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Local Lie derivations of factor von Neumann algebras
Linear Algebra and its Applications 519 (2017) 208–218
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Local Lie derivations of factor von Neumann algebras ✩ Dan Liu ∗ , Jianhua Zhang School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, People’s Republic of China
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Article history: Received 3 September 2016 Accepted 2 January 2017 Available online 9 January 2017 Submitted by M. Bresar
a b s t r a c t Let A be a factor von Neumann algebra acting on a complex Hilbert space H with dim(A) ≥ 2. We show that each local Lie derivation from A into itself is a Lie derivation. © 2017 Elsevier Inc. All rights reserved.
MSC: 47B47 47L10 Keywords: Derivation Local Lie derivation Factor von Neumann algebra
1. Introduction Let A be an algebra over the complex field C. A linear map ϕ : A → A is called a derivation if ϕ(AB) = ϕ(A)B + Aϕ(B) for all A, B ∈ A, and ϕ is called a local derivation if for each A ∈ A, there exists a derivation ϕA of A, depending on A, such that ϕ(A) = ϕA (A). Several authors have considered the relationship between local ✩ This research was supported by the NNSF of China (No. 11471199) and the IFGP of Shaanxi Normal University (No. 2015CXB007). * Corresponding author. E-mail addresses: [email protected] (D. Liu), [email protected] (J. Zhang).
http://dx.doi.org/10.1016/j.laa.2017.01.004 0024-3795/© 2017 Elsevier Inc. All rights reserved.
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derivations and derivations on self-adjoint algebras or non-self-adjoint algebras, see for example [1,3,5,8,11,15,16] and the references therein. In [10], Kadison proved that every norm-continuous local derivation from a von Neumann algebra into its dual normal bimodule is a derivation. In [12], Larson and Sourour obtained the same result for B(X), the algebra of all bounded linear operators on a Banach space X. In [9], Johnson extended Kadison’s result to local derivations from any C ∗ -algebra into its Banach bimodule. In [6], Hadwin and Li proved that any bounded local derivation from a CSL algebra into its Banach bimodule is a derivation. A linear map ϕ of A is called a Lie derivation if ϕ([A, B]) = [ϕ(A), B] + [A, ϕ(B)] for all A, B ∈ A, where [A, B] = AB − BA is the usual Lie product. We say that a linear map from A into itself is a local Lie derivation if for each A ∈ A, there exists a Lie derivation ϕA of A, depending on A, such that ϕ(A) = ϕA (A). Chen et al. [2] studied local Lie derivations of operator algebras on Banach spaces. Noticing that the proof in [2] depends heavily on rank one operators in B(X), but von Neumann algebras need not contain rank one operators. It is clear that any attempt to extend Chen’s results to general operator algebras must use different techniques. The purpose of the present paper is to study local Lie derivations of factor von Neumann algebras. Let A be a factor von Neumann algebra acting on a complex Hilbert space H. By factor we mean that the center of A is CI, where I is the identity of A. It follows from [4] that every operator A ∈ A can be written as finite linear combinations of projections in A. We refer the reader to [14] for the basic theory of von Neumann algebras. We close this section with a well known result concerning Lie derivations. Proposition 1.1. ([13]) Let A be a von Neumann algebra. If ϕ : A → A is a Lie derivation, then ϕ = d + τ , where d is an associative derivation and τ is a linear map from A into its center vanishing on each commutator. 2. Main results Our main result reads as follows. Theorem 2.1. Let A be a factor von Neumann algebra acting on a complex Hilbert space H with dim(A) ≥ 2. Then each local Lie derivation ϕ from A into itself is a Lie derivation. If dim(A) < ∞, then A is isomorphic to Mn (C). It follows from Theorem 2.1 of [2] that each local Lie derivation of A is a Lie derivation. In the following, A is a infinite dimensional factor von Neumann algebra. For any A ∈ A, the symbol ϕA stands for a Lie derivation from A into itself such that ϕ(A) = ϕA (A). Claim 2.2. For every idempotents P, Q ∈ A and X ∈ A, there exist λ1 , λ2 , λ3 , λ4 ∈ C such that
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ϕ(P XQ) = ϕ(P X)Q + P ϕ(XQ) − P ϕ(X)Q + λ1 P ⊥ Q⊥ − λ2 P Q⊥ + λ3 P Q − λ4 P ⊥ Q, where P ⊥ = I − P and Q⊥ = I − Q. Proposition 1.1 implies that for every idempotents P, Q ∈ A and X ∈ A, there exist derivations d1 , d2 , d3 , d4 : A → A and linear maps τ1 , τ2 , τ3 , τ4 : A → CI vanishing on each commutator such that ϕ(P XQ) = ϕP XQ (P XQ) = d1 (P XQ) + τ1 (P XQ), ⊥
⊥
⊥
⊥
(2.1)
ϕ(P XQ) = ϕP ⊥ XQ (P XQ) = d2 (P XQ) + τ2 (P XQ),
(2.2)
ϕ(P ⊥ XQ⊥ ) = ϕP ⊥ XQ⊥ (P ⊥ XQ⊥ ) = d3 (P ⊥ XQ⊥ ) + τ3 (P ⊥ XQ⊥ ),
(2.3)
ϕ(P XQ⊥ ) = ϕP XQ⊥ (P XQ⊥ ) = d4 (P XQ⊥ ) + τ4 (P XQ⊥ ).
