2-Local automorphisms on A W ∗ -algebras
Shavkat Ayupov, Karimbergen Kudaybergenov, Turabay Kalandarov
aa r X i v : . [ m a t h . OA ] A ug AW ∗ -ALGEBRAS SHAVKAT AYUPOV, KARIMBERGEN KUDAYBERGENOV, AND TURABAY KALANDAROV
With the deep respect, we dedicate the article to the 65-th anniversary of Professor Ben de Pagter.
Abstract.
The paper is devoted to 2-local automorphisms on AW ∗ -algebras. Usingthe technique of matrix algebras over a unital Banach algebra we prove that any 2-localautomorphism on an arbitrary AW ∗ -algebra without finite type I direct summands isa global automorphism. Introduction and the Main Theorem
In 1990, Kadison [9] and Larson and Sourour [11] independently introduced the con-cept of a local derivation. A linear map ∆ :
A → M is called a local derivation if forevery x ∈ A there exists a derivation D x (depending on x ) such that ∆( x ) = D x ( x ) . Itis natural to consider under which conditions local derivations automatically becomederivations. Many partial results have been done in this problem. In [9] Kadison showsthat every norm-continuous local derivation from a von Neumann algebra M into a dual M -bimodule is a derivation. In [8] Johnson extends Kadison’s result and proves everylocal derivation from a C ∗ -algebra A into any Banach A -bimodule is a derivation.In 1997, ˇSemrl [12] initiated the study of so-called 2-local derivations and 2-localautomorphisms on algebras. Namely, he described such maps on the algebra B ( H ) ofall bounded linear operators on an infinite dimensional separable Hilbert space H .In the above notations, a map ∆ : A → A (not necessarily linear) is called a if, for every x, y ∈ A , there exists an automorphism Φ x,y : A → A suchthat Φ x,y ( x ) = ∆( x ) and Φ x,y ( y ) = ∆( y ) . Afterwards local derivations and 2-local derivations have been investigated by manyauthors on different algebras and many results have been obtained in [1, 2, 3, 9, 10, 12].In [6] it was established that every 2-local ∗ -homomorphism from a von Neumannalgebra into a C ∗ -algebra is a linear ∗ -homomorphism. These authors also proved thatevery 2-local Jordan ∗ -homomorphism from a JBW*-algebra into a JB*-algebra is aJordan *-homomorphism.In the present paper we extend the result obtained in [1] for 2-local derivations on AW ∗ -algebras to the case of 2-local automorphisms on AW ∗ -algebras .If ∆ : A → A is a 2-local automorphism, then from the definition it easily followsthat ∆ is homogenous. At the same time,∆( x ) = Φ x,x ( x ) = Φ x,x ( x )Φ x,x ( x ) = ∆( x ) for each x ∈ A . This means that additive (and hence, linear) 2-local automorphism isa Jordan automorphism.The following Theorem is the main result of this paper.
Date : November 3, 2018.1991
Mathematics Subject Classification.
Primary 46L57; Secondary 47B47; 47C15.
Key words and phrases. AW ∗ -algebra; matrix algebra; automorphism; 2-local automorphism. Theorem 1.1.
Let M be an arbitrary AW ∗ -algebra without finite type I direct sum-mands. Then any 2-local automorphism ∆ on M is an automorphism. The proof of this Theorem is based on representations of AW ∗ -algebras as matrixalgebras over a unital Banach algebra with the following two properties: (J) : for any Jordan automorphism Φ on A there exists a decomposition A = A ⊕ A such that x ∈ A 7→ p (Φ( x )) ∈ A is a homomorphism and x ∈ A 7→ p (Φ( x )) ∈ A is an anti-homomorphism, where p i is a projection from A onto A i , i = 1 , (M) : There exist elements x, y ∈ A such that xy = 0 and yx = 0 . Remark 1.2.
Note that if an algebra A contains a subalgebra isomorphic to the matrixalgebra M ( C ) , then it satisfies the condition (M) . Indeed, for matrices x = (cid:18) (cid:19) and y = (cid:18) (cid:19) , we have xy = 0 and yx = 0 . The proof of the main result
The key tool for the proof of Theorem 1.1 is the following.
