐽𝑊(𝑀⊗𝑁)  of𝑀 and 𝑁  is the 𝐽𝑊-algebra generated by 𝑀⊗𝑁 in 𝑊^∗(𝑀)⊗𝑊^∗(𝑁). The reader is referred to [10] for the properties of the 𝐽𝑊-tensor product of 𝐽𝑊-algebras.

Theorem 1.3 (see [10,Theorem  2.9]). Let  𝑀  and 𝑁 be  𝐽𝑊- algebras.  If  𝐽𝑊(𝑀⊗𝑁)  is universally reversible, then 𝑊^∗(𝐽𝑊(𝑀⊗𝑁))=𝑊^∗(𝑀)⊗𝑊^∗(𝑁). (1.5)

2. Maximal Abelian 𝐽𝑊-algebras

Let 𝔄 and 𝔅 be maximal abelian von Neumann subalgebras of von Neumann algebras 𝔐 and 𝔑, respectively, then 𝔄⊗𝔅 is a maximal abelian von Neumann algebra of 𝔐⊗𝔑 (see [6, 11.2.18]). In Example 2.2, we show that the Jordan analog of this result, in the context of 𝐽𝑊-algebras and the 𝐽𝑊-tensor product, is not true in general. However, it is shown in Theorem 2.11 that the result does hold in special circumstances. Remark 2.1. (i) Note that any 𝐽𝑊-subalgebra of a spin factor which is not a spin factor is of dimension at most 2.   Indeed, let A be a 𝐽𝑊-subalgebra of a spin factor 𝑉⊆𝐵(𝐻)_𝑠.𝑎. If 1_𝐴≠1_𝑉,then 1_𝐴 is the only projection in 𝐴,    since every projection in 𝑉 is minimal, and hence  dim𝐴=1.   If 1_𝐴=1_𝑉, then any family of orthogonal central projections of 𝐴  contains at most two projections. Indeed if 𝑒₁+𝑒₂+𝑒₃=1_𝐴,𝑒_𝑖∈𝑍(𝐴),𝑖=1,2,3, then 𝑒₂+𝑒₃≤1_𝐴−𝑒₁.   Since 1_𝐴−𝑒₁ is a minimal projection, we see that one of 𝑒_𝑖,𝑖=1,2,3   must be zero. It is clear that if𝐴  is a factor, then it is of Type  𝐼₂, and hence it is a spin factor. (ii)Recall that 𝑊^∗(𝑉)=𝑀₂(ℂ)⊕𝑀₂(ℂ), where 𝑉 is the 4-dimensional spin factor 𝑀₂(ℂ)_𝑠.𝑎  [1, 6.2.1]: 𝑀₂(ℂ)=span{,,,1001100−101100𝑖−𝑖0}=ℂ⊗_ℝ𝑀₂(ℝ), (2.1) which is an 8-dimensional real  𝐶^∗-algebra.

Example 2.2. Let 𝐴 be a maximal abelian 𝐽𝑊-subalgebra of 𝑉=𝑀₂(𝐶)_𝑠.𝑎. Then 𝐽𝑊(𝐴⊗𝐴) is not a maximal abelian subalgebra of 𝐽𝑊(𝑉⊗𝑉).

Proof. By the above remark, dim𝐴=2, and hence 𝐴=ℝ𝑒+ℝ𝑓 for some minimal projections 𝑒,𝑓. Therefore, 𝐽𝑊(𝐴⊗𝐴)=𝐴⊗_ℝ𝐴=ℝ(𝑒⊗𝑒)⊕ℝ(𝑒⊗𝑓)⊕ℝ(𝑓⊗𝑒)⊕ℝ(𝑓⊗𝑓), (2.2) and hence dim𝐽𝑊(𝐴⊗𝐴)=4, since dim𝐴⊗_ℝ𝐴=dim𝐴·dim𝐴 (see [11, Corollary  7.5]). On the other hand, 𝐽𝑊(𝑉⊗𝑉) is universally reversible, by [10, Proposition  2.7] which implies that 𝐽𝑊(𝑉⊗𝑉)=𝑅𝑊^∗(𝐽𝑊(𝑉⊗𝑉))_𝑠.𝑎=(𝑅𝑊^∗(𝑉)⊗_ℝ𝑅𝑊^∗(𝑉))_𝑠.𝑎=(𝑀₂(ℂ)⊗_ℝ𝑀₂(ℂ))_𝑠.𝑎=𝑀₂²(ℂ)_𝑠.𝑎⊕𝑀₂²(ℂ)_𝑠.𝑎, (2.3) since 𝑅𝑊^∗(𝑀₂(ℂ)_𝑠.𝑎)=𝑀₂(ℂ) [3, page 385]. It can be seen that a maximal abelian 𝐽𝑊-subalgebra of 𝐽𝑊(𝑉⊗𝑉) is of dimension 8, which implies that 𝐽𝑊(𝐴⊗𝐴) is not maximal abelian in 𝐽𝑊(𝑉⊗𝑉).Remark 2.3. Note that if 𝐵 is an associative 𝐽𝑊-subalgebra of a 𝐽𝑊-algebra 𝑀 such that 𝑊^∗(𝐵)  is a maximal abelian subalgebra of 𝑊^∗(𝑀), then 𝐵  is a maximal associative𝐽𝑊-subalgebra of 𝑀,  since 𝐵=𝑊^∗(𝐵)∩𝑀.

