2-Local derivations of real AW*-algebras are derivation

2-Local derivations on real matrix algebras over unital semi-prime Banach algebras are considered. Using the real analogue of the result that any 2-local derivation on the algebra M2n(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{2^n}(A)$$\end{document} (n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}) is a derivation, it is shown that any 2-local derivation on real AW∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^*$$\end{document}-algebra for which the enveloping algebra is (complex) AW*-algebra, is a derivation, where A is a unital semi-prime Banach algebra with the inner derivation property.


Introduction
Given an algebra A, a linear operator D : A → A is called a derivation, if D(x y) = D(x)y + x D(y), for all x, y ∈ A. Each element a ∈ A implements a derivation D a on A defined as D a (x) = [a, x] = ax − xa, x ∈ A. Such derivations are said to be inner derivations. A map Δ : A → A (not linear in general) is called a 2-local derivation, if for every x, y ∈ A, there exists a derivation D x,y : A → A such that Δ(x) = D x,y (x) and Δ(y) = D x,y (y).
In the paper [ dimensional separable (complex) Hilbert space H . A similar description for the finitedimensional case appeared later in [2]. In the papers [3][4][5] and [6] the authors extended the Semrl's result for arbitrary finite, semi-finite and purely infinite von Neumann algebras, respectively. The real analogue of Semrl's result is received in the paper [7], i.e. it is described 2-local derivations on the real W*-algebra B(H ) of all bounded linear operators on the infinite-dimensional separable real Hilbert space H .
In the paper [8] the authors isvestigated 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property it is proved that any 2-local derivation on the algebra M 2 n (A), n ≥ 2, is a derivation. They apply this result to AW * -algebras and it is showed that any 2-local derivation on an arbitrary (complex) AW * -algebra is a derivation. In the present paper step by step we will prove analogous result for real AW * -algebras.
The authors would like to express his thanks to Professor Shavkat A. Ayupov for suggesting the derivation problem, and for many helpful discussions.

Preliminaries
Let B(H ) be the algebra of all bounded linear operators on a complex Hilbert space H . A weakly closed *-subalgebra M containing the identity operator 1 I in B(H ) is called a W*-algebra. A real *-subalgebra R ⊂ B(H ) is called a real W*-algebra if it is closed in the weak operator topology, 1 I ∈ R and R ∩ i R = {0}. A real W*-algebra R is called a real factor if its center Z (R) consists of the elements {λ1 I, λ ∈ R}. We say that a real W*-algebra R is of the type I f in , I ∞ , II 1 , II ∞ , or III λ , (0 ≤ λ ≤ 1) if the enveloping W*-algebra R + i R has the corresponding type in the ordinary classification of W*-algebras.
We say that an algebra A has the inner derivation property if every derivation on A is inner. Recall that an algebra A is said to be semi-prime if a Aa = 0 implies that a = 0.
In work [8] (Theorem 2.1) it is proved, that if A is a unital semi-prime (complex) Banach algebra with the inner derivation property and M 2 n (A) is the algebra of 2 n ×2 nmatrices over A, then any 2-local derivation on M 2 n (A) is a derivation.
The proof of this theorem without changes to pass for real Banach algebras. Therefore we shall formulate this result in a real case.

Theorem 1 Let A be a unital semi-prime real Banach algebra with the inner derivation property and let M 2 n (A) is the real algebra of 2 n × 2 n -matrices over A. Then any 2-local derivation on M 2 n (A) is a derivation.
We apply Theorem 1 to the description of 2-local derivations on real AW*-algebras.

2-Local derivations on real AW*-algebras
Firstly, we shall remind some definitions and the facts from the theory of complex and real AW * -algebras. Let A be a real or complex *-algebra and let S be a nonempty subset of A. Put denotes the left-annihilator of S. Following [12] we introduce the following notions.
Definition 1 A *-algebra A is called a Baer *-algebra if for any nonempty S ⊂ A, R(S) = g A for an appropriate projection g.
Since L(S) = (R(S * )) * = (h A) * = Ah the definition is symmetric and can be given in terms of the left-annihilator and a suitable projection h. Here S * = {s * | s ∈ S}.
In the particular case, where we consider only one point sets S = {x}, x ∈ A, we obtain the more general definition of a Rickart *-algebra. It is known that a Rickart *-algebra is a Baer *-algebra if and only if its projections form a complete lattice or every orthogonal family of projections has a supremum (i.e. a least upper bound).
Let A be a Banach *-algebra over the field C. The algebra A is called a C*-algebra, if aa * = a 2 for any a ∈ A. A real Banach *-algebra R is called a real C*-algebra, if aa * = a 2 and an element 1 I + aa * is invertible for any a ∈ R. It is easy to see that R is a real C*-algebra if and only if a norm on R can be extended onto the complexification A = R + i R of the algebra R so that algebra A is a C*-algebra.
This definition is equivalent to the definition given by Kaplansky [9], namely a C * -algebra is an AW * -algebra if and only if it satisfies the following conditions: (A) In the partially ordered set of projections, any set of orthogonal projections has a supremum; (B) Any maximal abelian *-subalgebra is generated by its projections.
Every W * -algebra is, of course, an AW * -algebra, however, the converse is not true as it was shown by Dixmier [10]. Given an AW * -algebra M, its center is Z M = {x ∈ M| x y = yx for all y ∈ M}. An AW * -algebra is called an AW * -factor, if its center consists of complex multiples of the identity 1 I, i.e. Z M = {λ1 I| λ ∈ C}. Now following Kaplansky [11, Appendix III] we introduce the main subject of the paper.