Theorem 2.1.
Let A be a unital Banach algebra with the properties (J) and (M) andlet M n ( A ) be the algebra of all n × n -matrices over A , where n ≥ . Then any 2-localautomorphism ∆ on M n ( A ) is an automorphism. The proof of Theorem 2.1 consists of two steps. In the first step we shall showadditivity of ∆ on the subalgebra of diagonal matrices from M n ( A ) . Let { e i,j } ni,j =1 be the system of matrix units in M n ( A ) . For x ∈ M n ( A ) by x i,j wedenote the ( i, j )-entry of x, where 1 ≤ i, j ≤ n. We shall, if necessary, identify thiselement with the matrix from M n ( A ) whose ( i, j )-entry is x i,j , other entries are zero,i.e. x i,j = e i,i xe j,j . Each element x ∈ M n ( A ) has the form x = n X i,j =1 x ij e ij , x ij ∈ A , i, j ∈ , n. Let ψ : A → A be an automorphism. Setting ψ ( x ) = n X i,j =1 ψ ( x ij ) e ij , x ij ∈ A , i, j ∈ , n (1)we obtain a well-defined linear operator ψ on M n ( A ) . Moreover ψ is an automorphism.For an invertible element a ∈ M n ( A ) setΦ a ( x ) = axa − , x ∈ M n ( A ) . Then Φ a is an automorphism and it is called a spatial automorphism.It is known [4, Corollary 3.14] that every automorphism Φ on M n ( A ) can be repre-sented as a product Φ = Φ a ◦ ψ, (2)where Φ a is a spatial automorphism implemented by an invertible element a ∈ M n ( A ) , while ψ is the automorphism of the form (1) generated by an automorphism ψ on A . -LOCAL AUTOMORPHISMS ON AW ∗ -ALGEBRAS 3 Consider the following two matrices: u = n X i =1 i e i,i , v = n X i =2 e i − ,i . (3)It is easy to see that an element x ∈ M n ( A ) commutes with u if and only if it isdiagonal, and if an element a ∈ M n ( A ) commutes with v, then a is of the form a = a a a . . . . a n a a . . . . a n − a . . . . a n − ... ... ... ... ... ...0 0 . . . . a a . . . . a . (4)Further in Lemmata 2.2–2.5 we assume that n ≥ . Lemma 2.2.
For every -local automorphism ∆ on M n ( A ) there exists an automor-phism Φ such that ∆ | sp { e i,j } ni,j =1 = Φ | sp { e i,j } ni,j =1 , where sp { e i,j } ni,j =1 is the linear span ofthe set { e i,j } ni,j =1 . Proof.
Take an automorphism Φ u,v on M n ( A ) such that∆( u ) = Φ u,v ( u ) , ∆( v ) = Φ u,v ( v ) , where u, v are the elements from (3). Replacing ∆ by Φ − u,v ◦ ∆, if necessary, we canassume that ∆( u ) = u, ∆( v ) = v. Let i, j ∈ , n. Take an automorphism Φ = Φ a ◦ ψ of the form (2) such that∆( e i,j ) = aψ ( e ij ) a − , ∆( u ) = aψ ( u ) a − . Since ∆( u ) = u and ψ ( u ) = u, it follows that [ a, u ] = 0 , and therefore a has a diagonalform, i.e. a = n P s =1 a s e s,s , a s ∈ A , s ∈ , n. In the same way, but starting with the element v instead of u , we obtain∆( e i,j ) = be i,j b − , where b has the form (4), depending on e i,j . So∆( e i,j ) = ae i,j a − = be i,j b − . Since ae i,j a − = a i a − j e i,j and [ be i,j b − ] i,j = 1 , it follows that ∆( e i,j ) = e i,j . Now let us take a matrix x = n P i,j =1 λ i,j e i,j ∈ M n ( C ) . Then e j,i ∆( x ) e j,i = ∆( e j,i )∆( x )∆( e j,i ) = Φ e j,i ,x ( e j,i )Φ e i,j ,x ( x )Φ e j,i ,x ( e j,i ) == Φ e j,i ,x ( e j,i xe j,i ) = Φ e j,i ,x ( λ i,j e j,i ) == λ i,j Φ e j,i ,x ( e j,i ) = λ i,j e j,i , i.e. e i,i ∆( x ) e j,j = λ i,j e i,j for all i, j ∈ , n. This means that ∆( x ) = x. The proof iscomplete. (cid:3)
