Maximal Abelian JW -Algebras

Volume 2009, Article ID 621386,7pages doi:10.1155/2009/621386

Research Article

On the Tensor Products of

Maximal Abelian JW -Algebras

F. B. H. Jamjoom

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Correspondence should be addressed to F. B. H. Jamjoom,[email protected] Received 24 April 2009; Accepted 7 June 2009

Recommended by Jie Xiao

It is well known in the work of Kadison and Ringrose1983that ifAandBare maximal abelian von Neumann subalgebras of von Neumann algebras Mand N, respectively, then A⊗B is a maximal abelian von Neumann subalgebra ofM⊗N. It is then natural to ask whether a similar result holds in the context ofJW-algebras and theJW-tensor product. Guided to some extent by the close relationship between aJW-algebra M and its universal enveloping von Neumann algebra W^∗M, we seek in this article to investigate the answer to this question.

Copyrightq2009 F. B. H. Jamjoom. 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.

1. Introduction

AJC-algebra Ais a normuniformlyclosed Jordan subalgebra of the Jordan algebraBH_s.a of all bounded self adjoint operators on a Hilbert spaceH.The Jordan product is given by a◦ b abba/2. A subspaceI of a JC-algebra A is called a Jordan ideal if a◦ b ∈ I for every a ∈ Aand everyb ∈ I. AJC-algebra is said to be simple if it has no nontrivial norm closed Jordan ideals. AJW-algebraM ⊆ BH_s.ais a weakly closedJC-algebra. If M is aJC-algebraresp.,JW-algebra, letC^∗M resp.,W^∗Mbe 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 regardM as a generating Jordan subalgebra ofC^∗MandW^∗Mso thatθMandΦMfix each point ofM.

The realC^∗-algebraR^∗M {x∈C^∗M:θMx x^∗}satisfies

R^∗M∩iR^∗M 0, C^∗M R^∗M⊕iR^∗M, 1.1 and the real von Neumann algebraRW^∗M {x∈W^∗M:ΦMx x^∗}satisfies

RW^∗M∩iRW^∗M 0, W^∗M RW^∗M⊕iRW^∗M. 1.2

The reader is refered to 1–5for a detailed account of the theory of JC-algebras andJW- algebras. The relevant background on the theory ofC^∗-algebras and von Neumann algebras can be found in6–8.

A projectioneof aJW-algebraMis said to be abelian ifeMeis associative, and it is called minimal if it is nonzero and contains no other nonzero projections ofM, or equivalently, eis minimal if and only ifeMeRe. AJW-factor is aJW-algebra with trivial centre; a Type I JW-factor is aJW-factor which contains a minimal projection. AJW-algebra is said to be of TypeInif there is a family of abelian projectionseα_α∈Jsuch that the central supportcMeα ofeαinMequals the unit 1_Mof M,

α∈Jeα1_Mand cardJ nsee1, Section 5.3. A spin factorV H⊕R1V is a real Jordan algebra with identity 1V, whereHis a real Hilbert space of dimension at least two. The Jordan product onV is defined by

aλ1V◦ bμ1V

μaλb

a, bλμ

1_V, a, b∈V, λ, μ∈R, 1.3

and the norm onVis given by

aλ1V a, a^1/2|λ|. 1.4 A spin factorVis 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 reflexiveJW-algebra and constitutes the TypeI2 JW-factorsee2, 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 aJC-algebra Ais 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 πA is reversible for every representation π of A 2, page 5. The only universally reversible spin factors are V2 M2R_s.a and V3 M2C_s.a2, Theorem 2.1. AJC-algebraA is universally reversible if and only if it has no spin factor representations other than ontoV2andV32, Theorems 2.2.

EveryJW-algebra without a direct summand of TypeI2 is universally reversible1, 5.1.5, 5.3.5, 6.2.3.