(2.4)
Set λ1 I = τ1 (P XQ), λ2 I = τ2 (P ⊥ XQ), λ3 I = τ3 (P ⊥ XQ⊥ ), λ4 I = τ4 (P XQ⊥ ). It follows from Eqs. (2.1)–(2.4) that P ⊥ ϕ(P XQ)Q⊥ = λ1 P ⊥ Q⊥ , P ϕ(P ⊥ XQ)Q⊥ = λ2 P Q⊥ , P ϕ(P ⊥ XQ⊥ )Q = λ3 P Q, P ⊥ ϕ(P XQ⊥ )Q = λ4 P ⊥ Q. Hence ϕ(P XQ)Q⊥ = P ϕ(P XQ)Q⊥ + P ⊥ ϕ(P XQ)Q⊥ = P ϕ(XQ)Q⊥ − P ϕ(P ⊥ XQ)Q⊥ + P ⊥ ϕ(P XQ)Q⊥ = P ϕ(XQ)Q⊥ + λ1 P ⊥ Q⊥ − λ2 P Q⊥ = P ϕ(XQ) − P ϕ(XQ)Q + λ1 P ⊥ Q⊥ − λ2 P Q⊥ , ϕ(P XQ⊥ )Q = P ϕ(P XQ⊥ )Q + P ⊥ ϕ(P XQ⊥ )Q = P ϕ(XQ⊥ )Q − P ϕ(P ⊥ XQ⊥ )Q + P ⊥ ϕ(P XQ⊥ )Q = P ϕ(XQ⊥ )Q − λ3 P Q + λ4 P ⊥ Q. Thus, ϕ(P XQ) = ϕ(P XQ)Q⊥ + ϕ(P XQ)Q = ϕ(P XQ)Q⊥ + ϕ(P X)Q − ϕ(P XQ⊥ )Q
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= ϕ(P X)Q + P ϕ(XQ) − P ϕ(X)Q + λ1 P ⊥ Q⊥ − λ2 P Q⊥ + λ3 P Q − λ4 P ⊥ Q. Fix a nontrivial projection P1 ∈ A and let P2 = I − P1 . In what follows, we write 2 Aij = Pi APj for i, j = 1, 2. Then each operator A ∈ A can be written as A = i,j=1 Aij . Note that the notation Aij denotes an arbitrary element of Aij . Claim 2.3. Let Aii ∈ Aii , i = 1, 2. If A11 B12 = B12 A22 for all B12 ∈ A12 , then A11 + A22 ∈ CI. For any X11 ∈ A11 and X12 ∈ A12 , we have A11 X11 X12 = X11 X12 A22 = X11 A11 X12 , which implies (A11 X11 − X11 A11 )X12 = 0. Since A is a factor von Neumann algebra, it follows that A11 X11 = X11 A11 . Then A11 = λ1 P1 , λ1 ∈ C. Similarly, A22 = λ2 P2 , λ2 ∈ C. So λ1 B12 = A11 B12 = B12 A22 = λ2 B12 , which implies λ1 = λ2 . Hence, A11 + A22 = λ1 P1 + λ1 P2 ∈ CI. It is easy to verify that for each derivation d : A → A, we have d(P1 ) = −d(P2 ) ∈ A12 ⊕ A21 and Pj d(Aii )Pj = Pj d(Aij )Pi = 0
(2.5)
for 1 ≤ i = j ≤ 2. Claim 2.4. P1 ϕ(P1 )P1 + P2 ϕ(P1 )P2 = μI for some μ ∈ C. For any A12 ∈ A12 , there exists a Lie derivation of A such that ϕP1 (A12 ) = ϕP1 ([P1 , A12 ]) = [ϕ(P1 ), A12 ] + [P1 , ϕP1 (A12 )] = ϕ(P1 )A12 − A12 ϕ(P1 ) + P1 ϕP1 (A12 ) − ϕP1 (A12 )P1 . Multiplying the above equality from the left by P1 and from the right by P2 , we arrive at P1 ϕ(P1 )A12 = A12 ϕ(P1 )P2 . It follows from Claim 2.3 that P1 ϕ(P1 )P1 + P2 ϕ(P1 )P2 = μI for some μ ∈ C. In the sequel, we define φ(A) = ϕ(A) − [A, P1 ϕ(P1 )P2 − P2 ϕ(P1 )P1 ]. One can verify that φ is also a local Lie derivation. Moreover, by Claim 2.4, we have φ(P1 ) = P1 ϕ(P1 )P1 + P2 ϕ(P1 )P2 = μI. Claim 2.5. φ(Aij ) ⊆ Aij , 1 ≤ i = j ≤ 2. Let B12 ∈ A12 and E be any projection in A11 . Taking P = E, X = P1 and Q = P1 + B12 in Claim 2.2, we have from Eqs. (2.3) and (2.4) that λ3 = λ4 = 0 and then
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φ(E + EB12 ) = φ(E)(P1 + B12 ) + Eφ(P1 + B12 ) − Eφ(P1 )(P1 + B12 ) + λ1 (I − E)(P2 − B12 ) − λ2 E(P2 − B12 ) = φ(E)P1 + φ(E)B12 + Eφ(B12 ) − Eφ(P1 )B12 + λ1 (P2 − B12 + EB12 ) + λ2 EB12 .
(2.6)
Multiplying the above equality from the right by P1 , we arrive at φ(EB12 )P1 = Eφ(B12 )P1 . Since every operator A11 ∈ A11 can be written as finite linear combinations of projections in A11 , it follows that φ(A11 B12 )P1 = A11 φ(B12 )P1
(2.7)
for all A11 ∈ A11 and B12 ∈ A12 . This implies that P2 φ(B12 )P1 = 0. Let F be any projection in A22 . By taking P = P2 + B12 , X = P2 and Q = F in Claim 2.2, we can obtain P2 φ(B12 F ) = P2 φ(B12 )F. Since every operator B22 ∈ A22 can be written as finite linear combinations of projections in A22 , it follows that P2 φ(B12 B22 ) = P2 φ(B12 )B22
(2.8)
for all B12 ∈ A12 and B22 ∈ A22 . Proposition 1.1 implies that there exists a derivation d : A → A and λ ∈ C such that φ(P1 + B12 ) = d(P1 + B12 ) + λI. It follows from the fact φ(P1 ) = μI that φ(B12 ) = d(P1 + B12 ) + (λ − μ)I. By Eqs. (2.5) and (2.9), 0 = P2 φ(B12 )P1 = P2 d(P1 )P1 and hence by Eqs. (2.5) and (2.9) again, P1 φ(B12 )P1 = P1 d(B12 )P1 + (λ − μ)P1 = P1 d(B12 P2 )P1 + (λ − μ)P1 = A12 d(P2 )P1 + (λ − μ)P1 = −A12 d(P1 )P1 + (λ − μ)P1 = (λ − μ)P1
(2.9)