SHAVKAT AYUPOV, KARIMBERGEN KUDAYBERGENOV, AND TURABAY KALANDAROV
Further in Lemmata 2.3–2.8 we assume that ∆ is a 2-local automorphism on M n ( A )such that ∆ | sp { e i,j } ni,j =1 = id | sp { e i,j } ni,j =1 . Let ∆ i,j be the restriction of ∆ onto A i,j = e i,i M n ( A ) e j,j , where 1 ≤ i, j ≤ n. Lemma 2.3. ∆ i,j maps A i,j into itself.Proof. Let us show that ∆ i,j ( x ) = e i,i ∆( x ) e j,j (5)for all x ∈ A i,j . Take x = x i,j ∈ A i,j , and consider an automorphism Φ = Φ a ◦ ψ of the form (2) suchthat ∆( x ) = aψ ( x ) a − , ∆( u ) = aψ ( u ) a − , where u is the element from (3). Since ∆( u ) = u and ψ ( u ) = u, it follows that[ a, u ] = 0 , and therefore a has a diagonal form. Then ∆( x ) = a i ψ ( x ij ) a − j e ij . Thismeans that ∆( x ) ∈ A i,j . The proof is complete. (cid:3)
Lemma 2.4.
Let x = n P i =1 x i,i be a diagonal matrix. Then e k,k ∆( x ) e k,k = ∆( x k,k ) (6) for all k ∈ , n. Proof.
Take an automorphism Φ of the form (2) such that∆( x ) = aψ ( x ) a − and ∆( x k,k ) = aψ ( x kk ) a − . If necessary, replacing x k,k by λe + x k,k ( | λ | > || x k,k || ) we can assume that x k,k isinvertible. Using the equality (5), we obtain that ∆( x k,k ) ∈ A k,k . Since ∆( x k,k ) a = aψ ( x kk ) , x k,k ) a ) k,i = x k,k a k,i , aψ ( x k,k )) i,k = a i,k ψ ( x k,k )for all i = k. Since x k,k and ψ ( x k,k ) are invertible, we have that a i,k = a k,i = 0 for all i = k. Further∆( x k,k ) = e k,k ∆( x k,k ) e k,k = e k,k aψ ( x k,k ) a − e k,k = a k,k ψ ( x k,k ) a − k,k . Since x is a diagonal matrix and a i,k = a k,i = 0 for all i = k. we get e k,k ∆( x ) e k,k = e k,k aψ ( x ) a − e k,k = a k,k ψ ( x k,k ) a − k,k . Thus e k,k ∆( x ) e k,k = ∆( x k,k ) . The proof is complete. (cid:3)
Lemma 2.5.
Let x = x i,i ∈ A i,i . Then e j,i ∆( x ) e i,j = ∆( e j,i xe i,j ) (7) for every j ∈ { , · · · , n } . Proof.
The case when i = j has been already proved (see Lemma 2.4).Suppose that i = j. For an arbitrary element x = x i,i ∈ A i,i , consider y = x + e j,i xe i,j ∈A i,i + A j,j . Take an automorphism Φ of the form (2) such that∆( y ) = aψ ( y ) a − and ∆( v ) = aψ ( v ) a − , -LOCAL AUTOMORPHISMS ON AW ∗ -ALGEBRAS 5 where v is the element from (3). Since ∆( v ) = v and δ ( v ) = v, it follows that a has theform (4). By Lemma 2.4 we obtain that e j,i ∆( x ) e i,j = e j,i e i,i ∆( y ) e i,i e i,j = a ψ ( y ) a − e j,j , ∆( e j,i xe i,j ) = e j,j ∆( y ) e j,j = a ψ ( x ) a − e j,j . The proof is complete. (cid:3)
Further in Lemmata 2.6–2.11 we assume that n ≥ . Lemma 2.6. ∆ i,i is additive for all i ∈ , n. Proof.
