Characterizing 𝜉-Lie Multiplicative Isomorphisms on Von Neumann Algebras

Abstract

Let and be von Neumann algebras without central summands of type . Assume that with . In this paper, all maps satisfying are characterized.

1. Introduction

Let and be two algebras over a field . Recall that a map is called a multiplicative map if for all ; a Lie multiplicative map if for all , where ; and a Jordan multiplicative map if for all .

The question when a multiplicative map is additive is studied by many mathematicians. As the first result in this line, Matindale [1] proved that every multiplicative bijective map from a prime ring containing a nontrivial idempotent onto an arbitrary ring is additive and thus is a ring isomorphism. Recently, Matindale’s result has been generalized in several directions, such as multiplicative maps and Jordan multiplicative maps between standard operator algebras or nest algebras (see [2, 3] and the references therein). For Lie multiplicative maps, Bai et al. [4] showed that if , are prime rings with being unital and containing a nontrivial idempotent and if is a Lie multiplicative bijective map, then for all , where is an element in the center of depending on and . This result reveals that the Lie multiplicativity of a map does not imply its additivity anymore. Note that factor von Neumann algebras are prime. Later, the similar results were obtained on triangular algebras and certain Banach space nest algebras, respectively, in [5, 6].

Let be an algebra over a field . For a scalar and for , we say that commutes with up to a factor if . The notion of commutativity up to a factor for pairs of operators is an important concept and has been studied in the context of operator algebras and quantum groups [7, 8]. Motivated by this, a binary operation , called -Lie product of and , was introduced in [9]. Moreover, a concept of -Lie multiplicative maps was introduced in [10], which unifies the above three kinds of maps. Recall that a map is called a -Lie multiplicative map if for all . In addition, is called a -Lie multiplicative isomorphism if is bijective and -Lie multiplicative and is called a -Lie ring isomorphism if is bijective, additive, and -Lie multiplicative. A linear (resp., conjugate linear) -Lie ring isomorphism between two algebras is called a -Lie isomorphism (resp., conjugate -Lie isomorphism).

Recall that a standard operator algebra on a Banach space is a subalgebra of the whole operator algebra containing the identity operator and the ideal of all finite rank operators. Qi and Hou in [10] gave a characterization of all -Lie multiplicative isomorphisms between standard operator algebras. Let and be standard operator algebras on infinite dimensional Banach spaces and over the real or complex field , respectively. Assume that is a unital bijection and is a scalar. The main result in [10] states that is -Lie multiplicative if and only if one of the following holds: (1) , there exists a functional with for all , and either there exists an invertible bounded linear or conjugate linear operator such that for all or there exists an invertible bounded linear or conjugate linear operator such that for all ; (2) , either there exists an invertible bounded linear or conjugate linear operator such that for all , or there exists an invertible bounded linear or conjugate linear operator such that for all ; (3) , there exists an invertible bounded linear operator such that for all if ; there exists an invertible bounded linear or conjugate linear operator such that for all if ; (4) , there exists an invertible bounded linear operator such that for all . A complete characterization of -Lie multiplicative isomorphisms on matrix algebras and certain nest algebras was given, respectively, in [10] and [6]. These results reveal the structural properties of the involved operator algebras from some new aspects. However, we have not seen any description on the structure of the -Lie multiplicative isomorphisms between nonfactor von Neumann algebras so far. The present paper considers this problem.

The purpose of this paper is to characterize the -Lie multiplicative isomorphisms with between certain quite general von Neumann algebras. Let and be two von Neumann algebras without central summands of type . Denote by and the unit operators in and , respectively. Assume that is a map and with . We show that is a -Lie multiplicative isomorphism if and only if one of the following statements is true: (1) , is a ring isomorphism; (2) , there exist central projections and such that , where is a ring isomorphism and is a ring anti-isomorphism; (3) , there exist central projections and such that , where is a ring isomorphism with for all and is a ring anti-isomorphism with for all (Theorem 1). It is clear from our result that, for , the -Lie multiplicativity of a bijective map on von Neumann algebras without central summands of type implies its additivity. Particularly, multiplicative isomorphisms are ring isomorphisms and the Jordan multiplicative isomorphisms are Jordan ring isomorphisms. Moreover, for rational real number , -Lie multiplicative isomorphisms are ring isomorphisms. However, -Lie ring isomorphisms with nonrational have some more complicated but still controllable algebraic structures. Assume further that act on some separable Hilbert spaces and has no central summands of type for any ; then is a -Lie multiplicative isomorphism if and only if one of the following statements is true: (1) , , where is an algebraic isomorphism and is a conjugate algebraic isomorphism; (2) , , where is an algebraic isomorphism, is a conjugate algebraic isomorphism, is an algebraic anti-isomorphism, and is a conjugate algebraic anti-isomorphism; (3) with , is an algebraic isomorphism; (4) with , , where is an algebraic isomorphism and is a conjugate algebraic anti-isomorphism.

