Maximal Injective Real W-Subalgebras of Tensor Products of Real W-Algebras

Abstract

It is known that injective (complex or real) W-algebras with particular factors have been studied well enough. In the arbitrary cases, i.e., in noninjective case, to investigate (up to isomorphism) W-algebras is hard enough, in particular, there exist continuum pairwise nonisomorphic noninjective factors of type II. Therefore, it seems interesting to study maximal injective W-subalgebras and subfactors. On the other hand, the study of maximal injective W-subalgebras and subfactors is also related to the well-known von Neumann’s bicommutant theorem. In the complex case, such subalgebras were investigated by S. Popa, L. Ge, R. Kadison, J. Fang, and J. Shen. In recent years, studies have also begun in the real case. Let us briefly recall the relevance of considering the real case. It is known that in the works of D. Topping and E. Stormer, it was shown that the study of JW-algebras (nonassociative real analogues of von Neumann algebras) of types II and III is essentially reduced to the study of real W-algebras of the corresponding type. It turned out that the structure of real W-algebras, generally speaking, differs essentially in the complex case. For example, in the finite-dimensional case, in addition to complex and real matrix algebras, quaternions also arise, i.e., matrix algebras over quaternions. In the infinite-dimensional case, it is proved that there exist, up to isomorphism, two real injective factors of type (), and a countable number of pairwise nonisomorphic real injective factors of type , whose enveloping (complex) W-factors are isomorphic, is constructed. It follows from the above that the study of the real analogue of problems in the theory of operator algebras is topical. Moreover, the real analogue is a generalization of the complex case, since the class of real linear operators is much wider than the class of complex linear operators. In this paper, the maximal injective real W-subalgebras of real W-algebras or real factors are investigated. For real factors , it is proven that if is a maximal injective W-subalgebra in , then also is a maximal injective real W-subalgebra in . The converse is proved in the case “”-factors, that is, it is shown that if is a real factor of type , then the maximal injectivity of implies the maximal injectivity of . Moreover, it is proven that a maximal injective real subfactor of a real factor is a maximal injective real W-subalgebra in if and only if is irreducible in , i.e., where is the unit. The “splitting theorem” of Ge-Kadison in the real case is also proven, namely, if is a finite real factor, is a finite real W-algebra, and is a real W-subalgebra of containing , then there is some real W -subalgebra such that . Moreover, it is given some affirmative answers to the question of S. Popa for the real case.

1. Introduction

Injective (complex or real) W-algebras with particular factors have been studied well enough. On the other hand, in the arbitrary cases, i.e., in noninjective case, to investigate (up to isomorphism) W-algebras is hard enough, in particular, there exist continuum pairwise nonisomorphic noninjective factors of type . Therefore, it seems interesting to study maximal injective W-subalgebras and subfactors. On the other hand, it will be described below that the study of maximal injective W-subalgebras and subfactors is also related to the well-known von Neumann’s bicommutant theorem. In the complex case, such subalgebras were investigated by S. Popa, L. Ge, R. Kadison, J. Fang, J. Shen, S. Stratila, and L. Zsido [1–4], [5–8], [9]. Now, in more detail, we present the main points of these articles. If is the algebra (more precisely, factor) of all bounded linear operators acting on a complex Hilbert space , i.e., if , and is a subfactor, then by the well-known theorem of von Neumann, the bicommutant of in is equal to , i.e., (see [8, 9]). But if is a continuous factor, then in general the bicommutant of in is not equal to (see [4]). If we denote the relative commutants of in and by and (), respectively, then in general, its relatives are not related, that is, and . Actually, it seems that the typical and more interesting type of imbedding as a subfactor of a continuous factor is such that the bicommutant of in is trivial, i.e., , and in this case is called irreducible in . For example, let be a properly outer action of a discrete group on . Then is naturally imbedded in the crossed product algebra , and by the relative commutant theorem, we have ([8, 9]). A sufficient condition for a subfactor of the factor to have a trivial relative commutant is that there exists an abelian -subalgebra which is maximal abelian in (see [1–3], [4, 6]). At the Baton Rouge (USA) conference in 1967, R.V. Kadison asked the question whether or not the converse is true, namely:

Ifis an irreducible subfactor of a factor, will some maximal abelian-subalgebra ofbe a maximal abelian-subalgebra of?