Lemma 2.4. Let 𝐵 be an associative 𝐽𝑊-subalgebra of a 𝐽𝑊-algebra 𝑀. Then, 𝑊^∗(𝐵)=[𝐵]=𝐵⊕𝑖𝐵, (2.4) is an abelian von Neumann algebra, where [𝐵] is the 𝑤𝑒𝑎𝑘^∗-closure of the 𝐶^∗-subalgebra [𝐵] of 𝑊^∗(𝑀)  generated by  𝐵.

Proof. Being associative, 𝐵 has no representation into a spin factor of the form 𝑉_4𝑛+1 and is, therefore, universally reversible. It follows from [3, page 383] that 𝐵=𝑅𝑊^∗(𝐵)_𝑠.𝑎. (2.5) Therefore, by [3, Corollary  3.2], 𝑅𝑊^∗(𝐵) is isomorphic to the weak^∗-closure 𝑅(𝐵) of the real 𝐶^∗-subalgebra 𝑅(𝐵) of 𝑊^∗(𝑀) generated by 𝐵, and the result follows.

Recall that if 𝑀 is a 𝐽𝑊-algebra isomorphic to the self adjoint part 𝒩_𝑠.𝑎 of a von Neumann algebra 𝒩 and has no one-dimensional representations, then 𝑊^∗(𝑀) is ∗-isomorphic to 𝒩⊕𝒩^∘, where 𝒩^∘ is the opposite algebra of 𝒩 [2, 7.4.15]. A real 𝐶^∗-algebra 𝔄 can be realized as a complex 𝐶^∗-algebra if there is a 𝐶^∗-algebra isomorphism 𝜙∶𝔅→𝔄 of a complex 𝐶^∗-algebra 𝔅 onto 𝔄. In this case, the real linear isometry 𝑗 on 𝔄 defined, for each 𝑎 in 𝔅, by 𝑗𝜙(𝑎)=𝜙(𝑖𝑎) (2.6) is such that 𝑗² and −𝑖𝑑_𝔄 coincide.

Lemma 2.5. Let 𝐵  be a maximal associative 𝐽𝑊-subalgebra of a 𝐽𝑊-algebra 𝑀. Suppose that 𝑀 is isomorphic to the self adjoint part 𝒩_𝑠.𝑎  of a von Neumann algebra 𝒩 and has no one-dimensional representations. Then 𝑊^∗(𝐵) is not a maximal abelian on Neumann subalgebra of  𝑊^∗(𝑀).

Proof. Identifying 𝑀 with 𝒩_𝑠.𝑎, [𝐵] is a von Neumann subalgebra of both 𝒩 and 𝒩^∘, and hence, the von Neumann  subalgebra [𝐵]⊕[𝐵] of 𝒩⊕𝒩^∘≅𝑊^∗(𝑀) is abelian and contains 𝑊^∗(𝐵)=[𝐵]≅[𝐵]⊕{0}, which implies that 𝑊^∗(𝐵)  is not maximal abelian in 𝑊^∗(𝑀).

Lemma 2.6. Let 𝐵  be a maximal associative 𝐽𝑊-subalgebra of a 𝐽𝑊-algebra 𝑀. If 𝑅𝑊^∗(𝑀) is ∗-isomorphic to a complex 𝐶^∗-algebra,   then  𝑊^∗(𝐵)  is not a maximal abelian von Neumann subalgebra of  𝑊^∗(𝑀).