For the case of , the result and the approach are quite different and we will discuss it in another paper.

2. Main Result and Corollary

The following is our main result and its proof will be presented in the next section.

Theorem 1. Let and be von Neumann algebras without central summands of type . Assume that is a map and with . Then, is a -Lie multiplicative isomorphism, that is, is bijective and satisfies for all , if and only if one of the following statements holds:
(1) , is a ring isomorphism.
(2) , there exist central projections and such that , where is a ring isomorphism and is a ring anti-isomorphism.
(3) , there exist central projections and such that , where is a ring isomorphism with for all and is a ring anti-isomorphism with for all . Here, denotes the unit in .

We remark that, from the above result, for bijective maps between von Neumann algebras without central summands of type , the -Lie multiplicativity () will imply the additivity; moreover, if is a rational real number and , then by (3), for all , which forces that and hence is a ring isomorphism.

In the sequel, we study in more detail on -Lie multiplicative isomorphisms and get a corollary of Theorem 1. Firstly, let us give a lemma which is interesting in itself.

Lemma 2. Let be a factor von Neumann algebra. Assume that is a von Neumann algebra and is a ring isomorphism (ring anti-isomorphism). If , then there exists a field automorphism such that is -linear; if , then is linear or conjugate linear.

Proof. We only deal in detail with the case that is a ring isomorphism. The other case can be proved similarly.
Assume that is a ring isomorphism. It is clear that , which implies that as is a factor. Then, since is a ring isomorphism, it is easy to check that is unital; that is, .
If is not a factor, there exists a nonzero central element such that is not invertible. Thus, is also not invertible, and so , a contradiction. Hence, is a factor and .
For any , let . Then, We claim that is a field automorphism. In fact, since is additive, is additive. Note that This implies that ; that is, is multiplicative.
In the following, we assume that is infinite dimensional. We will show that is continuous. As , there exists a sequence of projections which are orthogonal to each other. So, we have for and . If is not continuous, then is unbounded on any neighborhood of . So is unbounded on and hence there exists with such that Considering gives with such that Generally, for any , there exists with such that Let ; then . This implies that and . However, which implies that for any , a contradiction. Hence, is continuous and by [11, pp. 52–57] is the identity or the conjugation. Therefore, is linear or conjugate linear.

Lemma 3. Let be any von Neumann algebra and a factor of infinite dimension. Assume that is a ring anti-isomorphism and with . Then, if and only if is a conjugate algebra anti-isomorphism and .

Proof. By Lemma 2, is linear or conjugate linear. So or . If , it follows that or , which imply that or . Since , we see that must be conjugate linear and . The converse is obvious.

Lemma 4. Let and be von Neumann algebras acting on separable Hilbert spaces and assume that has no central summands of type for any .
(1) is a ring isomorphism (resp., a ring anti-isomorphism) if and only if there exist central projections and , an algebraic isomorphism (resp., an algebraic anti-isomorphism) and a conjugate algebraic isomorphism (resp., a conjugate algebraic anti-isomorphism) such that .
(2) Assume that is a ring anti-isomorphism and . Then, if and only if is a conjugate algebraic anti-isomorphism and .