This problem has been solved in some particular cases. Namely, in 1981, Popa [2] proved that if is a separable factor, is a semifinite irreducible subfactor, and there exists a normal conditional expectation , then there is an abelian W-subalgebra which is a maximal abelian W-subalgebra in . Kadison also conjectured that every separable abelian W-subalgebra of a factor is contained in some hyperfinite subfactor. Howeover, Kadison’s conjecture turned out to be wrong. In 1983, Popa [1] gave a counterexample in which an abelian W-algebra can be imbedded in a factor of type as a maximal injective W-subalgebra, i.e., it is not an imbedding as a hyperfinite subfactor. This was also mentioned in [10]. Recall that a W-subalgebra is called maximal injective in if it is injective and maximal, i.e., there is no other injective W-subalgebra containing . Note that every W-algebra has one maximal injective W-subalgebra at least. Indeed, if is a family of injective W-subalgebras of which is inductively ordered by inclusion, then the weak operator closure of is an injective W-subalgebra of which contains all . By Zorn’s lemma, has a maximal injective W-subalgebra. We also note that the maximality of an injective W-subalgebra and the maximality of a hyperfinite subfactor are not connected with each other, i.e., one does not imply the other. In this line, at first, Murray and von Neumann assumed that any separable factor of type contains a hyperfinite subfactor as a maximal injective W-subalgebra. This result was proved by Sorin Popa [1] in 1983. Without the condition of separability, the problem was affirmatively solved by Fang [5].

On the other hand, it turns out that the maximality of an injective W -subalgebra is related to its irreducibility. Namely, in 2007, Fang [5] (see also [6]) showed that if is a maximal hyperfinite subfactor of a factor , then is a maximal injective W-subalgebra if and only if it is irreducible, i.e., . Thus, if we look chronologically, the maximality of an injective W-subalgebra is concerned with the bicommutant theorem.

In this paper, we will investigate the real analogues of the considered notions and the results above. Let us explain the topicality of considering the real analogue of problems in the theory of W-algebras. In the mid-1960s, in the works of D. Topping and E. Stormer, Jordan (nonassociative real) analogues of von Neumann algebras JW-algebras were first considered, i.e., real linear spaces of self-adjoint algebraic operators , that contain , are closed under Jordan multiplication and are closed in the weak operator topology. The structure of these algebras turned out to be close to the structure of von Neumann algebras (that is, W-algebras), and in the study of JW-algebras, it turned out to be possible to apply ideas similar to those of the theory of W-algebras. In these works, an initial classification of JW-algebras by types was obtained, JW-algebras of type I were completely studied, and it was shown that the study of types II and III is essentially reduced to the study of real W-algebras of the corresponding type. Since the mid-1980s, the theory of real W-algebras has received significant development. It turned out that the structure of real W-algebras, generally speaking, differs essentially in the complex case. For example, in the finite-dimensional case, in addition to complex and real matrix algebras, quaternions also arise, i.e., matrix algebras over quaternions. In the infinite-dimensional case, it is proved that there exist, up to isomorphism, two real injective factors of type (), and a countable number of pairwise nonisomorphic real injective factors of type , whose enveloping W-factors are isomorphic, is constructed. It follows from the above that the study of the real analogue of problems in the theory of operator algebras is topical. Moreover, the real analogue is a generalization of the complex case, since the class of real linear operators is much wider than the class of complex linear operators.

2. Preliminaries

Let be a Banach -algebra over the field . The algebra is called a C- algebra if for any . A C-algebra is called a W-algebra if there exists a Banach space , so-called a predual of such that . Let be the algebra of all bounded linear operators acting on a complex Hilbert space and let be a -subalgebra. The subset is called the commutant of . It is easy to see that where . If , then it is called a von Neumann algebra. The set is called the center of . It is obvious that . A von Neumann algebra is called a factor if its center is trivial, i.e., , where is the unit of . By von Neumann’s bicommutant theorem, any von Neumann algebra is W- algebra, i.e., is weakly closed with . The converse is also true. Therefore, W-algebras are also called von Neumann algebras (see [11]).

Now, a real -subalgebra with unit is called a real W- algebra if it is weakly closed and . The smallest (complex) W-algebra containing coincides with its complexication , i.e., . It is known that generates a natural involutive (i.e., of order 2) -antiautomorphism of , namely, , where , . In this case . Conversely, given a W-algebra and any involutive -antiautomorphism on , the set is a real W-algebra.

Let be a real W-algebra. It is known that there is a real Hilbert space with

The commutant of a -algebra is defined similarly to the complex case:

It is directly verified that (see [11]).

