# On the Relationship between Jordan Algebras and Their Universal Enveloping Algebras

Authors: F. B. H. Jamjoom; A. H. Al Otaibi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

The relationship between JW-algebras (resp. JC-algebras) and their universal enveloping von Neumann algebras (resp.C∗-algebras) can be described as significant and influential. Examples of numerous relationships have been established. In this article, we established a relationship between the set of split faces of the state space (resp. normal states) of a JC-algebra (resp. a JW-algebra) and the set of split faces of the state space (resp. normal states) of its universal enveloping C∗-algebra (resp. von Neumann algebra), and we tied up this relationship with the correspondence between the classes of invariant faces, closed ideals, and central projections of these Jordan algebras and of their universal enveloping algebras.


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## Body

## 1. Introduction

LetA be a C∗-algebra with a unit element denoted by I. As an important consequence of the GNS-construction, A is contained in the algebra BH of all bounded linear operators on some complex Hilbert space H ([1], 4.5.6; [2], 1.9.8). The weak closure of A in BH will be denoted by A¯w. We denote by As.a and A+ the set of all self-adjoint elements and the set of all positive elements in A, respectively. A bounded linear functional ρ:A⟶ℂ on A is said to be positive if ρx∗x≥0 for every x∈A. It is called a state if ρ is positive and ρI=1. The set of all states of A will be denoted by SA and is called the state space of A; it is convex and weak∗ compact. A von Neumann algebraℳ is a strongly (=weakly) closed C∗-subalgebra of the algebra BH of bounded linear operators on a complex Hilbert space H. Every von Neumann algebra has a unit, and it is the norm closure of the linear span of its projections ([3], 2.2.6). A bounded linear functional φ on a von Neumann algebra ℳ is said to be normal if for each bounded increasing net xi in ℳs.a with limit x, the net φxi converges to φx. It is known that the set ℳ∗ of all normal bounded linear functionals on ℳ is the predual of ℳ, that is, ℳ=ℳ∗∗ ([3], 3.6.5). If A⊆BH is a C∗-algebra, its weak closure A¯w⊆BH and its second dual A∗∗ are von Neumann algebras ([4], 7.1.11). Every bounded linear functional on A extends to a normal bounded linear functional on A¯w and A∗∗. The set of all normal states on a von Neumann algebra ℳ will be denoted by Sℳ.AJC-algebraA (with a unit element 1) is a norm (uniformly) closed Jordan subalgebra of the Jordan algebra BHs.a of all bounded self-adjoint operators on a complex Hilbert space H. The Jordan product is given by a∘b=ab+ba/2. The self-adjoint part of a C∗- algebra is a JC-algebra. A JW-algebraM⊆BHs.a is a weakly closed JC-algebra. If A⊆BHs.a is a JC-algebra, its weak closure A¯w⊆BHs.a and its second dual A∗∗ are JW-algebras ([4], 7.1.11). As in the context of C∗-algebras, every bounded linear functional on a JC-algebra A extends to a normal bounded linear functional on A¯w and A∗∗ ([5], 4.7.3). The set of all states (resp. normal states) of a JC-algebra (resp. JW-algebra) A will be denoted by SA. An element a∈A is called positive, written as a≥0, if a is of a square ([4], 3.3.3). The set of all positive elements of A is denoted by A+. A linear map φ:A⟶B between JC-algebras A and B is called a (Jordan) homomorphism if it preserves the Jordan product. A representation of a JC-algebra A is a (Jordan) homomorphism π:A⟶BHs.a, for some complex Hilbert space H. It is known that a (Jordan) homomorphism φ between JC-algebras A and B is continuous, and φA is a JC-subalgebra B ([4], 3.3.3). 