Local Lie derivations of generalized matrix algebras

1.   Introduction and main results

A well-known and active direction in the study of derivations is the local derivations problem, which was initiated by Kadison ^[8] and Larson and Sourour ^[9]. Recall that a linear map $\varphi$ of an algebra $A$ is called a local derivation if for each $x\in A$, there exists a derivation $\varphi_x$ of $A$, depending on $x$, such that $\varphi(x) = \varphi_{x}(x)$. The question of determining under what conditions every local derivation must be a derivation has been studied by many authors (see ^{[4,6,7,13,15]}). Recently, Brešar ^[2] proved that each local derivation of algebras generated by all their idempotents is a derivation.

A linear map $\varphi$ of an algebra $A$ is called a Lie derivation if $\varphi([x, y]) = [\varphi(x), y]+[x, \varphi(y)]$ for all $x, y\in A$, where $[x, y] = xy-yx$ is the usual Lie product, also called a commutator. A Lie derivation $\varphi$ of $A$ is standard if it can be decomposed as $\varphi = d+\tau$, where $d$ is a derivation from $A$ into itself and $\tau$ is a linear map from $A$ into its center vanishing on each commutator. The classical problem, which has been studied for many years, is to find conditions on $A$ under which each Lie derivation is standard or standard-like. We say that a linear map $\varphi$ from $A$ into itself is a local Lie derivation if for each $x \in A$, there exists a Lie derivation $\varphi_x$ of $A$ such that $\varphi(x) = \varphi_x(x)$. In ^[3], Chen et al. studied local Lie derivations of operator algebras on Banach spaces. We remark that the methods in [3] depend heavily on rank one operators in $B(X)$. Later, Liu and Zhang ^[10] proved that each local Lie derivation of factor von Neumann algebras is a Lie derivation. Liu and Zhang ^[11] investigated local Lie derivations of a certain class of operator algebras. An et al. ^[1] proved that every local Lie derivation on von Neumann algebras is a Lie derivation.

It is quite common to study local derivations in algebras that contain many idempotents, in the sense that the linear span of all idempotents is 'large'. The main novelty of this paper is that we shall deal with the subalgebra generated by all idempotents instead of their span. Let $\mathcal{M}_{2}$ be the algebra of $2\times 2$ matrices over $L^{\infty}[0, 1]$. By ^[6], $\mathcal{M}_{2}$ is generated by, but not spanned by, its idempotents. In what follows, we denote by ${\mathcal J} (A)$ the subalgebra of $A$ generated by all idempotents in $A$. The purpose of the present paper is to study local Lie derivations of a certain class of generalized matrix algebras. Finally we apply the main result to full matrix algebras and unital simple algebras with nontrivial idempotents.

Let $A$ and $B$ be two unital algebras with unit elements $1_{A}$ and $1_{B}$, respectively. A Morita context consists of $A, B$, two bimodules $_{A}M_{B}$ and $_{B}N_{A}$, and two bimodule homomorphisms called the pairings $\Phi_{MN}:M\otimes_{B}N\rightarrow A$ and $\Psi_{NM}:N\otimes_{A}M\rightarrow B$ satisfying the following commutative diagrams:

and

If $(A, B, M, N, \Phi_{MN}, \Psi_{NM})$ is a Morita context, then the set

forms an algebra under matrix-like addition and multiplication. Such an algebra is called a generalized matrix algebras. We further assume that $M$ is faithful as an $(A, B)$-bimodule. The most common examples of generalized matrix algebras are full matrix algebras and triangular algebras.

Consider algebra ${\mathcal G}$. Any element of the form

will be denoted by $a\oplus b$. Let us define two natural projections $\pi_{A}:{\mathcal G}\rightarrow A$ and $\pi_{B}:{\mathcal G}\rightarrow B$ by

The center of ${\mathcal G}$ is

Furthermore, $\pi_{A}(Z({\mathcal G}))\subseteq Z(A)$ and $\pi_{B}(Z({\mathcal G}))\subseteq Z(B)$, and there exists a unique algebra isomorphism $\eta$ from $\pi_{B}(Z({\mathcal G}))$ to $\pi_{A}(Z({\mathcal G}))$ such that $\eta(b)m = mb$ and $n\eta(b) = bn$ for all $m\in M, n\in N$ (see ^[14]). Set

We immediately notice that $e$ and $f$ are orthogonal idempotents of ${\mathcal G}$ and so ${\mathcal G}$ may be represented as ${\mathcal G} = (e + f) {\mathcal G} (e + f) = e{\mathcal G}e + e{\mathcal G}f +f {\mathcal G}e+ f {\mathcal G}f$. Then each element $x = exe + exf + fxe+ fxf \in {\mathcal G}$ can be represented in the form $x = eae + emf + fne+ fbf = a + m +n+b$, where $a \in A, b \in B, m \in M, n \in N$.