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and P2 φ(B12 )P2 = P2 d(B12 )P2 + (λ − μ)P2 = P2 d(P1 )B12 + (λ − μ)P2 = (λ − μ)P2 . For arbitrary A11 ∈ A11 and B22 ∈ A22 , it follows from Eqs. (2.7) and (2.8) that (λ − μ)A11 = A11 φ(B12 )P1 = P1 φ(A11 B12 )P1 ∈ CP1 and (λ − μ)B22 = P2 φ(B12 )B22 = P2 φ(B12 B22 )P2 ∈ CP2 . If λ −μ is not zero, then A11 ⊆ CP1 and A22 ⊆ CP2 . It follows that dim(A) ≤ 4, contrary to the assumption that dim(A) = ∞. So λ −μ = 0. Thus, P1 φ(B12 )P1 = P2 φ(B12 )P2 = 0. Hence φ(A12 ) ⊆ A12 . With the same argument, we can obtain that φ(A21 ) ⊆ A21 . Claim 2.6. There exist linear functionals τi on Aii such that φ(Aii ) − τi (Aii )I ∈ Aii for all Aii ∈ Aii , i = 1, 2. Let E be any projection in A11 . It follows from Claim 2.2 that there exist λ1 , λ2 , λ3 , λ4 ∈ C such that φ(E) = φ(EP1 P1 ) = φ(E)P1 + Eφ(P1 ) − Eφ(P1 )P1 + λ1 (I − E)P2 − λ2 EP2 + λ3 EP1 − λ4 (I − E)P1 . On the other hand, by Claim 2.2, there exist λ1 , λ2 , λ3 , λ4 ∈ C such that φ(E) = φ(P1 P1 E) = φ(P1 )E + P1 φ(E) − P1 φ(P1 )E + λ1 P2 (I − E) − λ2 P1 (I − E) + λ3 P1 E − λ4 P2 E. Eqs. (2.1)–(2.4) imply that λ1 = λ1 and λ3 = λ4 = λ2 = λ3 = 0. Hence φ(E) = φ(E)P1 + λ1 P2 = P1 φ(E) + λ1 P2 . This implies that
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P2 φ(E)P2 = λ1 P2 and P1 φ(E)P2 = P2 φ(E)P1 = 0. Since every operator A11 ∈ A11 can be written as finite linear combinations of projections in A11 , it follows that P1 φ(A11 )P2 = P2 φ(A11 )P1 = 0 and there exists a scalar τ1 (A11 ) such that P2 φ(A11 )P2 = τ1 (A11 )P2 . Thus, we get φ(A11 ) = P1 φ(A11 )P1 + τ1 (A11 )P2 = P1 φ(A11 )P1 + τ1 (A11 )I − τ1 (A11 )P1 . Consequently, φ(A11 ) − τ1 (A11 )I = P1 φ(A11 )P1 − τ1 (A11 )P1 ∈ A11 . One can verify that τ1 is linear. With the similar argument, we can define a functional τ2 on A22 such that φ(A22 ) − τ2 (A22 )I ∈ A22 for all A22 ∈ A22 . 2 Now for any A = i,j=1 Aij ∈ A, we define two linear maps τ : A → CI and δ : A → A by τ (A) = (τ1 (A11 ) + τ2 (A22 ))I and δ(A) = φ(A) − τ (A). It is easy to verify that δ(P1 ) = 0. Moreover, we have δ(Aij ) ⊆ Aij for i, j = 1, 2 and δ(Aij ) = φ(Aij ) for 1 ≤ i = j ≤ 2. Claim 2.7. δ is a derivation. By Claims 2.5–2.6, it is sufficient to show that δ(Aij Bjk ) = δ(Aij )Bjk + Aij δ(Bjk ) for 1 ≤ i, j, k ≤ 2. We divide it into the following three steps. Step 1. Let B12 ∈ A12 and E be any projection in A11 . Taking P = E, X = P1 and Q = P1 + B12 in Eq. (2.2), we have φ(P1 + B12 − E − EB12 ) = d2 (P1 + B12 − E − EB12 ) + λ2 I. It follows from Eqs. (2.5) and (2.10) that 0 = P2 d2 (P1 − E)P1 = P2 d2 (P1 (P1 − E))P1 = P2 d2 (P1 )(P1 − E) and hence by Eqs. (2.5) and (2.10) again,
(2.10)
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P2 φ(P1 − E)P2 = P2 d2 (B12 − EB12 )P2 + λ2 P2 = P2 d2 (P1 )(P1 − E)B12 + λ2 P2 = λ2 P2 .
(2.11)
Multiplying Eq. (2.6) by P2 from both sides, we arrive at P2 φ(E)P2 = λ1 P2 .
(2.12)
By Eqs. (2.11) and (2.12), λ1 B12 = B12 φ(E) and λ2 EB12 = EB12 φ(P1 − E) = EB12 φ(P1 ) − EB12 φ(E) = EB12 φ(P1 ) − λ1 EB12 . Hence Eq. (2.6) implies that δ(EB12 ) = φ(EB12 ) = φ(E)B12 + Eφ(B12 ) − B12 φ(E) = (δ(E) + τ (E))B12 + Eδ(B12 ) − B12 (δ(E) + τ (E)) = δ(E)B12 + Eδ(B12 ). Note that in the fourth equality, we apply the fact that δ(Aij ) ⊆ Aij (i, j = 1, 2). Since every operator in A11 can be written as finite linear combinations of projections in A11 , it follows that δ(A11 B12 ) = δ(A11 )B12 + A11 δ(B12 ) for all A11 ∈ A11 and B12 ∈ A12 . Similarly, we can get δ(A12 B22 ) = δ(A12 )B22 + A12 δ(B22 ), δ(A21 B11 ) = δ(A21 )B11 + A21 δ(B11 ) and δ(A22 B21 ) = δ(A22 )B21 + A22 δ(B21 ). Step 2. Let A11 and B11 ∈ A11 . For any C12 ∈ A12 , on one hand, by Step 1, we have δ(A11 B11 C12 ) = δ(A11 )B11 C12 + A11 δ(B11 C12 ) = δ(A11 )B11 C12 + A11 δ(B11 )C12 + A11 B11 δ(C12 ). On the other hand, δ(A11 B11 C12 ) = δ(A11 B11 )C12 + A11 B11 δ(C12 ). Comparing these two equalities, we have (δ(A11 B11 ) − δ(A11 )B11 − A11 δ(B11 ))C12 = 0 for all C12 ∈ A12 . Since A is prime, we get
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δ(A11 B11 ) = δ(A11 )B11 + A11 δ(B11 ). Similarly, by considering δ(A22 B22 C21 ), we can get δ(A22 B22 ) = δ(A22 )B22 + A22 δ(B22 ). Step 3. Let A12 , B12 ∈ A12 and B21 ∈ A21 . Taking P = Q = P1 − B12 and X = B12 B21 + B21 in Claim 2.2, we have from P X = P ⊥ Q = P Q⊥ = 0 that 0 = (P1 − B12 )φ(B12 B21 − B12 B21 B12 + B21 − B21 B12 ) − (P1 − B12 )φ(B12 B21 + B21 )(P1 − B12 ) + λ1 (P2 + B12 ) + λ3 (P1 − B12 ) = −φ(B12 B21 B12 ) − P1 φ(B21 B12 ) − B12 φ(B12 B21 ) + B12 φ(B21 B12 ) + φ(B12 B21 )B12 − B12 φ(B21 )B12 + λ1 P2 + λ1 B12 + λ3 P1 − λ3 B12 .