Let i ∈ , n. Since n ≥ , we can take different numbers k, s such that( k − i )( s − i ) = 0 . For arbitrary x, y ∈ A i,i consider the diagonal element z ∈ A i,i + A k,k + A s,s suchthat z i,i = x + y, z k,k = x, z s,s = y. Take an automorphism Φ of the form (2) such that∆( z ) = aψ ( z ) a − and ∆( v ) = aψ ( v ) a − , where v is the element from (3). Since ∆( v ) = v and δ ( v ) = v, it follows that a has theform (4). Using Lemmata 2.4 and 2.5 we obtain that∆ i,i ( x + y ) (6) = e i,i ∆( z ) e i,i = a ψ ( x + y ) a − e i,i , ∆ i,i ( x ) (7) = e i,k ∆( e k,i xe i,k ) e k,i (6) = e i,k e k,k ∆( z ) e k,k e k,i == a ψ ( x ) a − e i,i , ∆ i,i ( y ) (7) = e i,s ∆( e s,i ye i,s ) e s,i (6) = e i,s e s,s ∆( z ) e s,s e s,i == a ψ ( y ) a − e i,i . Hence ∆ i,i ( x + y ) = ∆ i,i ( x ) + ∆ i,i ( y ) . The proof is complete. (cid:3)
As it was mentioned in the beginning of the section any additive 2-local automorphismis a Jordan automorphism. Since A i,i ∼ = A has the property (J) , by Lemma 2.6 thereexists a decomposition A = A ⊕ A such that x ∈ A 7→ p (∆ i,i ( x )) ∈ A is a homomorphism and x ∈ A 7→ p (∆ i,i ( x )) ∈ A is an anti-homomorphism.Suppose that p = 0 . By the condition (M) we can find elements x, y ∈ A such that xy = 0 and yx = 0 . Then0 = p (∆ i,i ( xy )) = p (∆ i,i ( y )) p (∆ i,i ( x )) . On the other hand, ∆ i,i ( y )∆ i,i ( x ) = Φ x,y ( y )Φ x,y ( x ) = Φ x,y ( yx ) = 0 . From this contradiction we obtain that p = 0 . So, we have the following
Lemma 2.7. ∆ i,i is an automorphism for all i ∈ , n. SHAVKAT AYUPOV, KARIMBERGEN KUDAYBERGENOV, AND TURABAY KALANDAROV
Denote by D n ( A ) the set of all diagonal matrices from M n ( A ) , i.e. the set of allmatrices of the following form x = x . . . x . . . . . . x n −
00 0 . . . x n . Let us consider an operator ∆ , of the form (1). By Lemmata 2.4 and 2.5 we obtainthat Lemma 2.8. ∆ | D n ( A ) = ∆ , | D n ( A ) and ∆ , | sp { e i,j } ni,j =1 = id | sp { e i,j } ni,j =1 . Now we are in position to pass to the second step of our proof. In this step we showthat if a 2-local automorphism ∆ satisfies the following conditions∆ | D n ( A ) ≡ id | D n ( A ) and ∆ | sp { e i,j } ni,j =1 ≡ id | sp { e i,j } ni,j =1 , then it is the identical map.In following five Lemmata 2.9-2.13 we shall consider 2-local automorphisms whichsatisfy the latter equalities.We denote by e the unit of the algebra A . Lemma 2.9.
Let x ∈ M n ( A ) . Then ∆( x ) k,k = x k,k for all k ∈ , n. Proof.