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 ([6], p. 5). A multiplication operatorTa:A⟶A, a∈A on a Jordan algebra A is a linear operator given by Tab=a∘b, for all b∈A. It is clear that for all a,b∈A, Tab=Tba. Two elements a and b in this algebra are said to operator commute if TaTb=TbTa. The Jordan triple operatorUa,c:A⟶A,a,c∈A is defined by Ua,cb=abc for all b∈A, where Ua,c=TaTc+TcTa−Ta∘c. The operator Ua,a is denoted by Ua. The centerZA of A is the set of all elements of A which operator commutes with every other element of A, that is, ZA=a∈A:TaTb=TbTa,for allb∈A. A JC-algebra A⊆BHs.a is called irreducible if A acts irreducibly on H, or equivalently, A¯w is a JW-factor (i.e., its center consists of scalar multiples of the identity). A Jordan subalgebra J of a JC-algebra A is called Jordan ideal if TbA⊆J for every b∈J, and it is called a quadratic ideal if UbA⊆J whenever b∈J (see [7]).LetA be a JC-algebra. Then, there exists a complex C∗-algebra C∗M and an embedding πA:A⟶C∗A such that πAA generates C∗A, and for any Jordan homomorphism π:A⟶As.a, where A is a complex C∗-algebra, there is a (unique) complex ∗-homomorphism π˜:C∗A⟶A such that π˜∘πA=π. The composition of πA:A⟶C∗A, and the identity map from C∗M onto its opposite algebra C∗Mop induces via the universal property an involutory ∗-anti-automorphism θA on C∗A. The C∗-algebra C∗A is called the universal enveloping complexC∗-algebra ofA and θA is called the canonical ∗ anti-automorphism ofC∗A ([4], 7.1.8). Also, given a JW-algebra M, there exists a von Neumann algebra W∗M and an embedding πM:M⟶W∗M such that πMM generates W∗M, and for any normal Jordan homomorphism π:M⟶Ns.a, where N is a von Neumann algebra, there is a (unique) normal ∗-homomorphism π˜:W∗M⟶N such that π˜∘πM=π. The composition of πM:M⟶W∗M, and the identity map W∗M⟶W∗Mop induces via the universal property an involutory ∗-anti-automorphism ΦM on W∗M. The von Neumann algebra W∗M is called the universal enveloping von Neumann algebra ofM and ΦM is called the canonical ∗ anti-automorphism ofW∗M ([4], 7.1.9).IfM 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 ∗-anti-automorphism of C∗M (resp. W∗M). Usually, we will regard M as a generating Jordan subalgebra of C∗M (resp. W∗M) so that θM (resp. ΦM) fixes each point of M. If M is a JC-algebra, the real C∗-algebra R∗M=x∈C∗M:θMx=x∗ satisfies(1)R∗M∩iR∗M=0,C∗M=R∗M⊕iR∗M,and if M is a JW-algebra, the real von Neumann algebra RW∗M=x∈W∗M:ΦMx=x∗ satisfies(2)RW∗M∩iRW∗M=0,W∗M=RW∗M⊕iRW∗M.It is known that the universal envelopingC∗-algebra C∗M of a JW-algebra M can be realized as the C∗-subalgebra of W∗M generated by M so that W∗M is the weak closure C∗M¯w of C∗M ([8], Theorem 2.7), and M=W∗MsaΦ=x∈W∗M:Φx=x=x∗ when M is universally reversible ([4], Lemma 7.3.3).Given a subspaceM of a Banach space E and a subset G of its dual E∗, let M⊥=ρ∈E∗:ρx=0,x∈M and G∘=x∈E:ρx=0,ρ∈G be the annihilators of M and G, respectively. It is easy to see that M⊥ is weak∗-closed (i.e., σE∗,E-closed) in E∗ and G∘ is a norm closed subspace of E. The closed unit ball E1=x∈E:x≤1 of a normed linear space E is weak∗-dense (i.e., σE∗∗,E∗-dense) in E∗∗1, and hence E is weak∗-dense in E∗∗ ([4], 1.1.19).A faceF of a convex subset X of a vector space E over ℂ is a nonempty convex subset of X such that the conditions 1−λx1+λx2∈F,0≤λ≤1,x1,x2∈X, imply that x1,x2∈F ([1], 1.4). A face F of X is called a split face if there exists a face F′ such that X is a direct convex sum of F and F′. That is, each x∈X can be written uniquely of the form x=λy+1−λz, where 0≤λ≤1, y∈F and z∈F′. The face F′ is called the complement ofF and is uniquely determined by F. If F is a subset of a cone C⊂X, then F is a face if and only if 0≤y≤x∈F⇒y∈F ([9], p. 3).Theorem 1 (see [9], Corollary 3.41, Corollary 3.63).