We close this section with a well known result concerning Lie derivations.

Proposition 1.1. (See ^[5],Theorem 1) Let ${\mathcal G}$ be a generalized matrix algebra. Suppose that

(1) $Z(A) = \pi_{A}(Z({\mathcal G}))$ and $Z(B) = \pi_{B}(Z({\mathcal G}))$;

(2) either $A$ or $B$ does not contain nonzero central ideals.

Then every Lie derivation $\varphi: {\mathcal G}\rightarrow {\mathcal G}$ is standard, that is, $\varphi$ is the sum of a derivation $d$ and a linear central-valued map $\tau$ vanishing on each commutator.

2.   Main results

Our main result reads as follows.

Theorem 2.1. Let ${\mathcal G}$ be a generalized matrix algebra. Suppose that

($1$) $A = {\mathcal J}(A)$ and $B = {\mathcal J}(B);$

($2$) $Z(A) = \pi_{A}(Z({\mathcal G}))$ and $Z(B) = \pi_{B}(Z({\mathcal G}));$

($3$) either $A$ or $B$ does not contain nonzero central ideals.

Then every local Lie derivation $\varphi$ from ${\mathcal G}$ into itself is a sum of a derivation $\delta$ and a linear central-valued map $h$ vanishing on each commutator.

To prove Theorem 2.1, we need some lemmas. In the following, $\varphi$ is a local Lie derivation and, for any $x\in {\mathcal G}$, the symbol $\varphi_x$ stands for a Lie derivation from ${\mathcal G}$ into itself such that $\varphi(x) = \varphi_x(x)$. It follows from $A = {\mathcal J} (A)$ that every $a$ in $A$ can be written as a linear combination of some elements $p_{1}p_{2}\cdots p_{i}$ ($i = 1,2,\ldots,k$), where $p_{1},p_{2},\ldots, p_{i}$ are idempotents in $A$.

Lemma 2.2. Let $p,q \in{\mathcal G}$ be idempotents, then for every $x\in{\mathcal G}$, there exist linear maps $\tau_{1},\tau_{2},\tau_{3},\tau_{4}: {\mathcal G}\rightarrow Z({\mathcal G})$ vanishing on each commutator such that

where $p^{\perp} = 1-p$ and $q^{\perp} = 1-q$.

Proof. Proposition 1.1 implies that for every idempotents $p, q \in{\mathcal G}$ and $x\in{\mathcal G}$, there exist derivations $d_{1}, d_{2}, d_3, d_4:{\mathcal G}\rightarrow {\mathcal G}$ and linear maps $\tau_{1}, \tau_{2}, \tau_3, \tau_4:{\mathcal G}\rightarrow Z({\mathcal G})$ vanishing on each commutator such that

It follows from (2.1)–(2.4) that

Hence

Thus,

It is easy to verify that for each derivation $d: {\mathcal G}\rightarrow {\mathcal G}$, we have

Lemma 2.3. $e\varphi(e)e+f\varphi(e)f\in Z({\mathcal G})$.

Proof. For any $m\in M$, there exists a Lie derivation $\varphi_{e}$ of ${\mathcal G}$ such that

Multiplying the above equality from the left by $e$ and from the right by $f$, we arrive at

Similarly, for any $n\in N$, we have from $\varphi_{e}(n) = \varphi_{e}([n, e]) = [\varphi_{e}(n), e]+[n, \varphi(e)]$ that

Hence

In the sequel, we define $\phi: {\mathcal G}\rightarrow {\mathcal G}$ by $\phi(x) = \varphi(x)-[x, e\varphi(e)f-f\varphi(e)e]$. One can verify that $\phi$ is also a local Lie derivation. Moreover, by Lemma 2.3, we have $\phi(e) = e\varphi(e)e+f\varphi(e)f\in Z({\mathcal G})$.

Lemma 2.4. $\phi(M) \subseteq M$ and $\phi(N) \subseteq N$.