(2.13)
This implies that λ1 = 0 and P1 φ(B21 B12 ) = λ3 P1 . Then φ(B21 B12 )B12 = λ3 B12 , and hence by Eq. (2.13), δ(B12 B21 B12 ) = φ(B12 B21 B12 ) = −B12 φ(B12 B21 ) + B12 φ(B21 B12 ) + φ(B12 B21 )B12 − B12 φ(B21 )B12 − φ(B21 B12 )B12 = −B12 (δ(B12 B21 ) + τ (B12 B21 )) + B12 (δ(B21 B12 ) + τ (B21 B12 )) + (δ(B12 B21 ) + τ (B12 B21 ))B12 − B12 δ(B21 )B12 − (δ(B21 B12 ) + τ (B21 B12 ))B12 = B12 δ(B21 B12 ) + δ(B12 B21 )B12 − B12 δ(B21 )B12 . Replacing B12 with A12 + B12 , we arrive at δ(B12 B21 A12 + A12 B21 B12 ) = δ(B12 B21 )A12 + B12 δ(B21 A12 ) − B12 δ(B21 )A12 + δ(A12 B21 )B12 + A12 δ(B21 B12 ) − A12 δ(B21 )B12 . On the other hand, by Steps 1 and 2, we have
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δ(B12 B21 A12 + A12 B21 B12 ) = δ(B12 B21 )A12 + B12 B21 δ(A12 ) + δ(A12 )B21 B12 + A12 δ(B21 B12 ). Comparing these two equalities, we have (δ(A12 B21 ) − δ(A12 )B21 − A12 δ(B21 ))B12 = −B12 (δ(B21 A12 ) − B21 δ(A12 ) − δ(B21 )A12 ).
(2.14)
Set F (A12 , B21 ) := δ(A12 B21 ) − δ(A12 )B21 − A12 δ(B21 ) and G(B21 , A12 ) := δ(B21 A12 ) − B21 δ(A12 ) − δ(B21 )A12 . By Claim 2.3, F (A12 , B21 ) − G(B21 , A12 ) = αI for some scalar α. Then F (A12 , B21 ) = αP1 and G(B21 , A12 ) = −αP2 . Then by Steps 1 and 2, we have F (A11 A12 , B21 ) = δ(A11 A12 B21 ) − δ(A11 A12 )B21 − A11 A12 δ(B21 ) = δ(A11 )A12 B21 + A11 δ(A12 B21 ) − δ(A11 )A12 B21 − A11 δ(A12 )B21 − A11 A12 δ(B21 ) = αA11 ∈ CP1 and G(B21 , A12 C22 ) = δ(B21 A12 C22 ) − B21 δ(A12 C22 ) − δ(B21 )A12 C22 = δ(B21 A12 )C22 + B21 A12 δ(C22 ) − B21 δ(A12 )C22 − B21 A12 δ(C22 ) − δ(B21 )A12 C22 = −αC22 ∈ CP2 . If α = 0, then A11 ⊆ CP1 and A22 ⊆ CP2 , contrary to the assumption that dim(A) = ∞. So α = 0. Consequently, δ(A12 B21 ) = δ(A12 )B21 + A12 δ(B21 ) and δ(B21 A12 ) = δ(B21 )A12 + B21 δ(A12 ) for all A12 ∈ A12 and B21 ∈ A21 . Claim 2.8. The theorem holds. Now, by Claim 2.7, we can infer that δ is a derivation. Since every derivation of A is inner, there exists an element T ∈ A such that δ(A) = T A − AT for every A ∈ A. Hence letting S = T − P1 ϕ(P1 )P2 + P2 ϕ(P1 )P1 , we have
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ϕ(A) = SA − AS + τ (A) for all A ∈ A. For each commutator R ∈ A, there exists an element SR ∈ A and a linear map τ1 from A into CI vanishing on each commutator such that τ (R) = ϕ(R) − [S, R] = [SR , R] + τ1 (R) − [S, R] = [SR − S, R] ∈ CI. Hence by [7, Problem 230], we have τ (R) = 0 for every commutator R ∈ A. The proof is complete. 2 References [1] M. Brešar, Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 9–21. [2] L. Chen, F. Lu, T. Wang, Local and 2-local Lie derivations of operator algebras on Banach spaces, Integral Equations Operator Theory 77 (2013) 109–121. [3] R.L. Crist, Local derivations on operator algebras, J. Funct. Anal. 135 (1996) 76–92. [4] P.A. Fillmore, D.M. Topping, Operator algebras generated by projections, Duke Math. J. 34 (1967) 333–336. [5] D. Hadwin, J. Li, Local derivations and local automorphisms, J. Math. Anal. Appl. 290 (2004) 702–714. [6] D. Hadwin, J. Li, Local derivations and local automorphisms on some algebras, J. Operator Theory 60 (2008) 29–44. [7] P.R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer-Verlag, New York, 1982. [8] W. Jing, Local derivations on reflexive algebras II, Proc. Amer. Math. Soc. 129 (2001) 1733–1737. [9] B.E. Johnson, Local derivations on C∗-algebras are derivations, Trans. Amer. Math. Soc. 353 (2001) 313–325. [10] R.V. Kadison, Local derivations, J. Algebra 130 (1990) 494–509. [11] S.O. Kim, J.S. Kim, Local automorphisms and derivations on Mn , Proc. Amer. Math. Soc. 132 (2004) 1389–1392. [12] D.R. Larson, A.R. Sourour, Local derivations and local automorphisms of B(X), Proc. Sympos. Pure Math. 51 (1990) 187–194. [13] C.R. Miers, Lie derivations of von Neumann algebras, Duke Math. J. 40 (1973) 403–409. [14] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York, 1979. [15] J. Zhang, G. Ji, H. Cao, Local derivations of nest subalgebras of von Neumann algebras, Linear Algebra Appl. 392 (2004) 61–69. [16] J. Zhang, F. Pan, A. Yang, Local derivations on certain CSL algebras, Linear Algebra Appl. 413 (2006) 93–99.