Letℳ be a von Neumann algebra (resp. a C∗-algebra). Then, there is a 1-1 correspondence between the set of σ-weakly closed (resp. norm closed two sided) ideals in ℳ and the set of norm closed split faces (resp. weak∗- closed split faces) of Sℳ given by J⟶ℱ=J⊥∩Sℳ=ρ∈Sℳ:ρx∗x=0,∀x∈J and ℱ⟶J=ℱ∘=x∈ℳ:ρx∗x=0,∀ρ∈ℱ.Theorem 2 (see [10], Proposition 5.36, Corollary 5.37; [11], Theorem 2.3).
LetM be a JW-algebra (resp. a JC-algebra). Then, there is a 1-1 correspondence between the set of weakly (resp. norm) closed Jordan ideals in M and the set of split faces (resp. weak∗- closed split faces) of SM given by J⟶F=J⊥∩SM=ρ∈SM:ρx=0,∀x∈J and F⟶J=F∘=x∈M:ρx=0,∀ρ∈F.Since JC-algebras and JW-algebras are so close toC∗-algebras and von Neumann algebras, respectively (see [3, 8, 12–17]), and in view of the similarity of the correspondences given in Theorems 1 and 2 above, a desire of studying the relationship of the sets in Theorem 2 for JW-algebras (resp. a JC-algebras) and the corresponding sets in Theorem 1 for their enveloping von Neumann algebras (resp. a C∗-algebras) arises naturally. One can expect that more similarities involving other classes of these algebras can be established. The most significant outcome of this paper is the collective correspondences given in Theorem 12 and Theorem 13.Throughout this paper, we keep the assumption that ourC∗-algebras and JC-algebras have units. The reader is referred to [4, 6, 7, 13, 14, 18] 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 [1, 2, 19, 20].


## 2. Main Results

In view of the remarkable connection between a JW-algebra (resp. a JC-algebra)M and its universal enveloping von Neumann algebra W∗M (resp. C∗-algebra C∗M), we investigate the relationship between certain classes of these Jordan algebras and the corresponding classes of their universal enveloping algebras. A nice feature of our discussions is that it is conceptually simple and notably involves well-known results in the literature.For sake of completeness and clarity, we state the following known lemmas which will be used frequently.Lemma 1 (see [21], Lemma 3.6).
LetM be a universally reversible JW-algebra such that the center ZW∗M of W∗M is pointwise fixed under the ∗-anti-automorphism ΦM:W∗M⟶W∗M. Then, each nonzero σ-weakly closed ideal J of W∗M intersects M, i.e., J∩M≠0.Lemma 2 (see[22], Proposition 2.10).
LetM be a universally reversible JW-algebra with no abelian part. If M contains no weakly closed Jordan ideal isomorphic to the self-adjoint part of a von Neumann algebra, then ZM=ZW∗Msa.Lemma 3 (see [21], Lemma 3.7).
LetA be an irreducible universally reversible JC-algebra. If J is a norm closed two-sided ideal in C∗A such that J∩A=0, then J=0.Theorem 3.
LetM be a universally reversible JW-algebra such that the center ZW∗M of W∗M is pointwise fixed under the ∗-anti-automorphism ΦM:W∗M⟶W∗M. Then, there is a 1-1 correspondence ℑMw⇆JW∗Mw between the set ℑMw of weakly closed Jordan ideals in M and the set JW∗Mw of σ-weakly closed two-sided ideals in W∗M.Proof.
LetJ be a weakly closed Jordan ideal of M; then, the C∗-algebra J generated by J in is an ideal in C∗A, J≅C∗J, and J=J∩M ([11], Theorem 2). Since C∗M can realized as the C∗-subalgebra of W∗M generated by M, W∗M is the weak closure C∗M¯w of C∗M, and W∗J≅C∗J¯w≅J¯w is the weakly closed ideal in W∗M corresponding to J.
Conversely, letJ≠0 be a weakly closed two-sided ideal of W∗M, and note that since M is universally reversible M=W∗MsaΦ=x∈W∗M:Φx=x=x∗, by ([4], Proposition 7.3.3), ZW∗Ms.a=ZM. Therefore, J=M∩J≠0 is a weakly closed Jordan ideal of M by ([21], Lemma 3.6), and the weak closure J¯w of the C∗-algebra J generated by J in W∗M is J (see [11], Theorem 2).Corollary 1.