Proof. Let $a\in A, m\in M$ and $p_{1}$ be any idempotent in $A$. Taking $p = p_{1}$, $x = a$ and $q = e+m$ in Lemma 2.2, it follows from the facts $p^{\perp}xq^{\perp}$ and $pxq^{\perp}$ can be written as commutators that $\tau_3(p^{\perp}xq^{\perp}) = \tau_4(pxq^{\perp}) = 0$, hence

Multiplying (2.6) from the right by $e$, we arrive at

In particular,

By the above two equations, then

for any idempotents $p_{1}, \ldots, p_{n} \in A$. It follows from $A = {\mathcal J}(A)$ that

for all $a\in A, m\in M$. This implies that $f\phi(M)e = 0$.

The hypothesis (2), (3) and Proposition 1.1 imply that there exist a derivation $d:{\mathcal G}\rightarrow {\mathcal G}$ and a linear map $\tau: {\mathcal G}\rightarrow Z({\mathcal G})$ vanishing on each commutator such that

It follows from (2.5), (2.8) and the fact $f\phi(M)e = 0$ that

and hence by (2.5) and (2.8) again,

and

Then we have from the fact $\phi(e) = e\phi(e)e+f\phi(e)f\in Z(\mathcal{G})$ that

We assume without loss of generality that $A$ does not contain nonzero central ideals. By (2.7) and (2.9) that $e\phi(m)e$ in the central ideal of $A$. Thus $e\phi(M)e = 0$. So, by (2.9), we get $f\phi(M)f = 0$. Hence, $\phi(M)\subseteq M$.

With the same argument, we can obtain that $\phi(N)\subseteq N$.

Lemma 2.5. There exist a linear map $h_{1}$ from $A$ into $Z({\mathcal G})$ such that $\phi(a)-h_{1}(a)\in A$ for all $a\in A$ and a linear map $h_{2}$ from $B$ into $Z({\mathcal G})$ such that $\phi(b)-h_{2}(b)\in B$ for all $b\in B$.

Proof. Taking $m = 0$ in (2.6), we have

In particular,

By the two equations above, we obtain

for all idempotents $p_{i}$ in $A$. It follows from $A = {\mathcal J}(A)$ that $e\phi(a)f = 0$. Similarly, by taking $p = e$, $x = a$ and $q = p_{1}$ in Lemma 2.2, we get

This implies that $f\phi(a)e = 0$. So $\phi(a) \in A\oplus B$.

By the hypothesis $(2)$ of Theorem 2.1, there exists a algebra isomorphism $\eta: Z(B)\rightarrow Z(A)$ such that $\eta(b)\oplus b \in Z({\mathcal G})$ for any $b\in Z(B)$.

It follows from (2.10) that $f\phi(a)f\in \pi_{B}(Z({\mathcal G})) = Z(B)$. We define $h_{1}: A\rightarrow Z({\mathcal G})$ by $h_{1}(a) = \eta(f\phi(a)f)\oplus f\phi(a)f$. It is clear that $h_{1}$ is linear and

With the similar argument, we can define a linear map $h_{2}:B\rightarrow Z({\mathcal G})$ such that $\phi(b)-h_{2}(b)\in B$ for all $b\in B$.

Now for any $x\in{\mathcal G}$, we define two linear maps $h: {\mathcal G} \rightarrow Z({\mathcal G})$ and $\delta: {\mathcal G} \rightarrow {\mathcal G}$ by

It is easy to verify that $\delta(e) = 0$. Moreover, we have

Lemma 2.6. $\delta$ is a derivation.

Proof. We divide the proof into the following three steps.

Step 1. We first prove that

for all idempotents $p_{i}$ in $A$ and $m\in M$.

Let $a\in A$, $m\in M$ and $p_{1}$ be any idempotent in $A$. Taking $p = p_{1}$, $x = a$ and $q = e+m$ in (2.2), we have

It follows from (2.5) and (2.12) that

and hence by (2.5) and (2.12) again,

Multiplying (2.6) by $f$ from both sides, we arrive at

By (2.13) and (2.14), then $m\tau_{1}(p_{1}a) = m\phi(p_{1}a)$ and

Hence (2.6) implies that

Taking $a = e$ in (2.15), we have from $\delta(e) = 0$ that

This shows that (2.11) is true for $n = 1$. One can verify that Eq (2.11) follows easily by induction based on (2.15). It follows from $A = \mathcal{J}(A)$ that $\delta(am) = \delta(a)m +a\delta(m)$.