Two elementsaandbof aJC-algebraAare said to operator commute ifTaTb TbTa, whereTa :A → Ais the multiplication operator defined byTax a◦x, for allx ∈A. A JW-algebraMis called associative if all its elements operators commute. AJW-subalgebraA of aJW-algebraMis called maximal associative if it is not contained in any larger associative JW-subalgebra of M. IfAis aJW-subalgebra of aJW-algebraM ⊆ BH_s.aandAis the set of all elements of BH_s.a which operator commutes with all elements of A, thenAis a maximal associativeJW-subalgebra ofMif and only ifA A∩M. Indeed, since Ais associative,A⊆A∩MandAtogether with any element ofA∩Mgenerates an associative JW-subalgebra ofMwhich implies thatA∩M⊆AsinceAis maximal abelian. In particular, if A ⊆ BH_s.a is an associativeJW-algebra, thenA is maximal associative if and only if AA.

This article aims to study the relationship between the maximality of an associative JW-subalgebraBof aJW-algebraMand that ofW^∗BinW^∗M. We give a counterexample which rules out the establishing of a result in the theory ofJW-tensor products analog to that

given in6, Theorem 11.2.18for von Neumann tensor productscf.Example 2.2. Then we prove that a Jordan analog of Theorem 11.2.18 in6can be established in some particular cases.

Theorem 1.1see9, Proposition 1. LetM⊆BH_s.abe aJW-algebra, and leta, b∈M. Then the following are equivalent:

iabba;

iiTaTb TbTa; iiia²◦baba.

That is, a and b operators commute if and only if they commute under ordinary operator multiplication.

Definition 1.2. LetMand Nbe a pair of JW-algebras canonically embedded in their respective universal enveloping von Neumann algebrasW^∗MandW^∗N.Then theJW- tensor productJWM⊗NofM and N is the JW-algebra generated by M ⊗ N in W^∗M⊗W^∗N. The reader is referred to10for the properties of theJW-tensor product ofJW-algebras.

Theorem 1.3 see 10, Theorem 2.9. LetMandNbeJW- algebras. IfJWM⊗N is universally reversible, then

W^∗ JW

M⊗N

W^∗M⊗W^∗N. 1.5

2. Maximal Abelian JW -algebras

LetAandB be maximal abelian von Neumann subalgebras of von Neumann algebrasM andN, respectively, thenA⊗Bis a maximal abelian von Neumann algebra ofM⊗Nsee 6, 11.2.18. In Example 2.2, we show that the Jordan analog of this result, in the context ofJW-algebras and theJW-tensor product, is not true in general. However, it is shown in Theorem 2.11that the result does hold in special circumstances.

Remark 2.1. i Note that anyJW-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 ⊆ BH_s.a. If 1_A/1_V,then 1A is the only projection inA, since every projection in V is minimal, and hence dimA1. If 1A 1V, then any family of orthogonal central projections ofAcontains at most two projections. Indeed ife1e2e3 1A, ei∈ZA, i1,2,3, thene2e3≤1A−e1. Since 1_A −e1 is a minimal projection, we see that one ofei, i 1,2,3 must be zero. It is clear that ifAis a factor, then it is of TypeI2, and hence it is a spin factor. iiRecall that W^∗V M2C⊕M2C, whereV is the 4-dimensional spin factorM2C_s.a 1, 6.2.1:

M2C span 1 0

0 1

, 1 0

0 −1

, 0 1

1 0

, 0 i

−i 0

C⊗

RM2R, 2.1 which is an 8-dimensional realC^∗-algebra.

Example 2.2. LetAbe a maximal abelianJW-subalgebra ofV M2C_s.a.ThenJWA⊗A is not a maximal abelian subalgebra ofJWV ⊗V.

Proof. By the above remark, dimA2, and henceAReRffor some minimal projections e, f. Therefore,

JW

A⊗A A⊗

RARe⊗e⊕R e⊗f

⊕R f⊗e

⊕R f⊗f

, 2.2

and hence dimJWA⊗A 4, since dimA⊗

RAdimA·dimAsee11, Corollary 7.5. On the other hand,JWV ⊗Vis universally reversible, by10, Proposition 2.7which implies that

JW V ⊗V

RW^∗

JWV ⊗V

s.a

RW^∗V⊗

RRW^∗V

s.a

M2C⊗

RM2C

s.a

M2²C_s.a⊕M2²C_s.a,

2.3

since RW^∗M2C_s.a M2C 3, page 385. It can be seen that a maximal abelian JW- subalgebra ofJWV ⊗Vis of dimension 8, which implies thatJWA⊗Ais not maximal abelian inJWV ⊗V.

