The aerodynamics analysis has grown in relevance for wind energy projects; this mechanism is focused on elucidating aerodynamic characteristics to maximize accuracy and practicability via the modelling of chaos in a wind turbine system's permanent magnet synchronous generator using short-memory methodologies. Fractional derivatives have memory impacts and are widely used in numerous practical contexts. Even so, they also require a significant amount of storage capacity and have inefficient operations. We suggested a novel approach to investigating the fractional-order operator's Lyapunov candidate that would do away with the challenging task of determining the indication of the Lyapunov first derivative. Next, a short-memory fractional modelling strategy is presented, followed by short-memory fractional derivatives. Meanwhile, we demonstrate the dynamics of chaotic systems using the Lyapunov function. Predictor-corrector methods are used to provide analytical results. It is suggested to use system dynamics to reduce chaotic behaviour and stabilize operation; the benefit of such a framework is that it can only be used for one state of the hybrid power system. The key variables and characteristics, i.e., the modulation index, pitch angle, drag coefficients, power coefficient, air density, rotor angular speed and short-memory fractional differential equations are also evaluated via numerical simulations to enhance signal strength.
Citation: Abdulaziz Khalid Alsharidi, Saima Rashid, S. K. Elagan. Short-memory discrete fractional difference equation wind turbine model and its inferential control of a chaotic permanent magnet synchronous transformer in time-scale analysis[J]. AIMS Mathematics, 2023, 8(8): 19097-19120. doi: 10.3934/math.2023975
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The aerodynamics analysis has grown in relevance for wind energy projects; this mechanism is focused on elucidating aerodynamic characteristics to maximize accuracy and practicability via the modelling of chaos in a wind turbine system's permanent magnet synchronous generator using short-memory methodologies. Fractional derivatives have memory impacts and are widely used in numerous practical contexts. Even so, they also require a significant amount of storage capacity and have inefficient operations. We suggested a novel approach to investigating the fractional-order operator's Lyapunov candidate that would do away with the challenging task of determining the indication of the Lyapunov first derivative. Next, a short-memory fractional modelling strategy is presented, followed by short-memory fractional derivatives. Meanwhile, we demonstrate the dynamics of chaotic systems using the Lyapunov function. Predictor-corrector methods are used to provide analytical results. It is suggested to use system dynamics to reduce chaotic behaviour and stabilize operation; the benefit of such a framework is that it can only be used for one state of the hybrid power system. The key variables and characteristics, i.e., the modulation index, pitch angle, drag coefficients, power coefficient, air density, rotor angular speed and short-memory fractional differential equations are also evaluated via numerical simulations to enhance signal strength.
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 φ of an algebra A is called a local derivation if for each x∈A, there exists a derivation φx of A, depending on x, such that φ(x)=φx(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 φ of an algebra A is called a Lie derivation if φ([x,y])=[φ(x),y]+[x,φ(y)] for all x,y∈A, where [x,y]=xy−yx is the usual Lie product, also called a commutator. A Lie derivation φ of A is standard if it can be decomposed as φ=d+τ, where d is a derivation from A into itself and τ 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 φ from A into itself is a local Lie derivation if for each x∈A, there exists a Lie derivation φx of A such that φ(x)=φx(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 M2 be the algebra of 2×2 matrices over L∞[0,1]. By [6], M2 is generated by, but not spanned by, its idempotents. In what follows, we denote by 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 1A and 1B, respectively. A Morita context consists of A,B, two bimodules AMB and BNA, and two bimodule homomorphisms called the pairings ΦMN:M⊗BN→A and ΨNM:N⊗AM→B satisfying the following commutative diagrams:
![]() |
and
![]() |
If (A,B,M,N,ΦMN,ΨNM) is a Morita context, then the set
G=(AMNB)={(amnb)∣a∈A,m∈M,n∈N,b∈B} |
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 G. Any element of the form
(a00b)∈G |
will be denoted by a⊕b. Let us define two natural projections πA:G→A and πB:G→B by
πA:(amnb)↦a and πB:(amnb)↦b. |
The center of G is
Z(G)={a⊕b∣am=mb,na=bn for all m∈M,n∈N}. |
Furthermore, πA(Z(G))⊆Z(A) and πB(Z(G))⊆Z(B), and there exists a unique algebra isomorphism η from πB(Z(G)) to πA(Z(G)) such that η(b)m=mb and nη(b)=bn for all m∈M,n∈N (see [14]). Set
e=(1A000), f=(0001B). |
We immediately notice that e and f are orthogonal idempotents of G and so G may be represented as G=(e+f)G(e+f)=eGe+eGf+fGe+fGf. Then each element x=exe+exf+fxe+fxf∈G can be represented in the form x=eae+emf+fne+fbf=a+m+n+b, where a∈A,b∈B,m∈M,n∈N.