A real Banach -algebra is called a real C- algebra if and the element is invertible for any . Examples of such algebras are the algebra of all linear bounded operators acting on a real Hilbert space , and the algebra of all compact operators on is a real C-algebra, but is not a real W-algebra. It is known that is a real C-algebra if and only if the norm on can be extended to the complexification of so that is a C-algebra (see [12], 5.1.1). Let be a C-algebra. Denote by the algebra of all matrices over which is also a C-algebra. Recall that a continuous linear mapping between two C-algebras and is called completely positive if for any , the natural map from the C-algebra to the C-algebra defined by is positive, where is the C-algebra of matrices over . We say that a W-algebra is injective if the following condition is held: for every C-algebra , for every self-adjoint linear subspace of containing the identity , and for every completely positive linear map , there is a completely positive linear map such that ([13, 14]). All the notions above are defined similarly for real C- and W-algebras. Recall [14] that a W-algebra is injective if and only if it has the extension property property of E in short (extension property), i.e., there exists a projection such that , . In this case, the map is completely positive (see [14]).

3. Maximal Injective Real W -Subalgebras

Let and be real W-algebras. Similarly to the complex case, we define: a real W-subalgebra is called maximal injective in if it is injective and maximal, i.e. there is no other injective real W-subalgebra of containing .

Theorem 1. Let be a real W-subalgebra of a real W-algebra . If is maximal injective in , then is also maximal injective in .

Proof. Let be a maximal injective W-subalgebra of . Then by Theorem 8 in [15], is injective. If there exists an injective real W-subalgebra of containing , then by Theorem 8 in [15], the algebra is injective and which contradicts the maximality of the injective subalgebra .

In general, the converse of Theorem 1 is still open, but we will prove in a particular case.

Theorem 2. Let be a real factor of type . If is a maximal injective real subfactor in , then is also a maximal injective subfactor in .

Proof. Let be an injective subfactor with . Since the class of injective factors of type is unique, then the subfactors and are isomorphic, i.e., there is an isomorphism . Let be involutive -antiautomorphism of , generating , i.e., . Put . It is clear that is an involutive -antiautomorphism on . Put . Then and is a real injective subfactor of . And, is also injective. But, since and is maximal injective, then , consequently, . Thus, is a maximal injective subfactor in .
Now, we prove one auxiliary result.

Lemma 3. Let be a real W-algebra and let be a maximal injective real W-subalgebra of . Then is singular in , i.e., for a unitary element with , it follows that . In particular , where is the center of .

Proof. Let be a unitary element in with . Then the real W-subalgebra of generated by and the element is injective. Since is maximal injective in , it follows that . In particular, it is held that and .
Below we prove the real analogue of the theorem of J. Fang which is the solution to the problem of Murray and von Neumann for real factors.

Theorem 4. Let be a real factor and let be a maximal injective real subfactor of . Then is a maximal injective real W-subalgebra in if and only if is irreducible in , i.e. .

Proof. If is a maximal injective real W-subalgebra of , then by Lemma 3, . Since is a real factor, , i.e., . Conversely, suppose that . For any injective real W-algebra with , we have . Therefore, is an injective real subfactor of . Since is a maximal injective real subfactor of , .

Recall that a positive linear mapping from onto is called a conditional expectation if for all , .

Lemma 5. Let be a real W-algebra and let be real W-subalgebras of such that . If is a conditional expectation from onto , then induces a conditional expectation from onto .

Proof. Let and . Then . Apply the conditional expectation to the both sides of the equality . Since . Thus, . Since , is a conditional expectation from onto when is restricted on .

Lemma 6. Let be a real W-algebra and let be a real W-subalgebra of . If is a real W-algebra such that and there is a conditional expectation , then induces a conditional expectation from onto .

Proof. By Lemma 1.7 in [5], we have that where is the enveloping W -algebra of the real W -algebra . In general, if and are real W -algebras, then it holds Then . By Lemma 5, induces a conditional expectation from onto .

From Lemma 6, we have the following:

Corollary 7. Let the condition of Lemma 6 be satisfied and . Consider the set . Then induces a conditional expectation from onto .

Using Lemma 6 and Corollary 7, we will prove the real analogue of Ge-Kadison’s theorem, so-called “splitting theorem”.

Theorem 8. If is a finite real factor, is a finite real W-algebra, and is a real W-subalgebra of containing , then there is some real W-subalgebra such that .