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Local Lie derivations of factor von Neumann algebras ✩ Dan Liu ∗ , Jianhua Zhang School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 3 September 2016 Accepted 2 January 2017 Available online 9 January 2017 Submitted by M. Bresar
a b s t r a c t Let A be a factor von Neumann algebra acting on a complex Hilbert space H with dim(A) ≥ 2. We show that each local Lie derivation from A into itself is a Lie derivation. © 2017 Elsevier Inc. All rights reserved.
MSC: 47B47 47L10 Keywords: Derivation Local Lie derivation Factor von Neumann algebra
1. Introduction Let A be an algebra over the complex field C. A linear map ϕ : A → A is called a derivation if ϕ(AB) = ϕ(A)B + Aϕ(B) for all A, B ∈ A, and ϕ is called a local derivation if for each A ∈ A, there exists a derivation ϕA of A, depending on A, such that ϕ(A) = ϕA (A). Several authors have considered the relationship between local ✩ This research was supported by the NNSF of China (No. 11471199) and the IFGP of Shaanxi Normal University (No. 2015CXB007). * Corresponding author. E-mail addresses: [email protected] (D. Liu), [email protected] (J. Zhang).
http://dx.doi.org/10.1016/j.laa.2017.01.004 0024-3795/© 2017 Elsevier Inc. All rights reserved.
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derivations and derivations on self-adjoint algebras or non-self-adjoint algebras, see for example [1,3,5,8,11,15,16] and the references therein. In [10], Kadison proved that every norm-continuous local derivation from a von Neumann algebra into its dual normal bimodule is a derivation. In [12], Larson and Sourour obtained the same result for B(X), the algebra of all bounded linear operators on a Banach space X. In [9], Johnson extended Kadison’s result to local derivations from any C ∗ -algebra into its Banach bimodule. In [6], Hadwin and Li proved that any bounded local derivation from a CSL algebra into its Banach bimodule is a derivation. A linear map ϕ of A is called a Lie derivation if ϕ([A, B]) = [ϕ(A), B] + [A, ϕ(B)] for all A, B ∈ A, where [A, B] = AB − BA is the usual Lie product. We say that a linear map from A into itself is a local Lie derivation if for each A ∈ A, there exists a Lie derivation ϕA of A, depending on A, such that ϕ(A) = ϕA (A). Chen et al. [2] studied local Lie derivations of operator algebras on Banach spaces. Noticing that the proof in [2] depends heavily on rank one operators in B(X), but von Neumann algebras need not contain rank one operators. It is clear that any attempt to extend Chen’s results to general operator algebras must use different techniques. The purpose of the present paper is to study local Lie derivations of factor von Neumann algebras. Let A be a factor von Neumann algebra acting on a complex Hilbert space H. By factor we mean that the center of A is CI, where I is the identity of A. It follows from [4] that every operator A ∈ A can be written as finite linear combinations of projections in A. We refer the reader to [14] for the basic theory of von Neumann algebras. We close this section with a well known result concerning Lie derivations. Proposition 1.1. ([13]) Let A be a von Neumann algebra. If ϕ : A → A is a Lie derivation, then ϕ = d + τ , where d is an associative derivation and τ is a linear map from A into its center vanishing on each commutator. 2. Main results Our main result reads as follows. Theorem 2.1. Let A be a factor von Neumann algebra acting on a complex Hilbert space H with dim(A) ≥ 2. Then each local Lie derivation ϕ from A into itself is a Lie derivation. If dim(A) < ∞, then A is isomorphic to Mn (C). It follows from Theorem 2.1 of [2] that each local Lie derivation of A is a Lie derivation. In the following, A is a infinite dimensional factor von Neumann algebra. For any A ∈ A, the symbol ϕA stands for a Lie derivation from A into itself such that ϕ(A) = ϕA (A). Claim 2.2. For every idempotents P, Q ∈ A and X ∈ A, there exist λ1 , λ2 , λ3 , λ4 ∈ C such that
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ϕ(P XQ) = ϕ(P X)Q + P ϕ(XQ) − P ϕ(X)Q + λ1 P ⊥ Q⊥ − λ2 P Q⊥ + λ3 P Q − λ4 P ⊥ Q, where P ⊥ = I − P and Q⊥ = I − Q. Proposition 1.1 implies that for every idempotents P, Q ∈ A and X ∈ A, there exist derivations d1 , d2 , d3 , d4 : A → A and linear maps τ1 , τ2 , τ3 , τ4 : A → CI vanishing on each commutator such that ϕ(P XQ) = ϕP XQ (P XQ) = d1 (P XQ) + τ1 (P XQ), ⊥
⊥
⊥
⊥
(2.1)
ϕ(P XQ) = ϕP ⊥ XQ (P XQ) = d2 (P XQ) + τ2 (P XQ),
(2.2)
ϕ(P ⊥ XQ⊥ ) = ϕP ⊥ XQ⊥ (P ⊥ XQ⊥ ) = d3 (P ⊥ XQ⊥ ) + τ3 (P ⊥ XQ⊥ ),
(2.3)
ϕ(P XQ⊥ ) = ϕP XQ⊥ (P XQ⊥ ) = d4 (P XQ⊥ ) + τ4 (P XQ⊥ ).
