On the Relationship between Jordan Algebras and Their Universal Enveloping Algebras

Jamjoom, F. B. H.; Al Otaibi, A. H.

doi:https://doi.org/10.1155/2020/6976084

International Journal of Mathematics and Mathematical Sciences

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Research Article | Open Access

Volume 2020 | Article ID 6976084 | https://doi.org/10.1155/2020/6976084

On the Relationship between Jordan Algebras and Their Universal Enveloping Algebras

F. B. H. Jamjoom¹and A. H. Al Otaibi¹

Academic Editor: Irena Lasiecka

Received08 May 2020

Revised30 Aug 2020

Accepted13 Sept 2020

Published23 Oct 2020

Abstract

The relationship between JW-algebras (resp. JC-algebras) and their universal enveloping von Neumann algebras (resp. -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 -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.

1. Introduction

Let be a -algebra with a unit element denoted by . As an important consequence of the GNS-construction, is contained in the algebra of all bounded linear operators on some complex Hilbert space ([1], 4.5.6; [2], 1.9.8). The weak closure of in will be denoted by . We denote by and the set of all self-adjoint elements and the set of all positive elements in , respectively. A bounded linear functional on is said to be positive if for every . It is called a state if is positive and . The set of all states of will be denoted by and is called the state space of ; it is convex and weak∗ compact. A von Neumann algebra is a strongly (=weakly) closed -subalgebra of the algebra of bounded linear operators on a complex Hilbert space . 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 on a von Neumann algebra is said to be normal if for each bounded increasing net in with limit , the net converges to . It is known that the set of all normal bounded linear functionals on is the predual of , that is, ([3], 3.6.5). If is a -algebra, its weak closure and its second dual are von Neumann algebras ([4], 7.1.11). Every bounded linear functional on extends to a normal bounded linear functional on and . The set of all normal states on a von Neumann algebra will be denoted by .

A JC-algebra (with a unit element ) is a norm (uniformly) closed Jordan subalgebra of the Jordan algebra of all bounded self-adjoint operators on a complex Hilbert space . The Jordan product is given by . The self-adjoint part of a - algebra is a JC-algebra. A JW-algebra is a weakly closed JC-algebra. If is a JC-algebra, its weak closure and its second dual are JW-algebras ([4], 7.1.11). As in the context of -algebras, every bounded linear functional on a JC-algebra extends to a normal bounded linear functional on and ([5], 4.7.3). The set of all states (resp. normal states) of a JC-algebra (resp. JW-algebra) will be denoted by . An element is called positive, written as , if is of a square ([4], 3.3.3). The set of all positive elements of is denoted by . A linear map between JC-algebras and is called a (Jordan) homomorphism if it preserves the Jordan product. A representation of a JC-algebra is a (Jordan) homomorphism , for some complex Hilbert space . It is known that a (Jordan) homomorphism between JC-algebras and is continuous, and is a JC-subalgebra ([4], 3.3.3). A JC-algebra is said to be reversible if whenever and is said to be universally reversible if is reversible for every representation of ([6], p. 5). A multiplication operator, on a Jordan algebra is a linear operator given by , for all . It is clear that for all , . Two elements and in this algebra are said to operator commute if . The Jordan triple operator is defined by for all , where . The operator is denoted by . The center of is the set of all elements of which operator commutes with every other element of , that is, . A JC-algebra is called irreducible if acts irreducibly on , or equivalently, is a JW-factor (i.e., its center consists of scalar multiples of the identity). A Jordan subalgebra of a JC-algebra is called Jordan ideal if for every , and it is called a quadratic ideal if whenever (see [7]).

Let be a JC-algebra. Then, there exists a complex -algebra and an embedding such that generates , and for any Jordan homomorphism , where is a complex -algebra, there is a (unique) complex ∗-homomorphism such that . The composition of , and the identity map from onto its opposite algebra induces via the universal property an involutory ∗-anti-automorphism on . The -algebra is called the universal enveloping complex-algebra of and is called the canonical ∗ anti-automorphism of ([4], 7.1.8). Also, given a JW-algebra , there exists a von Neumann algebra and an embedding such that generates , and for any normal Jordan homomorphism , where is a von Neumann algebra, there is a (unique) normal ∗-homomorphism such that . The composition of , and the identity map induces via the universal property an involutory ∗-anti-automorphism on . The von Neumann algebra is called the universal enveloping von Neumann algebra of and is called the canonical ∗ anti-automorphism of ([4], 7.1.9).

If is a JC-algebra (resp. JW-algebra), let (resp. ) be the universal enveloping -algebra (resp. von Neumann algebra) of , and let (resp. ) be the canonical involutive ∗-anti-automorphism of (resp. ). Usually, we will regard as a generating Jordan subalgebra of (resp. ) so that (resp. ) fixes each point of . If is a JC-algebra, the real -algebra satisfiesand if is a JW-algebra, the real von Neumann algebra satisfies

It is known that the universal enveloping -algebra of a JW-algebra can be realized as the -subalgebra of generated by so that is the weak closure of ([8], Theorem 2.7), and when is universally reversible ([4], Lemma 7.3.3).