Proof. (1) We consider the case of ring anti-isomorphism; the case of ring isomorphism is treated similarly.
Assume that is a ring anti-isomorphism. By [12, pp. 209, 236], there exists a positive measure space such that where every is a factor and is a ring anti-isomorphism. Since has no central summands of type for any , is a factor of infinite dimensional a.e. . By Lemma 2, is linear or conjugate linear. If is not a conjugate algebraic anti-isomorphism a.e. for all , then there exists a measurable subset with nonzero measure such that is an algebraic anti-isomorphism a.e. on it. It follows that there exists a proper central projection such that is an algebraic anti-isomorphism. Note that is a central projection. Now it is clear (e.g., using Zorn’s Lemma) that there exist central projections and , an algebraic anti-isomorphism and a conjugate algebraic anti-isomorphism such that .
Conversely, if has the mentioned decomposition, then is clearly a ring anti-isomorphism.
(2) The “if” part is clear. To check the “only if” part, assume that is a ring anti-isomorphism and . For any , writing , we have . Hence, It follows that a.e. . By Lemma 3, we have and is conjugate linear a.e. , and so is a conjugate algebraic anti-isomorphism.

Now, we are in a position to give the following corollary of Theorem 1.

Corollary 5. Let and be any von Neumann algebras acting on separable Hilbert spaces without central summands of type . Assume further that has no any central summands of type for . Let be a map and with . Then, is a -Lie multiplicative isomorphism if and only if one of the following statements is true.
(1) , there exist central projections and , and an algebraic isomorphism and a conjugate algebraic isomorphism such that .
(2) , there exist central projections and with and such that , where is an algebraic isomorphism, is a conjugate algebraic isomorphism, is an algebraic anti-isomorphism, and is a conjugate algebraic anti-isomorphism.
(3) with , is an algebraic isomorphism.
(4) with , there exist central projections and such that , where is an algebraic isomorphism and is a conjugate algebraic anti-isomorphism.

Proof. We only need to check the “only if” part. Assume that is a -Lie multiplicative isomorphism. By Theorem 1 and Lemma 4(1), if , (2) is true; if , (1) holds; if , there exists central projections and such that , where is a ring isomorphism with for all and is a ring anti-isomorphism with for all .
For , by Lemma 4(1), there exist central projections and such that , where is linear and is conjugate linear. Note that for all . This implies that if .
For , by Lemma 4(2), if and is a conjugate algebraic anti-isomorphism if . Thus, if , is a ring isomorphism and has the form (1); if with , then is an algebraic isomorphism and , and consequently, is an algebraic isomorphism, which implies the form (3); if with , then is an algebraic isomorphism and is a conjugate algebraic anti-isomorphism, which implies (4).

3. Proof of the Main Result

In this section, we present a proof of the main result Theorem 1. Before doing this, we need some notions. Let be any von Neumann algebra and . Recall that the central carrier of , denoted by , is the intersection of all central projections such that . If is self-adjoint, then the core of , denoted by , is sup. Particularly, if is a projection, it is clear that is the largest central projection ≤. A projection is core-free if . It is easy to see that if and only if [13].

The following two lemmas are needed.

Lemma 6 (see [13]). Let be a von Neumann algebra without central summands of type . Then, each nonzero central projection is the carrier of a core-free projection in . Particularly, there exists a nonzero core-free projection with .

Lemma 7 (see [14]). Let be a von Neumann algebra without central summands of type or . Then, the ideal of generated algebraically by is equal to .