(2.4)
Set λ1 I = τ1 (P XQ), λ2 I = τ2 (P ⊥ XQ), λ3 I = τ3 (P ⊥ XQ⊥ ), λ4 I = τ4 (P XQ⊥ ). It follows from Eqs. (2.1)–(2.4) that P ⊥ ϕ(P XQ)Q⊥ = λ1 P ⊥ Q⊥ , P ϕ(P ⊥ XQ)Q⊥ = λ2 P Q⊥ , P ϕ(P ⊥ XQ⊥ )Q = λ3 P Q, P ⊥ ϕ(P XQ⊥ )Q = λ4 P ⊥ Q. Hence ϕ(P XQ)Q⊥ = P ϕ(P XQ)Q⊥ + P ⊥ ϕ(P XQ)Q⊥ = P ϕ(XQ)Q⊥ − P ϕ(P ⊥ XQ)Q⊥ + P ⊥ ϕ(P XQ)Q⊥ = P ϕ(XQ)Q⊥ + λ1 P ⊥ Q⊥ − λ2 P Q⊥ = P ϕ(XQ) − P ϕ(XQ)Q + λ1 P ⊥ Q⊥ − λ2 P Q⊥ , ϕ(P XQ⊥ )Q = P ϕ(P XQ⊥ )Q + P ⊥ ϕ(P XQ⊥ )Q = P ϕ(XQ⊥ )Q − P ϕ(P ⊥ XQ⊥ )Q + P ⊥ ϕ(P XQ⊥ )Q = P ϕ(XQ⊥ )Q − λ3 P Q + λ4 P ⊥ Q. Thus, ϕ(P XQ) = ϕ(P XQ)Q⊥ + ϕ(P XQ)Q = ϕ(P XQ)Q⊥ + ϕ(P X)Q − ϕ(P XQ⊥ )Q
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= ϕ(P X)Q + P ϕ(XQ) − P ϕ(X)Q + λ1 P ⊥ Q⊥ − λ2 P Q⊥ + λ3 P Q − λ4 P ⊥ Q. Fix a nontrivial projection P1 ∈ A and let P2 = I − P1 . In what follows, we write 2 Aij = Pi APj for i, j = 1, 2. Then each operator A ∈ A can be written as A = i,j=1 Aij . Note that the notation Aij denotes an arbitrary element of Aij . Claim 2.3. Let Aii ∈ Aii , i = 1, 2. If A11 B12 = B12 A22 for all B12 ∈ A12 , then A11 + A22 ∈ CI. For any X11 ∈ A11 and X12 ∈ A12 , we have A11 X11 X12 = X11 X12 A22 = X11 A11 X12 , which implies (A11 X11 − X11 A11 )X12 = 0. Since A is a factor von Neumann algebra, it follows that A11 X11 = X11 A11 . Then A11 = λ1 P1 , λ1 ∈ C. Similarly, A22 = λ2 P2 , λ2 ∈ C. So λ1 B12 = A11 B12 = B12 A22 = λ2 B12 , which implies λ1 = λ2 . Hence, A11 + A22 = λ1 P1 + λ1 P2 ∈ CI. It is easy to verify that for each derivation d : A → A, we have d(P1 ) = −d(P2 ) ∈ A12 ⊕ A21 and Pj d(Aii )Pj = Pj d(Aij )Pi = 0
(2.5)
for 1 ≤ i = j ≤ 2. Claim 2.4. P1 ϕ(P1 )P1 + P2 ϕ(P1 )P2 = μI for some μ ∈ C. For any A12 ∈ A12 , there exists a Lie derivation of A such that ϕP1 (A12 ) = ϕP1 ([P1 , A12 ]) = [ϕ(P1 ), A12 ] + [P1 , ϕP1 (A12 )] = ϕ(P1 )A12 − A12 ϕ(P1 ) + P1 ϕP1 (A12 ) − ϕP1 (A12 )P1 . Multiplying the above equality from the left by P1 and from the right by P2 , we arrive at P1 ϕ(P1 )A12 = A12 ϕ(P1 )P2 . It follows from Claim 2.3 that P1 ϕ(P1 )P1 + P2 ϕ(P1 )P2 = μI for some μ ∈ C. In the sequel, we define φ(A) = ϕ(A) − [A, P1 ϕ(P1 )P2 − P2 ϕ(P1 )P1 ]. One can verify that φ is also a local Lie derivation. Moreover, by Claim 2.4, we have φ(P1 ) = P1 ϕ(P1 )P1 + P2 ϕ(P1 )P2 = μI. Claim 2.5. φ(Aij ) ⊆ Aij , 1 ≤ i = j ≤ 2. Let B12 ∈ A12 and E be any projection in A11 . Taking P = E, X = P1 and Q = P1 + B12 in Claim 2.2, we have from Eqs. (2.3) and (2.4) that λ3 = λ4 = 0 and then
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φ(E + EB12 ) = φ(E)(P1 + B12 ) + Eφ(P1 + B12 ) − Eφ(P1 )(P1 + B12 ) + λ1 (I − E)(P2 − B12 ) − λ2 E(P2 − B12 ) = φ(E)P1 + φ(E)B12 + Eφ(B12 ) − Eφ(P1 )B12 + λ1 (P2 − B12 + EB12 ) + λ2 EB12 .
(2.6)
Multiplying the above equality from the right by P1 , we arrive at φ(EB12 )P1 = Eφ(B12 )P1 . Since every operator A11 ∈ A11 can be written as finite linear combinations of projections in A11 , it follows that φ(A11 B12 )P1 = A11 φ(B12 )P1
(2.7)
for all A11 ∈ A11 and B12 ∈ A12 . This implies that P2 φ(B12 )P1 = 0. Let F be any projection in A22 . By taking P = P2 + B12 , X = P2 and Q = F in Claim 2.2, we can obtain P2 φ(B12 F ) = P2 φ(B12 )F. Since every operator B22 ∈ A22 can be written as finite linear combinations of projections in A22 , it follows that P2 φ(B12 B22 ) = P2 φ(B12 )B22
(2.8)
for all B12 ∈ A12 and B22 ∈ A22 . Proposition 1.1 implies that there exists a derivation d : A → A and λ ∈ C such that φ(P1 + B12 ) = d(P1 + B12 ) + λI. It follows from the fact φ(P1 ) = μI that φ(B12 ) = d(P1 + B12 ) + (λ − μ)I. By Eqs. (2.5) and (2.9), 0 = P2 φ(B12 )P1 = P2 d(P1 )P1 and hence by Eqs. (2.5) and (2.9) again, P1 φ(B12 )P1 = P1 d(B12 )P1 + (λ − μ)P1 = P1 d(B12 P2 )P1 + (λ − μ)P1 = A12 d(P2 )P1 + (λ − μ)P1 = −A12 d(P1 )P1 + (λ − μ)P1 = (λ − μ)P1
(2.9)