Proof. Since 𝐶^∗(𝑀) is the complex 𝐶^∗-algebra [𝑀] generated by 𝑀 in 𝑊^∗(𝑀) [12, Theorem  2.7], 𝑅𝑊^∗(𝑀) is the weak^∗-closure of 𝑅^∗(𝑀) in 𝑊^∗(𝑀). Therefore, 𝑅^∗(𝑀) is a complex 𝐶^∗-algebra, which implies that 𝐶^∗(𝑀)=ℐ⊕Φ_𝑀(ℐ) for some norm closed ideal ℐ of 𝐶^∗(𝑀) isomorphic to 𝑅^∗(𝑀) [13, Lemma  1], so that 𝑊^∗(𝑀)=𝒥⊕Φ_𝑀(𝒥), where 𝒥 is the weak^∗-closure ℐ of ℐ in 𝑊^∗(𝑀). Hence, 𝒥 is isomorphic to 𝑅𝑊^∗(𝑀). Let 𝜙 be the isomorphism of 𝒥 onto 𝑅𝑊^∗(𝑀), and let 𝑗 be the corresponding real linear operator on 𝑅𝑊^∗(𝑀), defind above. Then, using Lemma 2.4, there exists an isomorphism 𝜋 from the W^∗-algebra 𝐵⊕𝑖𝐵 into 𝑅𝑊^∗(𝑀) such that, for elements 𝑏₁ and 𝑏₂ in 𝐵, 𝜋(𝑏₁+𝑖𝑏₂)=𝑏₁+𝑗𝑏₂. (2.7) It follows that 𝜙⁻¹∘𝜋 and Φ_𝑀∘𝜙⁻¹∘𝜋 are ∗-isomorphisms of [𝐵]=𝐵⊕𝑖𝐵 into 𝒥 and Φ_𝑀(𝒥), respectively. Since a ∗-isomorphism between 𝐶^∗-algebras is an isometry [7, Corollary  1.5.4], we may identify [𝐵] with 𝜙⁻¹∘𝜋([𝐵]) and Φ_𝑀∘𝜙⁻¹∘𝜋([𝐵]). It follows that [𝐵]⊕[𝐵] is an abelian von Neumann subalgebra of 𝑊^∗(𝑀), proving that 𝑊^∗(𝐵)=[𝐵]≅[𝐵]⊕{0} is not maximal abelian in 𝑊^∗(𝑀).

Proposition 2.7. Let  𝑀  be a universally reversible 𝐽𝑊-algebra not isomorphic to the self adjoint part of a von Neumann algebra and without direct summands of type  𝐼₁. If  𝐵  is a maximal associative subalgebra of  𝑀,  then 𝑊^∗(𝐵) is a maximal abelian von Neumann subalgebra of   𝑊^∗(𝑀).

Proof. By Lemma 2.4, 𝑊^∗(𝐵)=[𝐵]=𝐵⊕𝑖𝐵↪𝑊^∗(𝑀). If 𝑊^∗(𝐵) is not maximal abelian in 𝑊^∗(𝑀), there exists an element 𝑧∈𝑊^∗(𝑀)=𝑅𝑊^∗(𝑀)⊕𝑖𝑅𝑊^∗(𝑀), 𝑧∉𝑊^∗(𝐵) such that 𝑧 together with 𝑊^∗(𝐵) generate an abelian von Neumann subalgebra 𝑌⊃_≠𝑊^∗(𝐵)⊇𝐵 of 𝑊^∗(𝑀)⊇𝑀. Let 𝑧=𝑥+𝑖𝑦, 𝑥,𝑦∈𝑊^∗(𝑀)_𝑠.𝑎. Since 𝑧∉𝑊^∗(𝐵), then either 𝑥 or 𝑦 (or both) does not belong to 𝑊^∗(𝐵). Suppose that 𝑥∉𝑊^∗(𝐵), since 𝑊^∗(𝑀)=𝑅𝑊^∗(𝑀)⊕𝑖𝑅𝑊^∗(𝑀), then 𝑥=𝑎+𝑖𝑏, for some 𝑎,𝑏∈𝑅𝑊^∗(𝑀). Then either 𝑎 or 𝑏 (or both) does not belong to 𝑊^∗(𝐵). Since 𝑥∈𝑊^∗(𝑀)_𝑠.𝑎, we have 𝑎=𝑎^∗, and 𝑏=−𝑏^∗, and so 𝑎∈𝑀=𝑅𝑊^∗(𝑀)_𝑠.𝑎, since 𝑀 is a universally reversible [3, page 383]. Therefore, 𝑎 must be the zero element, since it obviously commutes with all elements in 𝐵. On the other hand, 𝑏²=−𝑏𝑏^∗∈𝑅𝑊^∗(𝑀)_𝑠.𝑎=𝑀. Since 𝑏𝑢=𝑢𝑏 for all 𝑢∈𝑊^∗(𝐵), 𝑏²𝑢=𝑏𝑢𝑏=𝑢𝑏² for all 𝑢∈𝐵, and so 𝑏² and 𝑢 operators commute relative to the Jordan product in 𝐵 [9, Proposition  1]. Hence 𝑏²∈𝐵⊆𝑊^∗(𝐵), since 𝐵  is a maximal associative subalgebra of 𝑀, which implies that 𝑏∈𝑊^∗(𝐵). Therefore, 𝑥=𝑖𝑏∈𝑊^∗(𝐵), a contradiction. This proves the result.