LetM be a universally reversible JW-algebra with no abelian part such that M contains no weakly closed Jordan ideal isomorphic to the self-adjoint part of a von Neumann algebra. Then, there is a 1-1 correspondence between the set ℑMw of weakly closed Jordan ideals in M and the set JWMw of σ-weakly closed two-sided ideals in W∗M.Proof.
SinceM contains no weakly closed Jordan ideal isomorphic to the self-adjoint part of a von Neumann algebra, ZW∗Ms.a=ZM by ([22], Theorem 7). The proof is completed by Theorem 3.
Applying Lemma3 and a similar argument of Theorem 3, we have the following.Theorem 4.
LetA be an irreducible universally reversible JC-algebra. Then, there is a 1-1 correspondence between the set ℑAn of norm closed Jordan ideals in A and the set JC∗An of norm closed two-sided ideals in C∗A.
Recall that ifM is a von Neumann algebra (or a JW-algebra) and F is a nonempty set of positive normal functionals on M, then the smallest projection p in M such that σp=σ for all σ in F is called the support projection or the carrier projection of F, denoted by carF. Given a projection p∈M, the set Fp=σ∈SM:σp=1 associated with p is a normed closed face of SM, called a projective face. Every norm closed face F of SM is a projective face ([9], Corollary 3.31; [10], Theorem 5.32), that is, F=Fp for some projection p∈M, and carFp=p, and F⊆SM is a norm closed split face of SM if and only if p is a central projection in M ([9], Proposition 3.40; [10], Corollary 5.35).Theorem 5.
LetM be a universally reversible JW-algebra such that the center ZW∗M of W∗M is pointwise fixed under the ∗-anti-automorphism ΦM:W∗M⟶W∗M. Then, there is a 1-1 correspondence FSMn⇆FSW∗Mn between the set FSMn of norm closed split faces of SM and the set FSW∗Mn of norm closed split faces of SW∗M.Proof.
Letℱ be a norm closed split face of SW∗M. Then, ℱ=ℱp=σ∈SW∗M:σp=1 where p=carℱ. Since ℱ is a split face, p∈ZW∗Ms.a by ([9], Proposition 3.40) which implies that p∈ZM since ZW∗Ms.a=ZM. Let G=σMp,∀σ∈ℱ; then, it is clear that G⊆Fp=ρ∈SM:ρp=1. On the other hand, if ρ∈Fp, then ρ extends to a normal state ρ∧ on W∗M, and ρ∧p=ρp=1 (see [4], Theorem 7.1.9), which implies that ρ∧∈ℱp=ℱ and ρ∈G. Hence, G=Fp, so G is a split face of SM.
Conversely, LetF be a norm closed split face of SM; then, F=Fp=ρ∈SM:ρp=1 where p=carF∈ZM=ZW∗Ms.a. Let G∧ be the set of all normal extensions ρ∧ of states ρ in F; then, ρ∧p=ρp=1,∀ρ∧∈G∧, which implies that G∧⊆ℱp=σ∈SW∗M:σp=1. Since σMp=1,∀σ∈ℱp, we see that ℱp⊆G∧. Hence, G∧=ℱp and G∧=ρ∧∈SW∗M:ρ∧M∈F is a split face of SW∗M.Remark 1.
(i)
Recall that ifM is a JW-algebra, then Uab=aba≥0 for all a∈M and b∈M+=a2:a∈M ([4], Proposition 3.3.6), that is, UaM+⊆M+ for all a∈M, and hence M+ is a proper convex cone of M (see [4], Lemma 3.3.7). Also, recall that a norm closed subspace J of a JC-algebra A is a Jordan ideal of A if and only if UaJ⊆J for all a∈A ([23], Lemma 2.4), and J is a quadratic ideal of A if and only if it is a hereditary subalgebra of A (a subalgebra B⊆A is said to be hereditary if the cone B+ is a face of the cone A+) ([23], Theorem 2.3). It is easy to see a Jordan ideal J of A is a quadratic ideal by applying the identity Ua=2Ta2+−Ta2. Hence, if J is a norm closed Jordan ideal J of a JC-algebra A, then J+ is a norm closed invariant face of A+ and vice versa (see [23], Corollary 2.5).(ii)
A faceD in the positive cone ℳ+ of a von Neumann algebra ℳ is said to be invariant if whenever x∈D, u∈ℳ is a unitary element, then uxu∗∈D, or equivalently, x∈ℳ, x∗x∈D⟹xx∗∈D (see [20], p. 83).