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and P2 φ(B12 )P2 = P2 d(B12 )P2 + (λ − μ)P2 = P2 d(P1 )B12 + (λ − μ)P2 = (λ − μ)P2 . For arbitrary A11 ∈ A11 and B22 ∈ A22 , it follows from Eqs. (2.7) and (2.8) that (λ − μ)A11 = A11 φ(B12 )P1 = P1 φ(A11 B12 )P1 ∈ CP1 and (λ − μ)B22 = P2 φ(B12 )B22 = P2 φ(B12 B22 )P2 ∈ CP2 . If λ −μ is not zero, then A11 ⊆ CP1 and A22 ⊆ CP2 . It follows that dim(A) ≤ 4, contrary to the assumption that dim(A) = ∞. So λ −μ = 0. Thus, P1 φ(B12 )P1 = P2 φ(B12 )P2 = 0. Hence φ(A12 ) ⊆ A12 . With the same argument, we can obtain that φ(A21 ) ⊆ A21 . Claim 2.6. There exist linear functionals τi on Aii such that φ(Aii ) − τi (Aii )I ∈ Aii for all Aii ∈ Aii , i = 1, 2. Let E be any projection in A11 . It follows from Claim 2.2 that there exist λ1 , λ2 , λ3 , λ4 ∈ C such that φ(E) = φ(EP1 P1 ) = φ(E)P1 + Eφ(P1 ) − Eφ(P1 )P1 + λ1 (I − E)P2 − λ2 EP2 + λ3 EP1 − λ4 (I − E)P1 . On the other hand, by Claim 2.2, there exist λ1 , λ2 , λ3 , λ4 ∈ C such that φ(E) = φ(P1 P1 E) = φ(P1 )E + P1 φ(E) − P1 φ(P1 )E + λ1 P2 (I − E) − λ2 P1 (I − E) + λ3 P1 E − λ4 P2 E. Eqs. (2.1)–(2.4) imply that λ1 = λ1 and λ3 = λ4 = λ2 = λ3 = 0. Hence φ(E) = φ(E)P1 + λ1 P2 = P1 φ(E) + λ1 P2 . This implies that
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P2 φ(E)P2 = λ1 P2 and P1 φ(E)P2 = P2 φ(E)P1 = 0. Since every operator A11 ∈ A11 can be written as finite linear combinations of projections in A11 , it follows that P1 φ(A11 )P2 = P2 φ(A11 )P1 = 0 and there exists a scalar τ1 (A11 ) such that P2 φ(A11 )P2 = τ1 (A11 )P2 . Thus, we get φ(A11 ) = P1 φ(A11 )P1 + τ1 (A11 )P2 = P1 φ(A11 )P1 + τ1 (A11 )I − τ1 (A11 )P1 . Consequently, φ(A11 ) − τ1 (A11 )I = P1 φ(A11 )P1 − τ1 (A11 )P1 ∈ A11 . One can verify that τ1 is linear. With the similar argument, we can define a functional τ2 on A22 such that φ(A22 ) − τ2 (A22 )I ∈ A22 for all A22 ∈ A22 . 2 Now for any A = i,j=1 Aij ∈ A, we define two linear maps τ : A → CI and δ : A → A by τ (A) = (τ1 (A11 ) + τ2 (A22 ))I and δ(A) = φ(A) − τ (A). It is easy to verify that δ(P1 ) = 0. Moreover, we have δ(Aij ) ⊆ Aij for i, j = 1, 2 and δ(Aij ) = φ(Aij ) for 1 ≤ i = j ≤ 2. Claim 2.7. δ is a derivation. By Claims 2.5–2.6, it is sufficient to show that δ(Aij Bjk ) = δ(Aij )Bjk + Aij δ(Bjk ) for 1 ≤ i, j, k ≤ 2. We divide it into the following three steps. Step 1. Let B12 ∈ A12 and E be any projection in A11 . Taking P = E, X = P1 and Q = P1 + B12 in Eq. (2.2), we have φ(P1 + B12 − E − EB12 ) = d2 (P1 + B12 − E − EB12 ) + λ2 I. It follows from Eqs. (2.5) and (2.10) that 0 = P2 d2 (P1 − E)P1 = P2 d2 (P1 (P1 − E))P1 = P2 d2 (P1 )(P1 − E) and hence by Eqs. (2.5) and (2.10) again,
(2.10)
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P2 φ(P1 − E)P2 = P2 d2 (B12 − EB12 )P2 + λ2 P2 = P2 d2 (P1 )(P1 − E)B12 + λ2 P2 = λ2 P2 .
(2.11)
Multiplying Eq. (2.6) by P2 from both sides, we arrive at P2 φ(E)P2 = λ1 P2 .
(2.12)
By Eqs. (2.11) and (2.12), λ1 B12 = B12 φ(E) and λ2 EB12 = EB12 φ(P1 − E) = EB12 φ(P1 ) − EB12 φ(E) = EB12 φ(P1 ) − λ1 EB12 . Hence Eq. (2.6) implies that δ(EB12 ) = φ(EB12 ) = φ(E)B12 + Eφ(B12 ) − B12 φ(E) = (δ(E) + τ (E))B12 + Eδ(B12 ) − B12 (δ(E) + τ (E)) = δ(E)B12 + Eδ(B12 ). Note that in the fourth equality, we apply the fact that δ(Aij ) ⊆ Aij (i, j = 1, 2). Since every operator in A11 can be written as finite linear combinations of projections in A11 , it follows that δ(A11 B12 ) = δ(A11 )B12 + A11 δ(B12 ) for all A11 ∈ A11 and B12 ∈ A12 . Similarly, we can get δ(A12 B22 ) = δ(A12 )B22 + A12 δ(B22 ), δ(A21 B11 ) = δ(A21 )B11 + A21 δ(B11 ) and δ(A22 B21 ) = δ(A22 )B21 + A22 δ(B21 ). Step 2. Let A11 and B11 ∈ A11 . For any C12 ∈ A12 , on one hand, by Step 1, we have δ(A11 B11 C12 ) = δ(A11 )B11 C12 + A11 δ(B11 C12 ) = δ(A11 )B11 C12 + A11 δ(B11 )C12 + A11 B11 δ(C12 ). On the other hand, δ(A11 B11 C12 ) = δ(A11 B11 )C12 + A11 B11 δ(C12 ). Comparing these two equalities, we have (δ(A11 B11 ) − δ(A11 )B11 − A11 δ(B11 ))C12 = 0 for all C12 ∈ A12 . Since A is prime, we get
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δ(A11 B11 ) = δ(A11 )B11 + A11 δ(B11 ). Similarly, by considering δ(A22 B22 C21 ), we can get δ(A22 B22 ) = δ(A22 )B22 + A22 δ(B22 ). Step 3. Let A12 , B12 ∈ A12 and B21 ∈ A21 . Taking P = Q = P1 − B12 and X = B12 B21 + B21 in Claim 2.2, we have from P X = P ⊥ Q = P Q⊥ = 0 that 0 = (P1 − B12 )φ(B12 B21 − B12 B21 B12 + B21 − B21 B12 ) − (P1 − B12 )φ(B12 B21 + B21 )(P1 − B12 ) + λ1 (P2 + B12 ) + λ3 (P1 − B12 ) = −φ(B12 B21 B12 ) − P1 φ(B21 B12 ) − B12 φ(B12 B21 ) + B12 φ(B21 B12 ) + φ(B12 B21 )B12 − B12 φ(B21 )B12 + λ1 P2 + λ1 B12 + λ3 P1 − λ3 B12 .