Definition 3
A real C * -algebra which is a Baer *-ring is called a real AW * -algebra.
It is clear that any real AW * -algebra contains an identity 1 I, and we say that a real AW * -algebra A is a real AW * − f actor if its center consists of real multiples of 1 I. Remark 1 1. Unlike the complex case in the real case it is not possible to give a definition in terms of conditions (A) and (B) above, because in maximal abelian *-subalgebras skew-hermitian elements can not be generated by projection.

2.
A slightly more general notion of real AW * -algebras was given also by Berberian [12, p. 26, Exercise 14A]. He defined a real AW * -algebra as a Banach *-algebra over the field of real numbers such that x * x = x 2 for all x ∈ A and such that A is a Baer *-ring. In this case the field C of complex numbers with the identical involution x * = x becomes a real AW * -algebra, but it is not a real C * -algebra, because it is not a symmetric *-algebra, which means 1 I + x * x is invertible for any x ∈ A.
Any real W * -algebra (real W * -factor) is a real AW * -algebra (resp. a real AW *factor). But the converse is not true. Any complex AW * -algebra is a real AW * -algebra. Complex AW * -factors are not real AW * -factors, because their centers are complex multiples of 1 I.
For real C * -algebras and W * -algebras we know that their complexification are C *and W * -algebras respectively. But in AW * -algebras case we have: there is a real AW *algebra R such that the complex C * -algebra R + i R is not an AW * -algebra (see [13,Proposition 4.2.3]). Now, we shall prove the main result of paper.

Theorem 2 Let R be an arbitrary real AW * -algebra and suppose that its complexification M = R + i R is a (complex) AW * -algebra. Then any 2-local derivation Δ on R is a derivation.
Proof Let us first note that any (complex) AW * -algebra is semi-prime, and it is clear that a real algebra A is semi-prime if and only if its complexification A + i A is semi-prime. Therefore, any real AW * -algebra is also semi-prime. It is also known [14, Theorem 2] that AW * -algebra has the inner derivation property. It is easy to shown that any real AW * -algebra has also the inner derivation property, i.e. every derivation of real AW * -algebra is an inner. Indeed, let R be a real AW *algebra and let D : R → R be a derivation. D can be extended by the linearity to a derivation on M = R + i R as D(x + iy) = D(x) + i D(y). Since D is an inner there is an element z = a + ib (a, b ∈ R) such that D( Now, let z be a central projection in R. Then z is a central projection in M. It is known that D(z) = 0, and therefore D(z) = 0. Then it is easy to see that Δ(z) = 0 for any 2-local derivation Δ on R. For x ∈ R we consider the elements x and zx. Then there is a derivation D on R such that Δ(zx) = D(zx) and Δ(x) = D(x). Then we have It means that every 2-local derivation Δ maps z R into z R for each central projection z ∈ R. Thus we may consider the restriction of Δ onto eR. By [13,Proposition 4.4.3] an arbitrary real AW * -algebra can be decomposed along a central projection into the direct sum of an abelian real AW * -algebra, and real AW * -algebras of type I n , n ≥ 2, type I ∞ , type II and type III. We will consider these cases separately.
Let R be an abelian real AW * -algebra. It is well-known that any derivation on an abelian (complex) W * -algebra R + i R is identically zero. Therefore, the derivation D(x + iy) = D(x) + i D(y) on R + i R is identically zero, where D is a derivation on R. Hence D is identically zero, i.e. any 2-local derivation on an abelian AW * -algebra is also identically zero.
If R is a real AW * -algebra of type I n , n ≥ 2, with the center Z (R), then it is isomorphic to the algebra M n (Z (R)). By Lemma 2.3 [8] (as it is already told above, that the proof of theorem 2.1 and Lemmas from [8] without changes to pass for real Banach algebras) there exists a derivation D on R ≡ M n (Z (R)) such that Δ ≡ D. So, Δ is a derivation.
Let the real AW * -algebra R have one of the types I ∞ , II or III. Then using the methods developed in [13, § §4.3-4.7] and similarly following the scheme of the proof of Lemmas 4.5 and 4.12 in [9], the algebra R can be represented as a sum of mutually equivalent orthogonal projections e 1 , e 2 , e 3 , e 4 from R. Then the map x → 4 i, j=1 e i xe j defines an isomorphism between the algebra R and the matrix algebra M 4 (Q), where Q = e 1,1 Re 1,1 . It is easy to see that Q and Q + i Q are real and complex C * -algebras, respectively. Since M = R + i R is AW * -algebra by [12, Proposition 8 (iii), 23p.] C * -algebra eMe is also AW * -algebra, where e is an arbitrary projection in M. Then by [13, Proposition 4.3.1] a real C * -algebra Q = e 1,1 Re 1,1 is a real AW * -algebra and its complexification Q + i Q = e 1,1 Me 1,1 is also a (complex) AW * -algebra. Therefore Q is a unital semi-prime real Banach algebra with the inner derivation property. Hence Theorem 1 implies that any 2-local derivation on R is a derivation.

Remark 2
Everywhere in the work we considered a real AW * -algebra with a (complex) AW * -algebra of its complexification. Moreover, in the definition of real C * -algebra the condition of convertibility of an element 1 I + x x * (for all x) is required. It is equivalent to that a norm on real C * -algebra can be extended onto its complexification so that it is a (complex) C*-algebra. But in the [12, Exercise 14A] in definition of real C *algebra convertibility of 1 I + x x * is not required. In this connection we shall formulate following questions.
Question let R be a real Baer *-ring. Suppose that (i) R is a real Banach *-algebra with x x * = x 2 , for any x ∈ R, or (ii) R is a real AW * -algebra (not necessary its complexification is a (complex) AW *algebra). Then (1) is any derivation of R is inner? (2) is any 2-local derivation on R is derivation? by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.