Proof. Let be a real W-algebra such that . Since and are finite, there is a conditional normal expectation from onto . By Corollary 7, induces a conditional expectation from onto which we denote by . Then we have , for any , . Since is normal, it follows that . Since , .

4. Maximal Injective Real W-Subalgebras of Tensor Products of Real W-Algebras

Let be a real W-algebra. It is obvious that if is a maximal injective real W -subalgebra of , then is the minimal injective real W-subalgebra containing the algebra . Therefore, for any such algebra, there always exists as a maximal injective real W-subalgebra and such a minimal injective W-extension.

Let us prove one auxiliary result.

Lemma 9. Let be an abelian real W -algebra, be a real W -algebra, and let be a minimal separable injective real W -algebra containing . If is an injective real W -algebra such that then .

Proof. By formulas (6) and (7), we have By Lemma 2.4 from [5], we have . Since then from , we get .

The Lemma 9 can be proved in another way. Indeed:

Proof. Let be an abelian real W-algebra and let be a minimal separable injective real W-extension of a real W-algebra . Then it is directly verified that is also abelian and is the minimal separable injective W -extension of . By (7) we have Then by Lemma 2.4 in [5], we have . Consequently, since and , it follows that .

Now, we will prove the main result of this section.

Theorem 10. Let be an injective real W-algebra and be a separable real W-algebra. If is a maximal injective real W-subalgebra of , then is a maximal injective real W-subalgebra of .

Proof. Without loss of generality, we can assume that and are real W-algebras acting on complex Hilbert spaces and , respectively. Then by the hypothesis of the theorem, is separable. Let be the center of . Suppose that is an injective real W-algebra such that . Then we have . Since is an injective real W -subalgebra of , there is a conditional expectation from onto . By Lemma 5, induces a conditional expectation from onto . Consequently, is an injective real W-algebra such that . Since is a maximal injective real W-subalgebra of , is the minimal injective real W -extension of . By Lemma 9, we get . Thus . So, and .

5. Some Particular Cases

Let us consider the following problem which was formulated by Popa (see [1]) in the complex case:

Ifandare real W-algebras,andare maximal injective real W-subalgebras ofand, respectively, then will the algebrabe a maximal injective real W-subalgebra of?

In some certain cases, we will give affirmative answers to the question. (1)The case when is a real factor and is separable

As examples of such algebras, one can take as an infinite tensor product of a matrix algebra , which is a real factor of type , and as an arbitrary real W -algebra acting on a separable real Hilbert space.

Theorem 11. Let be a real W-algebra and let be a maximal injective real W-subalgebra of . If is separable and is a real factor, then is a maximal injective real W-subalgebra of .

Proof. Let be an injective real W-subalgebra with . Then we obtain . Since is a maximal injective real W-subalgebra of , is the minimal injective real W-extension of . Since is an injective real W-subalgebra of the W-subalgebra , there is a conditional expectation

Let . Then . By Lemma 3 we have

By Corollary 7, the mapping induces a conditional expectation from onto . Thus, is injective. Since and is the minimal injective real W-extension of , . So, . This implies that and hence, . By Theorem 10, we have . (2)The case when is separable and the center of is atomic

Recall that a real or complex von Neumann algebra is called atomic if every nonzero projection of majorizes a nonzero minimal projection (atom), i.e., under each nonzero projection, there is an atom.

As examples of such algebras, one can take as an arbitrary real W-algebra acting on a separable real Hilbert space, and as an arbitrary real W-algebra whose center is a commutative algebra.

Let us prove one auxiliary result which in itself is interesting.

Theorem 12. Let be a real W-algebra and be the minimal injective real W-algebra containing . If is a -automorphism of such that , , then That is, if a -automorphism is identical to a real W algebra, then it is also identical to its minimal injective real W-extension.

Proof. Let act on a Hilbert space , i.e., . denotes a linear extension of the -automorphism on the W-algebra that is defined as . As is known, every -automorphism of is inner, i.e., there exists some unitary element with , . By the condition of the theorem, we have for all , consequently, for all . Hence, we can get: . Put , for . It is easy to see that , for all , . Since , for all , i.e., . Note that is also a -automorphism on . Similar reasoning for the -automorphism , we can obtain . So, . It implies that is a normalizer of . By Lemma 1.1 in [5], . Consequently, for all , i.e., . In particular, .
Now, using Theorem 12, we prove the following auxiliary results.