(2.13)
This implies that λ1 = 0 and P1 φ(B21 B12 ) = λ3 P1 . Then φ(B21 B12 )B12 = λ3 B12 , and hence by Eq. (2.13), δ(B12 B21 B12 ) = φ(B12 B21 B12 ) = −B12 φ(B12 B21 ) + B12 φ(B21 B12 ) + φ(B12 B21 )B12 − B12 φ(B21 )B12 − φ(B21 B12 )B12 = −B12 (δ(B12 B21 ) + τ (B12 B21 )) + B12 (δ(B21 B12 ) + τ (B21 B12 )) + (δ(B12 B21 ) + τ (B12 B21 ))B12 − B12 δ(B21 )B12 − (δ(B21 B12 ) + τ (B21 B12 ))B12 = B12 δ(B21 B12 ) + δ(B12 B21 )B12 − B12 δ(B21 )B12 . Replacing B12 with A12 + B12 , we arrive at δ(B12 B21 A12 + A12 B21 B12 ) = δ(B12 B21 )A12 + B12 δ(B21 A12 ) − B12 δ(B21 )A12 + δ(A12 B21 )B12 + A12 δ(B21 B12 ) − A12 δ(B21 )B12 . On the other hand, by Steps 1 and 2, we have
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δ(B12 B21 A12 + A12 B21 B12 ) = δ(B12 B21 )A12 + B12 B21 δ(A12 ) + δ(A12 )B21 B12 + A12 δ(B21 B12 ). Comparing these two equalities, we have (δ(A12 B21 ) − δ(A12 )B21 − A12 δ(B21 ))B12 = −B12 (δ(B21 A12 ) − B21 δ(A12 ) − δ(B21 )A12 ).
(2.14)
Set F (A12 , B21 ) := δ(A12 B21 ) − δ(A12 )B21 − A12 δ(B21 ) and G(B21 , A12 ) := δ(B21 A12 ) − B21 δ(A12 ) − δ(B21 )A12 . By Claim 2.3, F (A12 , B21 ) − G(B21 , A12 ) = αI for some scalar α. Then F (A12 , B21 ) = αP1 and G(B21 , A12 ) = −αP2 . Then by Steps 1 and 2, we have F (A11 A12 , B21 ) = δ(A11 A12 B21 ) − δ(A11 A12 )B21 − A11 A12 δ(B21 ) = δ(A11 )A12 B21 + A11 δ(A12 B21 ) − δ(A11 )A12 B21 − A11 δ(A12 )B21 − A11 A12 δ(B21 ) = αA11 ∈ CP1 and G(B21 , A12 C22 ) = δ(B21 A12 C22 ) − B21 δ(A12 C22 ) − δ(B21 )A12 C22 = δ(B21 A12 )C22 + B21 A12 δ(C22 ) − B21 δ(A12 )C22 − B21 A12 δ(C22 ) − δ(B21 )A12 C22 = −αC22 ∈ CP2 . If α = 0, then A11 ⊆ CP1 and A22 ⊆ CP2 , contrary to the assumption that dim(A) = ∞. So α = 0. Consequently, δ(A12 B21 ) = δ(A12 )B21 + A12 δ(B21 ) and δ(B21 A12 ) = δ(B21 )A12 + B21 δ(A12 ) for all A12 ∈ A12 and B21 ∈ A21 . Claim 2.8. The theorem holds. Now, by Claim 2.7, we can infer that δ is a derivation. Since every derivation of A is inner, there exists an element T ∈ A such that δ(A) = T A − AT for every A ∈ A. Hence letting S = T − P1 ϕ(P1 )P2 + P2 ϕ(P1 )P1 , we have
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ϕ(A) = SA − AS + τ (A) for all A ∈ A. For each commutator R ∈ A, there exists an element SR ∈ A and a linear map τ1 from A into CI vanishing on each commutator such that τ (R) = ϕ(R) − [S, R] = [SR , R] + τ1 (R) − [S, R] = [SR − S, R] ∈ CI. Hence by [7, Problem 230], we have τ (R) = 0 for every commutator R ∈ A. The proof is complete. 2 References [1] M. Brešar, Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 9–21. [2] L. Chen, F. Lu, T. Wang, Local and 2-local Lie derivations of operator algebras on Banach spaces, Integral Equations Operator Theory 77 (2013) 109–121. [3] R.L. Crist, Local derivations on operator algebras, J. Funct. Anal. 135 (1996) 76–92. [4] P.A. Fillmore, D.M. Topping, Operator algebras generated by projections, Duke Math. J. 34 (1967) 333–336. [5] D. Hadwin, J. Li, Local derivations and local automorphisms, J. Math. Anal. Appl. 290 (2004) 702–714. [6] D. Hadwin, J. Li, Local derivations and local automorphisms on some algebras, J. Operator Theory 60 (2008) 29–44. [7] P.R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer-Verlag, New York, 1982. [8] W. Jing, Local derivations on reflexive algebras II, Proc. Amer. Math. Soc. 129 (2001) 1733–1737. [9] B.E. Johnson, Local derivations on C∗-algebras are derivations, Trans. Amer. Math. Soc. 353 (2001) 313–325. [10] R.V. Kadison, Local derivations, J. Algebra 130 (1990) 494–509. [11] S.O. Kim, J.S. Kim, Local automorphisms and derivations on Mn , Proc. Amer. Math. Soc. 132 (2004) 1389–1392. [12] D.R. Larson, A.R. Sourour, Local derivations and local automorphisms of B(X), Proc. Sympos. Pure Math. 51 (1990) 187–194. [13] C.R. Miers, Lie derivations of von Neumann algebras, Duke Math. J. 40 (1973) 403–409. [14] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York, 1979. [15] J. Zhang, G. Ji, H. Cao, Local derivations of nest subalgebras of von Neumann algebras, Linear Algebra Appl. 392 (2004) 61–69. [16] J. Zhang, F. Pan, A. Yang, Local derivations on certain CSL algebras, Linear Algebra Appl. 413 (2006) 93–99.