Similarly, we can get $\delta(mb) = \delta(m)b+m\delta(b)$, $\delta(mb) = \delta(m)b+m\delta(b)$ and $\delta(na) = \delta(n)a+n\delta(a)$.

Step 2. Let $a, a'\in A$. For any $m\in M$, on one hand, by Step 1, we have

On the other hand,

Comparing these two equalities, we have

for any $m\in M$. Since $M$ is a faithful left $A$-module, we get

Similarly, by considering $\delta(mbb')$, we can get

Step 3. Let $m, m'\in M$ and $n\in N$. Taking $p = e-m'$, $x = n+m'n$ and $q = e-m'$ in Lemma 2.2, we have from $pxq = pxq^{\perp} = 0$ that

This implies that

Then $e\phi(nm')m' = \tau_{3}(nm')m'$ and hence by (2.16),

Replacing $m'$ with $m+m'$, we arrive at

On the other hand, by Steps 1 and 2, we have

Comparing these two equalities, we have

Set

and

We assume without loss of generality that $A$ does not contain nonzero central ideals. For any $a\in A$, by (2.17),

which is equivalent to $(f(m, n)a-af(m, n))m' = 0$. Since $M$ is a faithful left $A$-module, we get $f(m, n)a = af(m, n)$. Then

By Steps 1 and 2, we have

The above two equalities show that $f(m, n)$ in the central ideal of $A$ and hence

that is

for all $m\in M, n\in N$. Since $M$ is a faithful right $B$-module, it follows from (2.17) that

for all $m\in M, n\in N$.

Lemma 2.7. The map $h:\mathcal{G}\rightarrow Z({\mathcal G})$ vanishes on each commutator.

Proof. Step 1. Let $a\in A$, $m\in M$, $n\in N$ and $b\in B$, by the definition of $h$, we have $h([a, m]) = h([m, b]) = h([n, a]) = h([b, n]) = 0$.

Step 2. Let $a, a'\in A$, we have $\phi([a, a']) = e\phi([a, a'])e+f\phi([a, a'])f\in A\oplus B$. On the other hand, Proposition 1.1 implies that $\phi([a, a']) = d([a, a'])\in A\oplus M\oplus N$, where $d$ is a derivation. Thus, $f\phi([a, a'])f = 0$. This implies that $h([a, a']) = h_{1}([a, a']) = \eta(f\phi([a, a'])f)+f\phi([a, a'])f = 0$.

Similarly, we can get $h([b, b']) = 0$, for all $b, b'\in B$.

Step 3. It follows from (2.18) that

Since $f\phi(a)f\in \pi_{B}(Z({\mathcal G}))$, $e\phi(b)e\in \pi_{A}(Z({\mathcal G}))$, we get that

It further follows from (2.19) that

Hence

Using an argument similar to that in the proof of (2.18), we arrive that

and

By (2.19) and (2.20), we get that $e\phi(nm)e = \eta(f\phi(mn)f)$. Note that $h([m, n]) = h_{1}(mn)-h_{2}(nm) = \eta(f\phi(mn)f)+f\phi(mn)f-e\phi(nm)e-\eta^{-1}(e\phi(nm)e)$, thus $h([m, n]) = 0$.

Therefore it is easily verify that $h$ vanishing on each commutator.

Proof of Theorem 1.1 By the definition of $\delta$, we have $\varphi(x) = \delta(x)+[x, e\varphi(e)f-f\varphi(e)e]+h(x)$ for all $x\in A$, where $\delta$ is a derivation and $h$ is a linear map from $A$ into its center vanishing on each commutator. The proof is complete.

Let $A$ be a unital algebra and $M_{k\times m}(A)$ be the set of all $k\times m$ matrices over $A$. For $n\geq 2$ and each $2\leq l < n-1$, the full matrix algebra $M_{n}(A)$ can be represented as a generalized matrix algebra of the form

Corollary 2.8. Let $M_{n}(A)$ be a full matrix algebra with $n\geq 4$. Then each local Lie derivation $\varphi$ on $M_{n}(A)$ is of the form $\varphi = d+\tau$, where $d$ is a derivation of $M_{n}(A)$ and $\tau$ is a linear map from $M_{n}(A)$ into its center $Z(A)\cdot I_{n}$ vanishing on each commutator.