Given a subspace of a Banach space and a subset of its dual , let and be the annihilators of and , respectively. It is easy to see that is weak∗-closed (i.e., -closed) in and is a norm closed subspace of . The closed unit ball of a normed linear space is weak^∗-dense (i.e., -dense) in , and hence is weak^∗-dense in ([4], 1.1.19).

A face of a convex subset of a vector space over is a nonempty convex subset of X such that the conditions , imply that ([1], 1.4). A face of is called a split face if there exists a face such that is a direct convex sum of and . That is, each can be written uniquely of the form , where , and . The face is called the complement of and is uniquely determined by . If is a subset of a cone , then is a face if and only if ([9], p. 3).

Theorem 1 (see [9], Corollary 3.41, Corollary 3.63). Let be a von Neumann algebra (resp. a -algebra). Then, there is a 1-1 correspondence between the set of -weakly closed (resp. norm closed two sided) ideals in and the set of norm closed split faces (resp. weak∗- closed split faces) of given by and .

Theorem 2 (see [10], Proposition 5.36, Corollary 5.37; [11], Theorem 2.3). Let 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 and the set of split faces (resp. weak∗- closed split faces) of given by and .

Since JC-algebras and JW-algebras are so close to -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 -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 our -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 -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) and its universal enveloping von Neumann algebra (resp. -algebra ), 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). Let be a universally reversible JW-algebra such that the center of is pointwise fixed under the -anti-automorphism . Then, each nonzero -weakly closed ideal of intersects , i.e., .

Lemma 2 (see[22], Proposition 2.10). Let be a universally reversible JW-algebra with no abelian part. If contains no weakly closed Jordan ideal isomorphic to the self-adjoint part of a von Neumann algebra, then .

Lemma 3 (see [21], Lemma 3.7). Let be an irreducible universally reversible JC-algebra. If is a norm closed two-sided ideal in such that , then .

Theorem 3. Let be a universally reversible JW-algebra such that the center of is pointwise fixed under the -anti-automorphism . Then, there is a 1-1 correspondence between the set of weakly closed Jordan ideals in and the set of -weakly closed two-sided ideals in .

Proof. Let be a weakly closed Jordan ideal of ; then, the -algebra generated by in is an ideal in , , and ([11], Theorem 2). Since can realized as the -subalgebra of generated by , is the weak closure of , and is the weakly closed ideal in corresponding to .
Conversely, let be a weakly closed two-sided ideal of , and note that since is universally reversible , by ([4], Proposition 7.3.3), . Therefore, is a weakly closed Jordan ideal of by ([21], Lemma 3.6), and the weak closure of the -algebra generated by in is (see [11], Theorem 2).

Corollary 1. Let be a universally reversible JW-algebra with no abelian part such that 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 of weakly closed Jordan ideals in and the set of -weakly closed two-sided ideals in .

Proof. Since contains no weakly closed Jordan ideal isomorphic to the self-adjoint part of a von Neumann algebra, by ([22], Theorem 7). The proof is completed by Theorem 3.
Applying Lemma 3 and a similar argument of Theorem 3, we have the following.

Theorem 4. Let be an irreducible universally reversible JC-algebra. Then, there is a 1-1 correspondence between the set of norm closed Jordan ideals in and the set of norm closed two-sided ideals in .
Recall that if is a von Neumann algebra (or a JW-algebra) and is a nonempty set of positive normal functionals on , then the smallest projection in such that for all in is called the support projection or the carrier projection of , denoted by . Given a projection , the set associated with is a normed closed face of , called a projective face. Every norm closed face of is a projective face ([9], Corollary 3.31; [10], Theorem 5.32), that is, for some projection , and , and is a norm closed split face of if and only if is a central projection in ([9], Proposition 3.40; [10], Corollary 5.35).

Theorem 5. Let be a universally reversible JW-algebra such that the center of is pointwise fixed under the -anti-automorphism . Then, there is a 1-1 correspondence between the set of norm closed split faces of and the set of norm closed split faces of .

Proof. Let be a norm closed split face of . Then, where . Since is a split face, by ([9], Proposition 3.40) which implies that since . Let ; then, it is clear that . On the other hand, if , then extends to a normal state on , and (see [4], Theorem 7.1.9), which implies that and . Hence, , so is a split face of .
Conversely, Let be a norm closed split face of ; then, where . Let be the set of all normal extensions of states in ; then, , which implies that . Since , we see that . Hence, and is a split face of .

