Abstract
This paper is devoted to the study of Jordan isomorphisms on nest subalgebras of factor von Neumann algebras. It is shown that every Jordan isomorphism between the two nest subalgebras and is either an isomorphism or an anti-isomorphism.
1. Introduction
Let and be two associative algebras. A Jordan isomorphism from onto is a bijective linear map such that for every . Obviously, isomorphisms and anti-isomorphisms are basic examples of Jordan isomorphisms. Jordan isomorphisms have been studied by many authors for various rings and algebras (see [1–5]). The standard problem is to determine whether a Jordan isomorphism is either an isomorphism or an anti-isomorphism. Using linear algebraic techniques, Molnár and Šemrl [6] proved that automorphisms and antiautomorphisms are the only linear Jordan automorphisms of , , where is a field with at least three elements. Later, Beidar et al. [7] generalized this result and proved that every linear Jordan isomorphisms of , , onto an arbitrary algebra over , is either an isomorphism or an anti-isomorphism, where is a 2-torsionfree commutative ring which is connected, that is, a ring in which the only idempotents are 0 and 1. Recently, Zhang [8] proved that every Jordan isomorphism between two nest algebras and is either an isomorphism or an anti-isomorphism. The same result was concluded in [9] for Jordan isomorphisms on nest algebras. The motivation for this paper is the work by Zhang. The aim of the present paper is to characterize Jordan isomorphisms of nest subalgebras.
In fact, the characterization of Jordan isomorphisms is closely related to isometric problems (see [10–12]). However, Jordan algebras or structures are related to vertex (operator) algebras or superalgebras and to representations of Kac-Moody and Virasoro algebras [13]. Note that the vertex operators of string theory also give reps of Lie algebras and Kac-Moody and Virasoro algebras (both infinite dimensional Lie algebras) as well as reps of the Fischer-Griess Monster group algebra. Jordan algebras or structures are also related to QFT, CFT, SCAs, and WZW models [14]. Another paper [15] about -theory with J3(O) and F4 makes use of the algebraic Kostant-Dirac operator, and a variant of this operator also appears on page 12 of a paper [16] about a gerbe based approach to supersymmetric WZW models (gerbes are useful in string theory; e.g., gerbes help in illuminating the geometry of mirror symmetry of CY threefolds, help in giving a noncommutative description of -branes in the presence of topologically nontrivial background fields, and provide a geometric way to unify properties of -form fields with gauge symmetries, etc.). There is more about how Jordan structures are involved with physics [17]. This paper studies the structure of Jordan isomorphisms of nest subalgebras. Our main result is that every Jordan isomorphism of nest subalgebras is either an isomorphism or an anti-isomorphism.
Now, let us introduce some notions and concepts that will be used later. Let be a von Neumann algebra acting on a separable Hilbert space . A nest in is totally a family of (self-adjoint) projections in which is closed in the strong operator topology and included and . The nest subalgebra of associated with a nest , denoted by , is the set . The diagonal of a nest subalgebra is the von Neumann subalgebra . Let denote the norm closed algebra generated by . It is clear that is a norm closed ideal of the nest subalgebra . If is a factor von Neumann algebra, it follows from [18] that is weakly dense in . When , is called a nest algebra and is denoted by . As a notational convenience, if is an idempotent, we let denote throughout this paper.
We refer the readers to [19] for background information about von Neumann algebras and to [18] for the theory of nest algebras and nest subalgebras.
2. Main Result
The following theorem is our main result.
Theorem 1. Let , be two nontrivial subalgebras in factor von Neumann algebra , and let be a Jordan isomorphism. Then, is either an isomorphism or an anti-isomorphism.
The proof is purely algebraic and will be organized in a series of lemmas. As a notational convenience, throughout this paper, we assume that and are nontrivial nests in a factor von Neumann algebra and that is a Jordan isomorphism.
Lemma 2. Let be a nest in factor von Neumann algebra , and then, for , if and only if there exists a projection such that , .
Proof. It is clear that if there exists a projection such that , , then .
Clearly, let be the closure of the space , and let be the projection onto . Then, and for all , and so . It is clear that , . Thus, , . The proof is complete.
Lemma 3. , and, for any , the following are equivalent:(a);(b);(c);(d).
Proof. As , let , and then, by (a), we have This shows that (b) is established.As , by (b), As ,Taking in (b), we get Multiplying the above equation on both sides by and noticing that , we have Thus, for any , . It follows from that . Hence, (a)–(d) are equivalent.
Lemma 4. If , then, for all , one has
Proof. By Lemma 3(b), for all ,Let , and then This implies that By (9),where Clearly, . Thus, by the above equation, we have This shows that, for any ,
Lemma 5. For any and any projection , either or .
Proof. If or , the result is clear. Suppose that . Let Then, by Lemma 4, for any ,By Lemma 2 and (15), there exists a projection such that There exists a projection such that By (16) and (18), we have Multiplying the above equation on both sides from the right side by , we get . If , then by (17) . So, . Hence, by (19), . Similarly, if , then . This implies that or . From the fact that , for all , one of the following is set up: Since is factor, then there exists a partial isometry operator such that , thus, . Therefore, either or . Suppose , if there exists such that , then . On the other hand, one of the following is set up: And , ; hence,So, . This shows that ; thus, . A contradiction. In conclusion, for any , we have . Similarly, suppose that , and then, for any , we have . Consequently, for any and any projection , either or .
Proof of Theorem 1. By Lemma 5, if, for any ,let , and then Thus, for all , we have Clearly, By (26)-(27), for all , we have Similar to the proof of (28), for any ,Let , and by (28), for any ,Thus, for all , we have Similarly, by (29), By the above two equations and Lemma 2, there exists a projection such that And there exists a projection such that Since is a Jordan isomorphism and , then, for any , we have Especially, ; that is, . Thus, by formula (34), we have This implies that , where is an orthogonal projection onto . Since is a nontrivial projection, then , thus, . Therefore, by (33), By (35) and similar discussion, we get This shows that ; thus, . So, by (36), In addition, by Lemma 3(c), we have Thus, for any , we have By (39) and (41), for all , we have . Hence, is an isomorphism.
If for any , we have , then, for any , we define , where is a conjugate linear involution operator defined in Lemma 2.3 of [11]. It is not difficult to verify that is a Jordan isomorphism and, for any projection , we have Thus, from the above discussion, is an isomorphism. Consequently, is an anti-isomorphism.
By Lemma 5, for all , one of the following holds: If , then Lemma 3(c) and the fact that imply that for all . Hence, for all , we have Similarly, it follows from Lemma 3(c) and that, for all , we have By (47), (48), and Lemma 2, there exists a projection such that Since , we have for all . Let , and denote the orthogonal projection from to . In particular, ; that is, . Thus, by formula (50), This shows that . Since is a nontrivial projection, then , so . By (49), In similar discussion, we have . This shows that , so . Thus, by (52), Because , by Lemma 3, we have for all . Since , similarly, By (56), (57), and Lemma 4, Thus, for all , we have By (54), (55), and (59), we have for any . Similarly, .
Consequently, is either an isomorphism or anti-isomorphism.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author wishes to thank the referees for their time and comments. He thanks Dr. Yongjian Xie from the College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, China. This paper is supported by the Fundamental Research Funds for the Central Universities (Grant no. GK201503017).