Let x ∈ M n ( A ) , and fix k ∈ , n. Since ∆ is homogeneous, we can assume that k x k,k k < , where k · k is the norm on A . Take a diagonal element y in M n ( A ) with y k,k = e + x k,k and y i,i = 0 otherwise. Since k x k,k k < , it follows that e + x k,k isinvertible in A . Take an automorphism Φ of the form (2) such that∆( x ) = aψ ( x ) a − and ∆( y ) = aψ ( y ) a − . Since y ∈ D n ( A ) we have that y = ∆( y ) = aψ ( y ) a − , and therefore0 = ∆( y ) i,k = a i,k ( e + x k,k ) , y ) k,i = − ( e + x k,k ) a k,i for all i = k. Thus a i,k = a k,i = 0for all i = k. The above equalities imply that∆( x ) k,k = ∆( y ) k,k = x k,k . The proof is complete. (cid:3)
Lemma 2.10.
Let x be a matrix with x k,s = λe. Then ∆( x ) k,s = λe. Proof.
We have e s,k ∆( x ) e s,k = ∆( e s,k )∆( x )∆( e s,k ) = Φ e s,k ,x ( e s,k )Φ e s,k ,x ( x )Φ e s,k ,x ( e s,k ) == Φ e s,k ,x ( e s,k xe s,k ) = Φ e s,k ,x ( λe s,k ) = λ ∆( e s,k ) = λe s,k . Thus e k,k ∆( x ) e s,s = e k,s e s,k ∆( x ) e s,k e k,s = λe k,s . This means that ∆( x ) k,s = λe. The proof is complete. (cid:3)
Lemma 2.11.
Let k, s be numbers such that k = s and let x be a matrix with x k,s = λe,λ = 0 . Then ∆( x ) s,k = x s,k . -LOCAL AUTOMORPHISMS ON AW ∗ -ALGEBRAS 7 Proof.
Take a diagonal element y such that y k,k = x s,k and y i,i = λ i e otherwise, where λ i ( i = k ) are distinct numbers with | λ i | > k x s,k k . Take an automorphism Φ such that∆( x ) = Φ( x ) and ∆( y ) = Φ( y ) . Then ya = aψ ( y ) , and therefore0 = ( ya − aψ ( y )) ij = λ j a i,j − λ i a i,j = a i,j ( λ j − λ i ) for ( i − j )( i − k )( j − k ) = 0 , ya − aψ ( y )) i,k = a i,k ψ ( y k,k ) − λ i a i,k = a i,k ( ψ ( x s,k ) − λ i ) for i = k, ya − aψ ( y )) k,j = a k,j λ j − ψ ( y kk ) a kj = ( λ j − ψ ( x s,k )) a k,j for j = k. Thus a i,j = 0 for all i = j, i.e. a is a diagonal element. Since λe = ∆( x ) ks = a kk λea − ss , it follows that a k,k = a s,s . Finally,∆( x ) s,k = a s,s ψ ( x s,k ) a − k,k == a k,k ψ ( y k,k ) a − k,k = ∆( y ) k,k = x s,k . The proof is complete. (cid:3)
In the next two Lemmata we assume that ∆ is a 2-local automorphism on M ( A ) . Lemma 2.12.
Let x = (cid:18) x , λex , x , (cid:19) and y = (cid:18) x , x , x , x , (cid:19) , where | λ | > || ∆( y ) , || . Then ∆( x ) , = ∆( y ) , . Proof.
Take an automorphism Φ such that∆( x ) = Φ( x ) and ∆( y ) = Φ( y ) . Then (cid:18) λe − ∆( y ) , (∆( x ) − ∆( y )) , (cid:19) (cid:18) a , a , a , a , (cid:19) = (cid:18) a , a , a , a , (cid:19) (cid:18) λe − x , (cid:19) . Thus (cid:26) ( λe − ∆( y ) , ) a , = 0 , (∆( x ) , − ∆( y ) , ) a , = 0 . Since | λ | > || ∆( y ) , || , it follows that λe − ∆( y ) , is invertible in A , and therefore thefirst equality implies that a , = 0 . Thus a , is invertible and the second equality givesus ∆( x ) , = ∆( y ) , . The proof is complete. (cid:3)
Lemma 2.13. ∆ = id.