Lemma 2.8. Let 𝑀 be a universally reversible 𝐽𝑊-algebra not isomorphic to the self adjoint part of a von Neumann algebra. If 𝐵 is a maximal associative subalgebra of 𝑀, then 𝑊^∗(𝐵) is a maximal abelian von Neumann subalgebra of 𝑊^∗(𝑀).

Proof. Splitting 𝑀=𝑀_𝐼1⊕𝑀_𝑛.𝑎 as the direct sum of a 𝐽𝑊-algebra 𝑀_𝐼1 of type 𝐼₁ (the abelian part) and a 𝐽𝑊-algebra 𝑀_𝑛.𝑎 without direct summands of type 𝐼₁(the nonabelian part). It is clear that 𝐵⊇𝑀_𝐼1,𝐵_𝑛.𝑎=𝐵∩𝑀_𝑛.𝑎 is a maximal associative subalgebra of 𝑀_𝑛.𝑎 and 𝐵=𝑀_𝐼1⊕𝐵_𝑛.𝑎. By Proposition 2.7, 𝑊^∗(𝐵_𝑛.𝑎) is a maximal abelian von Neumann subalgebra of 𝑊^∗(𝑀_𝑛.𝑎), and hence 𝑊^∗(𝐵)=𝑊^∗(𝑀_𝐼1)⊕𝑊^∗(𝐵_𝑛.𝑎) is a maximal abelian von Neumann subalgebra of 𝑊^∗(𝑀), since 𝑊^∗(𝑀)=𝑊^∗(𝑀_𝐼1)⊕𝑊^∗(𝑀_𝑛.𝑎) [12, Lemma  2.6].

Proposition 2.9. Let 𝐵_𝑖  be a maximal associative subalgebra of a 𝐽𝑊-algebra  𝑀_𝑖,𝑖=1,2, and suppose that  𝑀_𝑖   is universally reversible   𝐽𝑊-algebra not isomorphic to the self adjoint part of a von Neumann algebra and without direct summands of type  𝐼₁. Then 𝐽𝑊(𝐵₁⊗𝐵₂) is a maximal associative 𝐽𝑊-subalgebra of  𝐽𝑊(𝑀₁⊗𝑀₂).

Proof. Note first that 𝑊^∗(𝐵₁)⊗𝑊^∗(𝐵₂) is a von Neumann ∗-subalgebra of 𝑊^∗(𝑀₁)⊗𝑊^∗(𝑀₂) [8, Theorem  11.2.10], and 𝐽𝑊(𝐵₁⊗𝐵₂) is a 𝐽𝑊-subalgebra of 𝐽𝑊(𝑀₁⊗𝑀₂), since 𝐵₁⊗𝐵₂⊆𝑀₁⊗𝑀₂. By Proposition  5.2, 𝑊^∗(𝐵_𝑖) is maximal abelian in 𝑊^∗(𝑀_𝑖), and hence, 𝑊^∗(𝐵₁)⊗𝑊^∗(𝐵₂) is maximal abelian in 𝑊^∗(𝑀₁)⊗𝑊^∗(𝑀₂) [8, Corollary  11.2.18] and [10, Theorem  2.9]. The result is now obvious, since 𝑊^∗(𝐽𝑊(𝐵₁⊗𝐵₂))=𝑊^∗(𝐵₁)⊗𝑊^∗(𝐵₂), and 𝑊^∗(𝐽𝑊(𝑀₁⊗𝑀₂))=𝑊^∗(𝑀₁)⊗𝑊^∗(𝑀₂) [10, Theorem  2.9].