It is known that ifℳ is a von Neumann algebra, then there is a 1-1 correspondence between the set Jℳw of σ-weakly closed two-sided ideals of ℳ and the set Dℳw of weakly closed invariant faces of ℳ+ ([20], Corollary 3.21.2). The following theorem is the Jordan analogue of this result.Theorem 6.
LetM be a JW-algebra. Then, the mapping J⟶J+ is a 1-1 correspondence between the set ℑMw of weakly closed Jordan ideals J of M and the set DMw of weakly closed invariant faces of M+.Proof.
LetJ be a weakly closed Jordan ideal of M and let a and b be elements of M+ such that 0≤a≤b∈J+. By ([23], Lemma 2.1), there is an element c∈M+ such that a=Ub1l3c, which implies that a=b1l3cb1/3≥0 ([4], Proposition 3.3.6), that is, a∈J+, and hence J+ is a an invariant face of M+. Clearly, J+ is weakly closed since M+ is weakly closed and J+=J∩M+. Therefore, J↦J+ is an injection from the set ℑMw of weakly closed Jordan ideals J of M into the set DMw of weakly closed invariant faces of M+. Conversely, given a weakly closed invariant face H of M+, H is norm closed since the weak topology is weaker than the norm topology and UaH⊆H for all a∈M. By ([23], Corollary 2.5), the set J≡JH=a∈M:a2∈H is a norm closed Jordan ideal in M and J+=H. Since the weak closure J¯w of a Jordan ideal J of M is ideal in M ([11], p. 314), there is a unique central projection e∈M such that J¯w=eM ([4], Proposition 4.3.6; [10], Proposition 2.39). Since the multiplication operator Ta is weakly continuous for all a∈M, J¯w+=J+¯w=H¯w=H ([4], Corollary 4.1.6), and so we have H=J¯w+=eM+. Now, let c∈J¯w=eM; then, c=ea for some a∈M, and c2∈J¯w+=H. Note that c=ea=e∘a by ([10], Proposition 1.47 and Proposition 1.49), which implies that(3)c2=a∘e2=a∘e∘a∘e=Te∘ae∘a=TeTaTea=TeTea2=TeTa2e=Ta2Tee=a2∘e=ea2.
Therefore,c2=ea2∈eM+=H, which implies that c∈J, and hence J is a weakly closed Jordan ideal of M, proving that the mapping H↦a∈M:a2∈H is an injection form the set DMw of weakly closed invariant faces of M+ into the set ℑMw of weakly closed Jordan ideals of M which is clearly the inverse of the injection map J↦J+:ℑMw⟶DMw, proving the theorem.
Note that given a von Neumann algebraℳ, there is a 1-1 correspondence between the set FSℳn of norm closed split faces of Sℳ and the set Dℳw of σ-weakly closed invariant faces of ℳ+, which is an immediate result of the reciprocal bijections Jℳw⇆Dℳw between the set Jℳw of σ-weakly closed two-sided ideals of ℳ and the set Dℳw of σ-weakly closed invariant faces of ℳ+ (see [2], Corollary 3.21.2) and the mutual bijections Jℳw⇆FSℳn between the set Jℳw of σ-weakly closed ideals in ℳ and the set FSℳn of norm closed split faces of Sℳ ([9], Corollary 3.41 and Corollary 3.63).
The Jordan analogue of the above result is given in the following.Theorem 7.
LetM be a JW-algebra. Then, there is a 1-1 correspondence between the set FSℳn of norm closed split faces of SM and the set DMw of weakly closed invariant faces of M+.Proof.
The result follows by Theorem6, Theorem 2, and the commutative diagram FSMn⇆ℑMw⇆DMw.Remark 2.
LetA be a JC-algebra with positive cone A+; using the reciprocal bijections ℑAn⇆DAn between the set ℑAn of norm closed Jordan ideals I in A and the set DAn of norm closed invariant faces D of A+ ([23], Corollary 2.5) and the reciprocal bijections ℑAn⇆FSAw∗ between the set ℑAn of norm closed Jordan ideals I in A and the set FSAw∗ of w∗-closed split faces of SA (cf. Theorem 2), we have the JC-algebra analogue of Theorem 7.Theorem 8.