Remark 1. (i)Recall that if is a JW-algebra, then for all and ([4], Proposition 3.3.6), that is, for all , and hence is a proper convex cone of (see [4], Lemma 3.3.7). Also, recall that a norm closed subspace of a JC-algebra is a Jordan ideal of if and only if for all ([23], Lemma 2.4), and is a quadratic ideal of if and only if it is a hereditary subalgebra of (a subalgebra is said to be hereditary if the cone is a face of the cone ) ([23], Theorem 2.3). It is easy to see a Jordan ideal of is a quadratic ideal by applying the identity . Hence, if is a norm closed Jordan ideal of a JC-algebra , then is a norm closed invariant face of and vice versa (see [23], Corollary 2.5).(ii)A face in the positive cone of a von Neumann algebra is said to be invariant if whenever , is a unitary element, then , or equivalently, , (see [20], p. 83).It is known that if is a von Neumann algebra, then there is a 1-1 correspondence between the set of -weakly closed two-sided ideals of and the set of weakly closed invariant faces of ([20], Corollary 3.21.2). The following theorem is the Jordan analogue of this result.

Theorem 6. Let be a JW-algebra. Then, the mapping is a 1-1 correspondence between the set of weakly closed Jordan ideals of and the set of weakly closed invariant faces of .

Proof. Let be a weakly closed Jordan ideal of and let and be elements of such that . By ([23], Lemma 2.1), there is an element such that , which implies that ([4], Proposition 3.3.6), that is, , and hence is a an invariant face of . Clearly, is weakly closed since is weakly closed and . Therefore, is an injection from the set of weakly closed Jordan ideals of into the set of weakly closed invariant faces of . Conversely, given a weakly closed invariant face of , is norm closed since the weak topology is weaker than the norm topology and for all . By ([23], Corollary 2.5), the set is a norm closed Jordan ideal in and . Since the weak closure of a Jordan ideal of is ideal in ([11], p. 314), there is a unique central projection such that ([4], Proposition 4.3.6; [10], Proposition 2.39). Since the multiplication operator is weakly continuous for all , ([4], Corollary 4.1.6), and so we have . Now, let ; then, for some , and . Note that by ([10], Proposition 1.47 and Proposition 1.49), which implies thatTherefore, , which implies that , and hence is a weakly closed Jordan ideal of , proving that the mapping is an injection form the set of weakly closed invariant faces of into the set of weakly closed Jordan ideals of which is clearly the inverse of the injection map , proving the theorem.
Note that given a von Neumann algebra , there is a 1-1 correspondence between the set of norm closed split faces of and the set of -weakly closed invariant faces of , which is an immediate result of the reciprocal bijections between the set of -weakly closed two-sided ideals of and the set of -weakly closed invariant faces of (see [2], Corollary 3.21.2) and the mutual bijections between the set of -weakly closed ideals in and the set of norm closed split faces of ([9], Corollary 3.41 and Corollary 3.63).
The Jordan analogue of the above result is given in the following.

Theorem 7. Let be a JW-algebra. Then, there is a 1-1 correspondence between the set of norm closed split faces of and the set of weakly closed invariant faces of .

Proof. The result follows by Theorem 6, Theorem 2, and the commutative diagram .

Remark 2. Let be a JC-algebra with positive cone ; using the reciprocal bijections between the set of norm closed Jordan ideals in and the set of norm closed invariant faces of ([23], Corollary 2.5) and the reciprocal bijections between the set of norm closed Jordan ideals in and the set of -closed split faces of (cf. Theorem 2), we have the JC-algebra analogue of Theorem 7.

Theorem 8. Let be a JC-algebra. Then, there is a 1-1 correspondence between the set of norm closed invariant faces of and the set of -closed split faces of .
An application of Theorems 6, 3, and ([9], Corollary 3.41 and Corollary 3.63) gives the following.

Theorem 9. Let be a universally reversible JW-algebra such that the center of is pointwise fixed under the -anti-automorphism . Then, there is a 1-1 correspondence between the set of weakly closed invariant faces of and the set of -weakly closed invariant faces of .

Proof. The result follows from the following commutative diagram:

Definition 1. Let be a unital -algebra; a face is said to be invariant if whenever and is a unitary element in .
Recall that if is a von Neumann algebra, then there is a reciprocal bijection between the set of all two-sided ideals and the set of all invariant faces ([20], Corollary 3.21.2). The -algebra analogue of this result is proved in the following theorem.

Theorem 10. Let be a -algebra. Then, there is a 1-1 correspondence between the set of norm closed invariant faces of and the set of norm closed two-sided ideals of .

