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-algebraA 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-algebraM⊆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 θM (resp., ΦM) 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 θM and ΦM fix each point of M. The real C*-algebra R*(M)={x∈C*(M):θM(x)=x*} satisfies
R*(M)∩iR*(M)=0,C*(M)=R*(M)⊕iR*(M),
and the real von Neumann algebra RW*(M)={x∈W*(M):ΦM(x)=x*} satisfies
RW*(M)∩iRW*(M)=0,W*(M)=RW*(M)⊕iRW*(M).
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 TypeIn if there is a family of abelian projections (eα)α∈J such that the central support cM(eα) of eα in M equals the unit 1M of M, ∑α∈Jeα=1M and card J=n (see [1, Section 5.3]). A spin factor V=H⊕ℝ1V is a real Jordan algebra with identity 1V, where H is a real Hilbert space of dimension at least two. The Jordan product on V is defined by
(a+λ1V)∘(b+μ1V)=(μa+λb)+(〈a,b〉+λμ)1V,a,b∈V,λ,μ∈ℝ,
and the norm on V is given by
∥a+λ1V∥=〈a,a〉1/2+|λ|.
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⋯an+anan-1⋯a1∈A whenever a1,a2,…,an∈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 TaTb=TbTa, where Ta:A→A is the multiplication operator defined by Ta(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 [<xref ref-type="bibr" rid="B13">9</xref>, Proposition 1]).

Let M⊆B(H)s.a be a JW-algebra, and let a,b∈M. Then the following are equivalent:

ab=ba;

TaTb=TbTa;

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 [<xref ref-type="bibr" rid="B8">10</xref>,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).

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 1A≠1V,then 1A is the only projection in A, since every projection in V is minimal, and hence dimA=1. If 1A=1V, then any family of orthogonal central projections of A contains at most two projections. Indeed if e1+e2+e3=1A,ei∈Z(A),i=1,2,3, then e2+e3≤1A-e1. Since 1A-e1 is a minimal projection, we see that one of ei,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{(1001),(100-1),(0110),(0i-i0)}=ℂ⊗ℝM2(ℝ),
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),
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)⊗ℝ̅RW*(V))s.a=(M2(ℂ)⊗ℝM2(ℂ))s.a=M22(ℂ)s.a⊕M22(ℂ)s.a,
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,
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 V4n+1 and is, therefore, universally reversible. It follows from [3, page 383] that
B=RW*(B)s.a.
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 𝒩s.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)
is such that j2 and -id𝔄 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 𝒩s.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 𝒩s.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⊕ΦM(ℐ) for some norm closed ideal ℐ of C*(M) isomorphic to R*(M) [13, Lemma 1], so that W*(M)=𝒥⊕ΦM(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.
It follows that ϕ-1∘π and ΦM∘ϕ-1∘π are *-isomorphisms of [B]¯=B⊕iB into 𝒥 and ΦM(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 ΦM∘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=MI1⊕Mn.a as the direct sum of a JW-algebra MI1 of type I1 (the abelian part) and a JW-algebra Mn.a without direct summands of type I1(the nonabelian part). It is clear that B⊇MI1,Bn.a=B∩Mn.a is a maximal associative subalgebra of Mn.a and B=MI1⊕Bn.a. By Proposition 2.7, W*(Bn.a) is a maximal abelian von Neumann subalgebra of W*(Mn.a), and hence W*(B)=W*(MI1)⊕W*(Bn.a) is a maximal abelian von Neumann subalgebra of W*(M), since W*(M)=W*(MI1)⊕W*(Mn.a) [12, Lemma 2.6].

Proposition 2.9.

Let Bi be a maximal associative subalgebra of a JW-algebra Mi,i=1,2, and suppose that Mi 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*(Bi) is maximal abelian in W*(Mi), 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=MI1⊕Mn.a be the decomposition of M into abelian part MI1 and nonabelian part Mn.a. Then B=MI1⊕Bn.a, where Bn.a=B∩Mn.a is obviously a maximal associative subalgebra of Mn.a. By [10, Remark 2.14],
JW(N⊗̅M)=JW(N⊗̅(MI1⊕Mn.a))=JW(A⊗̅MI1)⊕JW(A⊗̅Mn.a),JW(N⊗̅B)=JW(N⊗̅(MI1⊕Bn.a))=JW(N⊗̅MI1)⊕JW(N⊗̅Bn.a).
It is clear now that JW(N⊗̅B) is a maximal associative JW-subalgebra of JW(N⊗̅M), since JW(N⊗̅MI1) is obviously associative, and JW(N⊗̅Bn.a) is maximal in JW(N⊗̅Mn.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=MI1⊕Mn.a,N=NI1⊕Nn.a be the decomposition of M,N into abelian parts (MI1,NI1), and nonabelian parts (Mn.a,Nn.a). Then A=MI1⊕An.a and B=NI1⊕Bn.a, where An.a=A∩Mn.a and Bn.a=B∩Nn.a. Therefore,
JW(M⊗̅N)=JW(MI1⊗̅NI1)⊕JW(MI1⊗̅Nn.a)⊕JW(Mn.a⊗̅NI1)⊕JW(Mn.a⊗̅Nn.a),JW(A⊗̅B)=JW(MI1⊗̅NI1)⊕JW(MI1⊗̅Bn.a)⊕JW(An.a⊗̅NI1)⊕JW(An.a⊗̅Bn.a),
by [13, Remark 2.14]. The proof is complete, by Propositions 2.9 and 2.10.

Hanche-OlsenH.On the structure and tensor products of JC-algebrasHanche-OlsenH.StørmerE.BunceL. J.WrightJ. D. M.Introduction to the K-theory of Jordan C∗-algebrasStørmerE.Jordan algebras of type IWrightJ. D. M.Jordan C∗-algebrasKadisonR. V.RingroseJ. R.TakesakiM.KadisonR. V.RingroseJ. R.ToppingD. M.Jordan algebras of self-adjoint operatorsJamjoomF. B. H.On the tensor products of JW-algebrasCohnP. M.JamjoomF. B. H.The connection between the universal enveloping C∗-algebra and the universal enveloping von Neumann algebra of a JW-algebraJamjoomF. B. H.On the tensor products of simple JC-algebras