Proof. It follows from the example $(C)$ of ^[2] that the matrix algebras $M_{l}(A)$ and $M_{n-l}(A)$ are generated by their idempotents for $2\leq l < n-1$. Since $Z(M_{n}(A)) = Z(A)\cdot I_{n}$, $Z(M_{l}(A)) = Z(A)\cdot I_{l}$ and $Z(M_{n-l}(A)) = Z(A)\cdot I_{n-l}$, the condition $(2)$ of Theorem 2.1 is satisfied. By [5,Lemma 1], $M_{k}(A)$ does not contain nonzero central ideals for $k\geq 2$. Hence by Theorem 2.1, every local Lie derivation of $M_{n}(A)$ is a sum of a derivation and a linear central-valued map vanishing on each commutator.

Corollary 2.9. Let $R$ be an unital simple algebra with a nontrivial idempotent. If $\varphi:R\rightarrow R$ is a local Lie derivation, then there exit a derivation $d$ and a linear central map $\tau$ vanishing on each commutator, such that $\varphi = d+\tau$.

Proof. Let $R$ be an unital simple algebra with a nontrivial idempotent $e_{0}$ and let $f_{0}$ denote the idempotent $1-e_{0}$. Then $R$ can be represented in the so-called Peirce decomposition form

where $e_{0}Re_{0}$ and $f_{0}Rf_{0}$ are subalgebras with unitary element $e_{0}$ and $f_{0}$, respectively, $e_{0}Rf_{0}$ is an $(e_{0}Re_{0}, f_{0}Rf_{0})$-bimodule.

Next, we will show that

and

That is $e_{0}Rf_{0}$ is faithful as an $(e_{0}Re_{0}, f_{0}Rf_{0})$-bimodule. Let $e = f_{0}+e_{0}Rf_{0}$, then $e^{2} = e$ and $[e, R]\subseteq eR(1-e)+(1-e)R e$. Note that

Furthermore, the assumption $e_{0}xe_{0}\cdot e_{0}Rf_{0} = \{0\}$ implies

and then

Let $r = [e, y]$ and $z, w \in R$. It follows from

that $e_{0}xe_{0}zrw = 0$. Then

It is clear that $I = R[e, R]R$ is a nonzero ideal of $R$. $R$ is a simple algebra, which implies $I = R$. By (2.21), $e_{0}xe_{0}R = 0$. Since $1\in R$, we get $e_{0}xe_{0} = 0$. Similarly, we can show that $e_{0}Rf_{0}\cdot f_{0}xf_{0} = \{0\}$ implies $f_{0}xf_{0} = 0.$ Now, we can conclude that $R$ can be represented as a generalized matrix algebra of the form $R = e_{0}Re_{0} + e_{0}Rf_{0} +f_{0}Re_{0}+ f_{0}Rf_{0}$.

It follows from the example $(A)$ of ^[2] that the unital simple algebra with a nontrivial idempotent is generated by its idempotents, the condition $(1)$ of Theorem 2.1 is satisfied. It is clear that $e_{0}Re_{0}$ and $f_{0}Rf_{0}$ satisfy the conditions $(2)$ and $(3)$ of Theorem 2.1. Hence by Theorem 2.1, every local Lie derivation of $R$ is the sum of a derivation and a linear central-valued map vanishing on each commutator.

Let $B(H)$ be the set of bounded linear operators acting on a complex Hilbert space $H$, and let $K(H)$ be the ideal of compact operators on $H$. If $H$ is an infinite-dimensional separable Hilbert space, by [12,Theorem 4.1.16], the Calkin algebra $B(H)/K(H)$ is a simple $C^{\ast}$-algebra.

Corollary 2.10. If $H$ is an infinite-dimensional separable Hilbert space, then every local Lie derivation of the Calkin algebra $B(H)/K(H)$ is the sum of a derivation and a linear central map vanishing on each commutator.

3.   Conclusions

In this paper, we investigate local Lie derivations of a certain class of generalized matrix algebras and show that, under certain conditions every local Lie derivation of a generalized matrix algebra is a sum of a derivation and a linear central-valued map vanishing on each commutator. The main result is then applied to full matrix algebras and unital simple algebras with nontrivial idempotents.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 11901248). Moreover, the authors express their sincere gratitude to the referee for reading this paper very carefully and specially for valuable suggestions concerning improvement of the manuscript.

Conflict of interest

All authors declare no conflicts of interest in this paper.