Remark 2.3. Note that if B is an associative JW-subalgebra of a JW-algebra M such that W^∗Bis a maximal abelian subalgebra of W^∗M, then Bis a maximal associativeJW- subalgebra ofM, sinceBW^∗B∩M.

Lemma 2.4. LetBbe an associativeJW-subalgebra of aJW-algebraM. 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,Bhas no representation into a spin factor of the formV4n1 and is, therefore, universally reversible. It follows from3, page 383that

BRW^∗B_s.a. 2.5

Therefore, by3, Corollary 3.2,RW^∗Bis isomorphic to the weak^∗-closureRBof the real C^∗-subalgebraRBofW^∗Mgenerated byB, 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 toN ⊕ N^◦, whereN^◦is the opposite algebra ofN2, 7.4.15. A realC^∗-algebraA

can be realized as a complexC^∗-algebra if there is aC^∗-algebra isomorphismφ:B → Aof a complexC^∗-algebraBontoA. In this case, the real linear isometryjonAdefined, for eacha inB,by

jφa φia 2.6

is such thatj²and−idAcoincide.

Lemma 2.5. LetBbe a maximal associativeJW-subalgebra of aJW-algebra M.Suppose thatM is isomorphic to the self adjoint partNs.a of a von Neumann algebraNand has no one-dimensional representations. ThenW^∗Bis not a maximal abelian on Neumann subalgebra of W^∗M.

Proof. IdentifyingMwithNs.a,Bis a von Neumann subalgebra of bothNand N^◦, and hence, the von Neumann subalgebraB⊕BofN ⊕ N^◦ ∼W^∗Mis abelian and contains W^∗B B∼ B⊕ {0}, which implies thatW^∗Bis not maximal abelian inW^∗M.

Lemma 2.6. LetBbe a maximal associativeJW-subalgebra of aJW-algebraM. If RW^∗Mis∗- isomorphic to a complexC^∗-algebra, then W^∗Bis not a maximal abelian von Neumann subalgebra of W^∗M.

Proof. SinceC^∗Mis the complexC^∗-algebraMgenerated byMinW^∗M 12, Theorem 2.7,RW^∗Mis the weak^∗-closure ofR^∗MinW^∗M. Therefore,R^∗Mis a complexC^∗- algebra, which implies that C^∗M I ⊕ΦMIfor some norm closed idealI ofC^∗M isomorphic toR^∗M 13, Lemma 1, so thatW^∗M J ⊕ΦMJ, whereJis the weak^∗- closure Iof IinW^∗M. Hence, Jis isomorphic to RW^∗M. Let φ be the isomorphism ofJontoRW^∗M, and letj be the corresponding real linear operator onRW^∗M, defind above. Then, usingLemma 2.4, there exists an isomorphismπ from the W^∗-algebraB⊕iB intoRW^∗Msuch that, for elementsb1andb2inB,

πb1ib2 b1jb2. 2.7 It follows thatφ⁻¹◦πandΦM◦φ⁻¹◦πare^∗-isomorphisms ofB B⊕iBintoJandΦMJ, respectively. Since a^∗-isomorphism betweenC^∗-algebras is an isometry7, Corollary 1.5.4, we may identifyBwithφ⁻¹◦πBandΦM◦φ⁻¹◦πB. It follows thatB⊕Bis an abelian von Neumann subalgebra ofW^∗M, proving thatW^∗B B ∼ B⊕ {0}is not maximal abelian inW^∗M.

Proposition 2.7. LetMbe a universally reversible JW-algebra not isomorphic to the self adjoint part of a von Neumann algebra and without direct summands of typeI1. IfBis a maximal associative subalgebra ofM, thenW^∗Bis a maximal abelian von Neumann subalgebra of W^∗M.