Proof.
Let x ∈ M ( A ) . By Lemma 2.9 we have that ∆( x ) k,k = x k,k for k = 1 , . Let now k = s. Take a matrix y with y s,k = λe and y i,j = x i,j otherwise. ByLemma 2.11 we have that ∆( y ) k,s = x k,s . Further Lemma 2.12 implies that∆( x ) k,s = ∆( y ) k,s = x k,s . Thus ∆( x ) k,s = ∆( y ) k,s = x k,s for all k, s = 1 , , and therefore ∆( x ) = x. The proof iscomplete. (cid:3)
SHAVKAT AYUPOV, KARIMBERGEN KUDAYBERGENOV, AND TURABAY KALANDAROV
Now we are in position to prove Theorem 2.1.
Proof of Theorem 2.1 . Let ∆ be a 2-local automorphism on M n ( A ) , where n ≥ . By Lemma 2.2 there exists an automorphism Φ on M n ( A ) such that ∆ | sp { e i,j } ni,j =1 =Φ | sp { e i,j } ni,j =1 . Replacing, if necessary, ∆ by Φ − ◦ ∆ , we may assume that ∆ is identicalon sp { e i,j } n i,j =1 . Further, by Lemma 2.8 there exists an automorphism Φ on M n ( A )such that ∆ | D n = Φ | D n . Now replacing ∆ by Φ − ◦ ∆ , we can assume that ∆ acts asthe identity on D n . So, we can assume that∆ | sp { e i,j } ni,j =1 ≡ id | sp { e i,j } ni,j =1 and ∆ | D n ≡ id | D n . Let us to show that ∆ ≡ id. We proceed by induction on n. Let n = 2 . We identify the algebra M ( A ) with the algebra of 2 × M ( B ) , over B = M ( A ) . Let { e i,j } i,j =1 be a system of matrix units in M ( A ) . Then p , = e , + e , , p , = e , + e , , p , = e , + e , , p , = e , + e , is the system of matrix units in M ( B ) . Since ∆ | sp { e i,j } i,j =1 ≡ id | sp { e i,j } i,j =1 , it followsthat ∆ | sp { p i,j } i,j =1 ≡ id | sp { p i,j } i,j =1 . Take an arbitrary element x ∈ p , M ( B ) p , ≡ B . Choose an automorphism Φ on M ( B ) such that ∆( x ) = Φ( x ) , ∆( p , ) = Φ( p , ) . Since ∆( p , ) = p , , we obtain that p , ∆( x ) p , = p , Φ( x ) p , = ∆( x ) . This means that the restriction ∆ , of ∆ onto p , M ( B ) p , ≡ B maps B = M ( A )into itself.If D is the subalgebra of diagonal matrices from M ( A ) , then p , D p , is the sub-algebra of diagonal matrices in the algebra M ( A ) . Since ∆ | D ≡ id | D , it follows that∆ , acts identically on diagonal matrices from M ( A ) . So,∆ , | sp { e i,j } i,j =1 ≡ id | sp { e i,j } i,j =1 and ∆ , | p , D p , ≡ id | p , D p , . By Lemma 2.13 it follows that ∆ , ≡ id. Let D be the set of diagonal matrices from M ( B ) . Since D = (cid:18) B B (cid:19) and ∆ , = id, Lemma 2.4 implies that ∆ | D ≡ id | D . Hence, ∆ is a 2-local derivationon M ( B ) such that∆ | sp { p i,j } i,j =1 ≡ id | sp { p i,j } i,j =1 and ∆ | D ≡ id | D . Again by Lemma 2.13 it follows that ∆ ≡ id. Now assume that the assertion of the Theorem is true for n − . Considering the algebra M n ( A ) as the algebra of 2 × M ( B ) over B = M n − ( A ) and repeating the above arguments we obtain that ∆ ≡ id. The proof iscomplete. ✷ Now we apply Theorem 2.1 to the proof of our main result which describes 2-localautomorphism on AW ∗ -algebras.First note that by [13, Theorem 3.3] (see also [7, Theorem 3.2.3]) any C ∗ -algebra, inparticular, AW ∗ -algebra, has the property (J) . Proof of Theorem 1.1 . Let M be an arbitrary AW ∗ -algebra without finite type Idirect summands. Then there exist mutually orthogonal central projections z , z , z in -LOCAL AUTOMORPHISMS ON AW ∗ -ALGEBRAS 9 M such that M = z M ⊕ z M ⊕ z M, where z M, z M, z M are algebras of types I ∞ , IIand III, respectively. Then the halving Lemma [5, P. 120, Theorem 1] applied to eachsummand implies that the unit z i of the algebra z i M, ( i = 1 , ,