Proof. Let be a norm closed two-sided ideal in . Then, is a norm closed face of by ([9], Theorem 3.46). To see that is invariant, let and let be a unitary element in . Since is two-sided ideal, . By ([1], Corollary 4.2.7), , and hence . That is, is a norm closed invariant face of . Conversely, let be a norm closed invariant face of . By ([9], Theorem 3.46), is a norm closed left ideal in , and . To see that is also a right ideal, first we show that whenever and let be a unitary element of . So, let and be a unitary element of . Then, for some , and (see [9], Proposition 4.2.9). Since and is invariant, we have . It follows that , since is a left ideal. By ([9], Theorem 4.1.7), each element in is a finite linear combination of unitary elements in , which implies that for any and any . Hence, is a right ideal.
Our next result is the JC-algebra analogue of Theorem 9.

Theorem 11. Let be an irreducible universally reversible JC-algebra. Then, there is a 1-1 correspondence between the set of norm closed invariant faces of and the set of norm closed invariant faces of .

Proof. By Theorem 10, we have , and from Theorem 4, we have . The result follows since by ([23], Corollary 2.5).

Remark 3. (i)It is well known that every -weakly closed two-sided ideal in a von Neumann algebra contains a unique central projection in such that (see ([2], Proposition 2.3.12); [19], Theorem 6.8.8; [9], Theorem 3.35 and Theorem 3.40). Obviously, given a central projection in , is a -weakly closed two-sided ideal in . The uniqueness of the central projection presenting the weakly closed ideal implies that there is a mutual bijection between the set of -weakly closed two-sided ideals in and the set of central projections in . On the other hand, since , by ([20], Corollary 3.21.2), there is a 1-1 correspondence between the set of -weakly closed invariant faces of and the set of central projections in .(ii)The similar known result in the context of JW-algebras asserts that a weakly closed subalgebra of a JW-algebra is a Jordan ideal if and only if it has the form for some central projection (necessarily unique) ([10], Proposition 2.39) (see also ([4], Proposition 4.3.6)). Hence, there is a mutual bijection between the set of weakly closed Jordan ideals of and the set of central projections in . By Theorem 6, there is a 1-1 correspondence between the set of weakly closed invariant faces of and the set of central projections in .Combining the results in Theorems 1–3 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. Let be a universally reversible JW-algebra such that the center of is pointwise fixed under the -anti-automorphism . Then, there is a 1-1 correspondence between the following sets:(i)The set of norm closed split faces of (ii)The set of norm closed split faces of (iii)The set of weakly closed Jordan ideals of (iv)The set of -weakly closed two-sided ideals of (v)The set of weakly closed invariant faces of (vi)The set of -weakly closed invariant faces of (vii)The set of central projections in (viii)The set of central projections in

Proof. It is immediate from following commutative diagram:

Remark 4. (i)Given a -algebra , we shall regard as a -subalgebra of its second dual and naturally identify the states of with the normal states of . Then, we identify the enveloping von Neumann algebra (i.e., the -weak closure of , where is the universal representation of on the Hilbert space ) with (see [2], Theorem 3.2.4, p.122; [9], Corollary 2.1.27). If is a norm closed two-sided ideal in , then the weak closure of in is a two-sided ideal, and hence for a unique central projection and . Conversely, if is -weakly closed two-sided ideal in , and since is weakly dense in , is a norm closed two-sided ideal in such that . Note that the map is a bijection from the set of norm closed two ideals of and the set of weakly closed ideals in . Since the map is a mutual bijection between the set of weakly closed two-sided ideals of and the set of central projections in , we have a 1-1 correspondence between the set of norm closed two-sided ideals in and the set of central projections in .(ii)A similar argument in the context of JC-algebras leads to a mutual correspondence between the set of norm closed Jordan ideals in a JC-algebra and the set of central projections in its second dual by using the fact that the weak closure in of a norm closed Jordan ideal in is an ideal, , for some central projection and .(iii)Since , then by Theorem 11 and (ii) above, there is a 1-1 correspondence between the set of norm closed invariant faces of and the set of central projections in .Collecting the results in Theorems 1, 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 -algebras.

Theorem 13. Let be an irreducible universally reversible JC-algebra. Then, there is a 1-1 correspondence between the following sets:(i)The set of weak^∗-closed split faces of (ii)The set of weak^∗- closed split faces of (iii)The set of norm closed Jordan ideals in (iv)The set of norm closed two-sided ideals in (v)The set of norm closed invariant faces of (vi)The set of norm closed invariant faces of (vii)The set of central projections in (viii)The set of central projections in

Proof. It is immediate from the following commutative diagram:

Data Availability

The data used to support the findings of this study are included within the article.

Disclosure

Most of the results in this article are partly contained in the second author’s MS.C. thesis written at King Abdulaziz University under the supervision of Professor F. B. H. Jamjoom.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

This study was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University. The authors, therefore, gratefully acknowledge the DSR for technical and financial support.

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Copyright © 2020 F. B. H. Jamjoom and A. H. Al Otaibi. 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.

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