Proof. ByLemma 2.4,W^∗B B B⊕iB → W^∗M. IfW^∗Bis not maximal abelian in W^∗M, there exists an elementz ∈ W^∗M RW^∗M⊕iRW^∗M,z /∈W^∗Bsuch that z together withW^∗B generate an abelian von Neumann subalgebraY ⊃

/ W^∗B ⊇ B of W^∗M⊇ M. Letz xiy,x, y ∈W^∗M_s.a. Sincez /∈W^∗B, then eitherxoryor both does not belong toW^∗B. Suppose thatx /∈W^∗B, sinceW^∗M RW^∗M⊕iRW^∗M, thenx aib, for somea, b ∈ RW^∗M. Then eitheraorbor bothdoes not belong to

W^∗B. Sincex∈W^∗M_s.a, we haveaa^∗, andb−b^∗, and soa∈MRW^∗M_s.a, since Mis a universally reversible3, page 383. Therefore, amust be the zero element, since it obviously commutes with all elements inB. On the other hand,b²−bb^∗∈RW^∗M_s.aM.

Sincebu ub for allu ∈ W^∗B,b²u bub ub² for allu ∈ B, and sob² anduoperators commute relative to the Jordan product in B 9, Proposition 1. Henceb² ∈ B ⊆ W^∗B, sinceBis a maximal associative subalgebra ofM, which implies thatb∈W^∗B. Therefore, xib∈W^∗B, a contradiction. This proves the result.

Lemma 2.8. LetMbe a universally reversibleJW-algebra not isomorphic to the self adjoint part of a von Neumann algebra. IfBis a maximal associative subalgebra ofM, then W^∗Bis a maximal abelian von Neumann subalgebra ofW^∗M.

Proof. SplittingMMI1⊕Mn.aas the direct sum of aJW-algebraMI1of typeI1the abelian partand a JW-algebraMn.a without direct summands of typeI1the nonabelian part. It is clear thatB ⊇ MI1, Bn.a B∩Mn.ais a maximal associative subalgebra ofMn.a andB MI1 ⊕Bn.a. By Proposition 2.7,W^∗Bn.ais 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 ofW^∗M, sinceW^∗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 typeI1. ThenJWB1 ⊗B2is a maximal associativeJW-subalgebra ofJWM1 ⊗M2.

Proof. Note first thatW^∗B1⊗W^∗B2is a von Neumann∗-subalgebra ofW^∗M1⊗W^∗M2 8, Theorem 11.2.10, and JWB1 ⊗B2is a JW-subalgebra of JWM1 ⊗M2, sinceB1 ⊗ B2 ⊆ M1 ⊗ M2. By Proposition 5.2, W^∗Bi is maximal abelian in W^∗Mi, and hence, W^∗B1⊗W^∗B2is maximal abelian inW^∗M1⊗W^∗M2 8, Corollary 11.2.18 and10, Theorem 2.9. The result is now obvious, sinceW^∗JWB1 ⊗B2 W^∗B1⊗W^∗B2,and W^∗JWM1 ⊗M2 W^∗M1⊗W^∗M2 10, Theorem 2.9.

Proposition 2.10. LetNbe an associativeJW-algebra, and letM be a universally reversibleJW- algebra not isomorphic to the self adjoint part of a von Neumann algebra and without direct summands of typeI1. IfBis a maximal associative subalgebra ofM, thenJWN⊗Bis a maximal associative JW-subalgebra ofJWN⊗M.

Proof. LetMMI1⊕Mn.abe the decomposition ofMinto abelian partMI1and nonabelian partMn.a. ThenB MI1⊕Bn.a, whereBn.a B∩Mn.ais obviously a maximal associative subalgebra ofMn.a. By10, 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

. 2.8

It is clear now that JWN⊗B is a maximal associative JW-subalgebra of JWN ⊗ M, since JWN ⊗MI1 is obviously associative, and JWN ⊗Bn.a is maximal inJWN ⊗Mn.a, byProposition 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. IfA and Bare maximal associative subalgebra ofMand N, respectively, thenJWA⊗Bis a maximal associativeJW-subalgebra ofJWM⊗N.

Proof. LetMMI1⊕Mn.a, N NI1⊕Nn.abe the decomposition ofM, Ninto abelian parts MI1, NI1, and nonabelian partsMn.a, Nn.a. ThenAMI1⊕An.aandBNI1⊕Bn.a, where An.aA∩Mn.aandBn.aB∩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

,

2.9

by13, Remark 2.14. The proof is complete, by Propositions2.9and2.10.

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