Lemma 13. If is the minimal injective real W-extension of a real W-algebra and are nonzero central orthogonal projections in , then there is not a -isomorphism with , .

Proof. Suppose that there is a -isomorphism with , . Then for all . Define the mapping as Since are mutually orthogonal central projections and the mapping is a -isomorphism, it is easy to see that the mapping is a -automorphism of . Moreover, for any , we have also That is, the -automorphism acts identically on . Then by Theorem 12, we have , for any . That is, the -automorphism is identical to . Consequently, . Hence, we obtain that . It contradicts to the assumption that .

Lemma 14. Let be the minimal injective real W-extension of a real W-algebra and let be nonzero central projections in . If there is a -isomorphism with , , then and for any .

Proof. Suppose that . Let . Without loss of generality, we can assume that . Let . Then are nonzero central projections in and . It is easy to see that the -isomorphism induces a -isomorphism with for all . Moreover, for any , we have: As we see that it contradicts Lemma 13. Thus . Now, define a mapping on as: It is directly shown that is -automorphism of and , for all . Then by Theorem 12, we have: for all . Hence, for all .
Now, let us prove the main result of the section of the paper.

Theorem 15. Let be a real W-algebra and let be a maximal injective real W-subalgebra of . If is separable and the center of is atomic, then is maximal injective in .

Proof. Let be an injective real W-subalgebra with To prove the theorem, we have to show that . By Theorem 10, it is sufficient to prove that . For that, it is enough to prove that . Put , i.e., let be the center of which is atomic according to the condition. Since the algebras are separable, then the set of minimal projections is countable, which we denote by . Put As in the proof of Theorem 10, it is similarly shown that is injective and .

Note that for every . Since is the minimal injective real W-extension of , we have for every . We denote by the smallest projection in satisfying the condition: . Then we can get a -isomorphism uniquely determined by the formula

Since , it follows that for all .

Thus for all and for all , we have . Since is the minimal injective real W-extension of , then by Lemma 14, the last equation is satisfied for all and all . Moreover, hence, for every there is an element such that for all . In particular , i.e., . Therefore, . (3)The case when , are real factors

Theorem 16. Let be a real factor and be a maximal injective real subfactor of , . If and , then is a maximal injective real subfactor of .

Proof. Let be an injective real W-algebra with Then we have . By Theorem 4, is a maximal injective real W-subalgebra of , and consequently, is the minimal injective real W-extension of . By assumption, . Let be a conditional expectation and . By Lemma 6, induces a conditional expectation from onto , where . Therefore, is an injective real W-algebra such that Similar arguments as the proof of Theorem 15 shows that . So, . By Theorem 8 (the real analogue of Ge-Kadison’s splitting theorem), for some real W -subalgebra we have: . Therefore, . Since is injective, it follows that is also injective and . Since is a maximal injective real subfactor, then . Consequently, .

Corollary 17. Let be a real factor, be an irreducible, maximal injective real subfactor of . Then is an irreducible, maximal injective real subfactor of . Moreover, by Theorem 4, is a maximal injective real W -subalgebra of the factor .

6. Conclusion

In the article, the maximal injective real W-subalgebras of real W-algebras or real factors are investigated. For real factors , it is proven that if is a maximal injective W-subalgebra in , then also is a maximal injective real W-subalgebra in . The converse may not be true. It is assumed that using the crossed product of a finite real W-algebra on an action, one can construct a maximal injective real subfactor of type III, for which the enveloping complex subfactor is not the maximal injective. Now the authors of the article are preparing a continuation of this article, where this example will be given. Nevertheless, in this article, the validity of the converse statement for finite factors is proved. Exactly, the converse is proved in the case “”-factors, that is, it is shown that if is a real factor of type , then the maximal injectivity of implies the maximal injectivity of . Moreover, it is proven that a maximal injective real subfactor of a real factor is a maximal injective real W-subalgebra in if and only if is irreducible in . The above show how the real case differs from the complex case, and this is the contribution of this article. Moreover, the “splitting theorem” of Ge-Kadison in the real case is also proven, namely, if is a finite real factor, is a finite real W-algebra, and is a real W-subalgebra of containing , then there is some real W-subalgebra such that . And also, it is given some affirmative answers to the question of S. Popa for the real case.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors jointly worked on the results and they read and approved the final manuscript.

Acknowledgments

The authors gratefully acknowledge the Research Management Office, Universiti Malaysia Terengganu that this research was partially supported.