LetA be a JC-algebra. Then, there is a 1-1 correspondence between the set DAn of norm closed invariant faces of A+ and the set FSAw∗ of w∗-closed split faces of SA.
An application of Theorems6, 3, and ([9], Corollary 3.41 and Corollary 3.63) gives the following.Theorem 9.
LetM be a universally reversible JW-algebra such that the center ZW∗M of W∗M is pointwise fixed under the ∗-anti-automorphism ΦM:W∗M⟶W∗M. Then, there is a 1-1 correspondence between the set DMw of weakly closed invariant faces of M+ and the set DW∗Mw of σ-weakly closed invariant faces of W∗M+.Proof.
The result follows from the following commutative diagram:(4)Definition 1.
LetA be a unital C∗-algebra; a face D⊂A+ is said to be invariant if uxu∗∈D whenever x∈D and u is a unitary element in A.
Recall that ifM is a von Neumann algebra, then there is a reciprocal bijection between the set of all two-sided ideals J⊂M and the set of all invariant faces D⊂M+ ([20], Corollary 3.21.2). The C∗-algebra analogue of this result is proved in the following theorem.Theorem 10.
LetA be a C∗-algebra. Then, there is a 1-1 correspondence between the set DAn of norm closed invariant faces of A+ and the set JAn of norm closed two-sided ideals of A.Proof.
LetJ be a norm closed two-sided ideal in A. Then, J+ is a norm closed face of A+ by ([9], Theorem 3.46). To see that J+ is invariant, let x∈J+ and let u be a unitary element in A. Since J is two-sided ideal, uxu∗∈J. By ([1], Corollary 4.2.7), uxu∗∈A+, and hence uxu∗∈J+. That is, J+ is a norm closed invariant face of A+. Conversely, let D be a norm closed invariant face of A+. By ([9], Theorem 3.46), J=d∈A:d∗d∈D is a norm closed left ideal in A, and D=J+. To see that J is also a right ideal, first we show that uxu∗∈J+ whenever x∈J+ and let u be a unitary element of A. So, let x∈J+ and u be a unitary element of A. Then, x=vy for some v∈A, and y∈J+ (see [9], Proposition 4.2.9). Since J+=D and D is invariant, we have uyu∗∈J+⊂J. It follows that xu=vyu=vuu∗yu=vuu∗yu∈J, since J is a left ideal. By ([9], Theorem 4.1.7), each element in A is a finite linear combination of unitary elements in A, which implies that xz∈J for any x∈J and any z∈A. Hence, J is a right ideal.
Our next result is the JC-algebra analogue of Theorem9.Theorem 11.
LetA be an irreducible universally reversible JC-algebra. Then, there is a 1-1 correspondence between the set DAn of norm closed invariant faces of A+ and the set DC∗Mn of norm closed invariant faces of C∗M+.Proof.
By Theorem 10, we haveDC∗An⇆JC∗Mn, and from Theorem 4, we have JC∗Mn⇆ℑAn. The result follows since ℑAn⇆DAn by ([23], Corollary 2.5).Remark 3.
(i)
It is well known that everyσ-weakly closed two-sided ideal J in a von Neumann algebra M contains a unique central projection e in M such that J=eM (see ([2], Proposition 2.3.12); [19], Theorem 6.8.8; [9], Theorem 3.35 and Theorem 3.40). Obviously, given a central projection e in M, eM is a σ-weakly closed two-sided ideal in M. The uniqueness of the central projection presenting the weakly closed ideal implies that there is a mutual bijection J↔eM:JMw⇆PZM between the set JMw of σ-weakly closed two-sided ideals in M and the set PZM of central projections in M. On the other hand, since J+=eM+=eM+, by ([20], Corollary 3.21.2), there is a 1-1 correspondence between the set DMw of σ-weakly closed invariant faces of M+ and the set PZM of central projections in M.(ii)
The similar known result in the context of JW-algebras asserts that a weakly closed subalgebraJ of a JW-algebra M is a Jordan ideal if and only if it has the form cM for some central projection c∈M (necessarily unique) ([10], Proposition 2.39) (see also ([4], Proposition 4.3.6)). Hence, there is a mutual bijection J⟷eM:JMw⇆PZM between the set JMw of weakly closed Jordan ideals of M and the set PZM of central projections in M. By Theorem 6, there is a 1-1 correspondence between the set DMw of weakly closed invariant faces of M+ and the set PZM of central projections in M.