Proof of Theorem 1. For , by Lemma 6, we can find a nonzero central core-free projection with central carrier . By the definitions of core and central carrier, is core-free and . For the convenience, denote , , where and . Then, . In the sequel, when we write , we always indicate .
If the statement (1) holds, it is clear that is a multiplicative isomorphism. If the statement (2) holds, it is easy to check that is a Jordan multiplicative isomorphism. If the statement (3) holds, then for any , we have and for any , we have which implies that . Hence, for any , one obtains that . This completes the proof of “if” part.
We will prove the “only if” part by checking a series of claims.
Claim 1. is additive. We will complete the proof of Claim  1 by nine steps.
Step 1. . Since is surjective, there exists an element such that . So .
In the sequel, we will use a so-called standard argument: suppose that are such that . Multiplying this equation by from the left and the right, respectively, we get and . Then, It follows that Moreover, if we have and , by the injectivity of , one gets Step 2. For any and , we have , .
By the surjectivity of , there is an element such that For any , applying the standard argument to (14), we obtain By (15) and the injectivity of , one gets for all , which implies that , and for all . Particularly, taking , one gets and, as , . This, combining (16) and the injectivity of , yields , and so .
For any , applying the standard argument to (14) again, we get It follows from the injectivity of that for every . Note that and . The above equation reduces to ; that is, holds for all . Note that . It follows from the definition of the central carrier that span is dense in . Hence, . Consequently, .
Similarly, one can check that the following Step  3 holds.
Step 3. For any and , we have , .
Step 4. is additive on , .
For any , since by Steps  2-3, one can obtain For any , note that By a similar computation as above, one can show .
Step 5. is additive on , .
Take any and choose such that Let . For , applying the standard argument to (22) and the injectivity of , we get and . Note that , these two equations imply that .
For any , applying the standard argument to (22), and by Step  4, one obtains Note that is injective and . The above equation implies for all ; that is, for all . It follows from that .
Step 6. Consider .
Choose such that For any , applying the standard argument to (24), and the injectivity of , we have for all . Multiplying this equation by from both sides, we get for all , which implies that .
Similarly, for any , applying the standard argument to (24), one can prove that .
For and , applying the standard argument to (24), respectively, we have Therefore, , which implies that . A simple computation reveals that . Consequently, .
Step 7. Consider .
Let be such that . Then, by Steps  2-3, we have For any , applying the standard argument to (27) and the injectivity of , we get Multiplying by from the left in (28), one obtains for each , and so . Multiplying by and from the left and the right, respectively, in (28), one gets Similarly, for any , applying the standard argument to (26), one can get .
For , applying the standard argument to (26), by Step  6, we have , which implies that . As and , a direct computation leads to . This fact and (29) yield for all , and so . Consequently, , as desired.
Step 8. Consider .
Let be such that For , applying the standard argument to (30), by Step  7, we have and . It follows that and . By a simple computation, one obtains , , and .
For any , applying the standard argument to (30), one gets Furthermore, for , applying the standard argument to the above equation, by Steps  2 and 4, one gets Thus, we have Note that , , and . It follows that for all , and hence . Consequently, .
Step 9. is additive, and so Claim  1 is true.
For any , write and . By Steps  2–8, we have
Claim 2. The statements (1)-(2) hold in the theorem. By Claim  1, is additive. So, in the case of , the statement (1) is true; in the case of , by [14] (also see [15]), it is also easy to see that the statement (2) is true.
In the sequel, we always assume that .
Claim 3. Consider .
For any , we have Note that, by Claim  1, is additive. Thus, the above two equations imply that As and , the above equation ensures that if and only if . So holds for all . It follows from the surjectivity of that .
Claim 4. is invertible.
For any , by Claim  3, we have Taking in the above equation, one gets . It follows from the fact that . So is invertible and is its inverse. The claim holds.
Note that as . For any , let . Since we get . So It follows that is a Jordan ring isomorphism and . Thus, by [14], there exist central projections and such that is a ring isomorphism and is a ring anti-isomorphism.
For the convenience, write , , , and . Then, we may write .
Note that , , and . It is easy to check that , , , and . Hence, Claim  5. is a ring isomorphism satisfying for all .
For any , since is a ring isomorphism, we have These imply that holds for all . It follows from the surjectivity of that holds for all . Furthermore, it is easily checked that holds for every . Note that is a von Neumann algebra without central summands of type . By Lemma 7, one obtains . Hence, for all ; that is, is a ring isomorphism.
Also note that, for any , we have This leads to which completes the proof of Claim  5.
Claim 6. is a ring anti-isomorphism and for all .
For every , we have Then, , and so holds for all . It is easily checked that holds for each . It follows from Lemma 7 that . Hence, , and is a ring anti-isomorphism.
Since we get for all . Hence, Claim  6 is true.
Combining Claims 3–6, one sees that the statement (3) in Theorem 1 holds.
The proof of the theorem is therefore completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors wish to express their thanks to the referees for suggesting many improvements of the original draft. This work is partially supported by the National Natural Science Foundation of China (11171249, 11101250, and 11271217) and Youth Foundation of Shanxi Province (2012021004).