Proposition 2.10. Let 𝑁 be an associative 𝐽𝑊-algebra, and let 𝑀  be a universally reversible 𝐽𝑊-algebra not isomorphic to the self adjoint part of a von Neumann algebra and without direct summands of type  𝐼₁. If 𝐵  is a maximal associative subalgebra of  𝑀,   then 𝐽𝑊(𝑁⊗𝐵) is a maximal associative 𝐽𝑊-subalgebra of  𝐽𝑊(𝑁⊗𝑀).

Proof. Let 𝑀=𝑀_𝐼1⊕𝑀_𝑛.𝑎 be the decomposition of 𝑀 into abelian part 𝑀_𝐼1 and nonabelian part 𝑀_𝑛.𝑎. Then 𝐵=𝑀_𝐼1⊕𝐵_𝑛.𝑎, where 𝐵_𝑛.𝑎=𝐵∩𝑀_𝑛.𝑎 is obviously a maximal associative subalgebra of 𝑀_𝑛.𝑎. By [10, Remark  2.14], 𝐽𝑊(𝑁⊗𝑀)=𝐽𝑊(𝑁⊗(𝑀_𝐼1⊕𝑀_𝑛.𝑎))=𝐽𝑊(𝐴⊗𝑀_𝐼1)⊕𝐽𝑊(𝐴⊗𝑀_𝑛.𝑎),𝐽𝑊(𝑁⊗𝐵)=𝐽𝑊(𝑁⊗(𝑀_𝐼1⊕𝐵_𝑛.𝑎))=𝐽𝑊(𝑁⊗𝑀_𝐼1)⊕𝐽𝑊(𝑁⊗𝐵_𝑛.𝑎). (2.8) It is clear now that 𝐽𝑊(𝑁⊗𝐵) is a maximal associative 𝐽𝑊-subalgebra of 𝐽𝑊(𝑁⊗𝑀), since 𝐽𝑊(𝑁⊗𝑀_𝐼1) is obviously associative, and 𝐽𝑊(𝑁⊗𝐵_𝑛.𝑎) is maximal in  𝐽𝑊(𝑁⊗𝑀_𝑛.𝑎), by Proposition 2.9.Theorem 2.11. Let 𝑀 and 𝑁 be universally reversible 𝐽𝑊-algebras not isomorphic to the self adjoint parts of von Neumann algebras. If 𝐴 and 𝐵 are maximal associative subalgebra of  𝑀 and 𝑁, respectively, then 𝐽𝑊(𝐴⊗𝐵) is a maximal associative 𝐽𝑊-subalgebra of  𝐽𝑊(𝑀⊗𝑁).

Proof. Let 𝑀=𝑀_𝐼1⊕𝑀_𝑛.𝑎,𝑁=𝑁_𝐼1⊕𝑁_𝑛.𝑎 be the decomposition of 𝑀,𝑁 into abelian parts (𝑀_𝐼1,𝑁_𝐼1), and nonabelian parts (𝑀_𝑛.𝑎,𝑁_𝑛.𝑎). Then 𝐴=𝑀_𝐼1⊕𝐴_𝑛.𝑎 and 𝐵=𝑁_𝐼1⊕𝐵_𝑛.𝑎, where 𝐴_𝑛.𝑎=𝐴∩𝑀_𝑛.𝑎 and 𝐵_𝑛.𝑎=𝐵∩𝑁_𝑛.𝑎. Therefore, 𝐽𝑊(𝑀⊗𝑁)=𝐽𝑊(𝑀_𝐼1⊗𝑁_𝐼1)⊕𝐽𝑊(𝑀_𝐼1⊗𝑁_𝑛.𝑎)⊕𝐽𝑊(𝑀_𝑛.𝑎⊗𝑁_𝐼1)⊕𝐽𝑊(𝑀_𝑛.𝑎⊗𝑁_𝑛.𝑎),𝐽𝑊(𝐴⊗𝐵)=𝐽𝑊(𝑀_𝐼1⊗𝑁_𝐼1)⊕𝐽𝑊(𝑀_𝐼1⊗𝐵_𝑛.𝑎)⊕𝐽𝑊(𝐴_𝑛.𝑎⊗𝑁_𝐼1)⊕𝐽𝑊(𝐴_𝑛.𝑎⊗𝐵_𝑛.𝑎), (2.9) by [13, Remark  2.14]. The proof is complete, by Propositions 2.9 and 2.10.