Combining the results in Theorems1–3 and 9, Corollary 1, and Remarks 1 and 3, we have the following collective overview correspondences between JW-algebras and their universal enveloping von Neumann algebras.Theorem 12.
LetM be a universally reversible JW-algebra such that the center ZW∗M of W∗M is pointwise fixed under the ∗-anti-automorphism ΦM:W∗M⟶W∗M. Then, there is a 1-1 correspondence between the following sets:(i)
The setFSMn of norm closed split faces of SM(ii)
The setFSW∗Mn of norm closed split faces of SW∗M(iii)
The setℑMw of weakly closed Jordan ideals of M(iv)
The setJW∗Mw of σ-weakly closed two-sided ideals of W∗M(v)
The setDMw of weakly closed invariant faces of M+(vi)
The setDW∗Mw of σ-weakly closed invariant faces of W∗M+(vii)
The setPZM of central projections in M(viii)
The setPZW∗M of central projections in W∗MProof.
It is immediate from following commutative diagram:(5)Remark 4.
(i)
Given aC∗-algebra A, we shall regard A as a C∗-subalgebra of its second dual A∗∗ and naturally identify the states of A with the normal states of A∗∗. Then, we identify the enveloping von Neumann algebra (i.e., the σ-weak closure A¯w≅πA¯w⊆BH of A, where π:A⟶BH is the universal representation of A on the Hilbert space H) with A∗∗ (see [2], Theorem 3.2.4, p.122; [9], Corollary 2.1.27). If J is a norm closed two-sided ideal in A, then the weak closure J¯w of J in A∗∗ is a two-sided ideal, and hence J¯w=eA∗∗ for a unique central projection e∈A∗∗ and J=A∩J¯w. Conversely, if J is σ-weakly closed two-sided ideal in A∗∗, and since A is weakly dense in A∗∗, ℐ=A∩J is a norm closed two-sided ideal in A such that ℐ¯w=J. Note that the map J↦J¯w is a bijection from the set JAn of norm closed two ideals of A and the set JA∗∗w of weakly closed ideals in A∗∗. Since the map J↔eA∗∗:JA∗∗w⇆PZA∗∗ is a mutual bijection between the set JA∗∗w of weakly closed two-sided ideals of A∗∗ and the set PZA∗∗ of central projections in A∗∗, we have a 1-1 correspondence JAn⇆PZA∗∗ between the set JA of norm closed two-sided ideals J in A and the set PZA∗∗ of central projections in A∗∗.(ii)
A similar argument in the context of JC-algebras leads to a mutual correspondenceℑAn⇆PZA∗∗ between the set ℑAn of norm closed Jordan ideals in a JC-algebra A and the set PZA∗∗ of central projections in its second dual A∗∗ by using the fact that the weak closure J¯w in A∗∗ of a norm closed Jordan ideal J in A is an ideal, J¯w=eA∗∗, for some central projection e∈A∗∗ and J=A∩J¯w.(iii)
SinceJ+¯w=J¯w+=eA∗∗+=eA∗∗+, then by Theorem 11 and (ii) above, there is a 1-1 correspondence DAn⇆PZA∗∗ between the set DAn of norm closed invariant faces of A+ and the set PZA∗∗ of central projections in A∗∗.
Collecting the results in Theorems1, 2, 8, 10, and 11 and Remarks 2 and 4, we have the following combined summary of correspondences between JC-algebras and their universal enveloping C∗-algebras.Theorem 13.
LetA be an irreducible universally reversible JC-algebra. Then, there is a 1-1 correspondence between the following sets:(i)
The setFSAw∗ of weak∗-closed split faces of SA(ii)
The setFC∗Aw∗ of weak∗- closed split faces of SC∗A(iii)
The setℑAn of norm closed Jordan ideals in A(iv)
The setJC∗An of norm closed two-sided ideals in C∗A(v)
The setDAn of norm closed invariant faces of A+(vi)
The setDC∗Mn of norm closed invariant faces of C∗M+(vii)
The setPZA∗∗ of central projections in A∗∗(viii)
The setPZC∗A∗∗ of central projections in C∗A∗∗Proof.
It is immediate from the following commutative diagram:(6)



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