3) can be representedas a sum of mutually equivalent orthogonal projections e ( i )1 , e ( i )2 , e ( i )3 , e ( i )4 from z i M. Set e k = P i =1 e ( i ) k , k = 1 , , , . Then the map x P i,j =1 e i xe j defines an isomorphism betweenthe algebra M and the matrix algebra M ( A ) , where A = e , M e , . Moreover, thealgebra A has the properties (J) and (M) (see the Remark 1.2 after the definition ofproperty (M) ). Therefore Theorem 2.1 implies that any 2-local automorphism on M isan automorphism. The proof is complete. ✷ . Acknowledgments
The authors are indebted to the reviewer for useful remarks.
References [1] Sh. A. Ayupov and K. K. Kudaybergenov, AW ∗ -algebras, Journal of Physics: Conference Series, (2016) 1–10.[2] Sh. A. Ayupov, K. K. Kudaybergenov,
Derivations, local and 2-local derivations on algebras ofmeasurable operators, in Topics in Functional Analysis and Algebra,
Contemporary Mathematics ,vol. 672, Amer. Math. Soc., Providence, RI, 2016, pp. 51-72.[3] Sh. A. Ayupov, K. K. Kudaybergenov and A. M. Peralta,
A survey on local and 2-local derivationson C ∗ - and von Neumann algebras, in Topics in Functional Analysis and Algebra, ContemporaryMathematics , vol. 672, Amer. Math. Soc., Providence, RI, 2016, pp. 73-126.[4] G.M.Benkart, J.M. Osborn,
Derivations and automorphisms of nonassociative matrix algebras,
Trans. AMS.
A Kowalski-Slodkowski theorem for2-local ∗ -homomorphisms on von Neumann algebras , Revista Serie A Matematicas , Issue 2(2015), Page 551-568.[7] O. Brattelli, D. Robinson, Operator algebras and quantum statistical mechanics,
Local derivations on C ∗ -algebras are derivations, Trans. Amer. Math. Soc., (200) 313–325.[9] R. V. Kadison,
Local derivations,
J. Algebra, (1990) 494–509.[10] S.O. Kim, J.S. Kim,
Local automorphisms and derivations on M n , Proc. Amer. Math. Soc. ,no. 5, 1389-1392 (2004).[11] D. R. Larson and A. R. Sourour,
Local derivations and local automorphisms of B ( X ) , Operatortheory: operator algebras and applications, part 2 (Durham,NH, 1988), 187–194, Proc. Sym-pos. Pure Math. 51, Part 2, Amer.Math.Soc., Providence, RI, (1990).[12] P. ˇSemrl,
Local automorphisms and derivations on B ( H ) , Proc. Amer. Math. Soc. , 2677-2680(1997).[13] E. Stormer,
On the Jordan structure of C ∗ -algebras , Trans. Amer. Math. Soc. (1965), 438-447. V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 81,Mirzo Ulughbek street, 100170, Tashkent, Uzbekistan
National University of Uzbekistan, 4, University str.,Tashkent, Uzbekistan
E-mail address : sh − [email protected] Ch. Abdirov 1, Department of Mathematics, Karakalpak State University, Nukus230113, Uzbekistan
E-mail address : [email protected] Ch. Abdirov 1, Department of Mathematics, Karakalpak State University, Nukus230113, Uzbekistan
E-mail address : turaboy −−