We close this section with a well known result concerning Lie derivations.
Proposition 1.1. (See [5],Theorem 1) Let G be a generalized matrix algebra. Suppose that
(1) Z(A)=πA(Z(G)) and Z(B)=πB(Z(G));
(2) either A or B does not contain nonzero central ideals.
Then every Lie derivation φ:G→G is standard, that is, φ is the sum of a derivation d and a linear central-valued map τ vanishing on each commutator.
Our main result reads as follows.
Theorem 2.1. Let G be a generalized matrix algebra. Suppose that
(1) A=J(A) and B=J(B);
(2) Z(A)=πA(Z(G)) and Z(B)=πB(Z(G));
(3) either A or B does not contain nonzero central ideals.
Then every local Lie derivation φ from G into itself is a sum of a derivation δ and a linear central-valued map h vanishing on each commutator.
To prove Theorem 2.1, we need some lemmas. In the following, φ is a local Lie derivation and, for any x∈G, the symbol φx stands for a Lie derivation from G into itself such that φ(x)=φx(x). It follows from A=J(A) that every a in A can be written as a linear combination of some elements p1p2⋯pi (i=1,2,…,k), where p1,p2,…,pi are idempotents in A.
Lemma 2.2. Let p,q∈G be idempotents, then for every x∈G, there exist linear maps τ1,τ2,τ3,τ4:G→Z(G) vanishing on each commutator such that
φ(pxq)=φ(px)q+pφ(xq)−pφ(x)q+p⊥τ1(pxq)q⊥−pτ2(p⊥xq)q⊥+pτ3(p⊥xq⊥)q−p⊥τ4(pxq⊥)q, |
where p⊥=1−p and q⊥=1−q.
Proof. Proposition 1.1 implies that for every idempotents p,q∈G and x∈G, there exist derivations d1,d2,d3,d4:G→G and linear maps τ1,τ2,τ3,τ4:G→Z(G) vanishing on each commutator such that
φ(pxq)=φpxq(pxq)=d1(pxq)+τ1(pxq), | (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(pxq⊥)=φpxq⊥(pxq⊥)=d4(pxq⊥)+τ4(pxq⊥). | (2.4) |
It follows from (2.1)–(2.4) that
p⊥φ(pxq)q⊥=p⊥τ1(pxq)q⊥, pφ(p⊥xq)q⊥=pτ2(p⊥xq)q⊥, |
pφ(p⊥xq⊥)q=pτ3(p⊥xq⊥)q, p⊥φ(pxq⊥)q=p⊥τ4(pxq⊥)q. |
Hence
φ(pxq)q⊥=pφ(pxq)q⊥+p⊥φ(pxq)q⊥=pφ(xq)q⊥−pφ(p⊥xq)q⊥+p⊥φ(pxq)q⊥=pφ(xq)q⊥+p⊥τ1(pxq)q⊥−pτ2(p⊥xq)q⊥=pφ(xq)−pφ(xq)q+p⊥τ1(pxq)q⊥−pτ2(p⊥xq)q⊥, |
φ(pxq⊥)q=pφ(pxq⊥)q+p⊥φ(pxq⊥)q=pφ(xq⊥)q−pφ(p⊥xq⊥)q+p⊥φ(pxq⊥)q=pφ(xq⊥)q−pτ3(p⊥xq⊥)q+p⊥τ4(pxq⊥)q. |
Thus,
φ(pxq)=φ(pxq)q⊥+φ(pxq)q=φ(pxq)q⊥+φ(px)q−φ(pxq⊥)q=φ(px)q+pφ(xq)−pφ(x)q+p⊥τ1(pxq)q⊥−pτ2(p⊥xq)q⊥+pτ3(p⊥xq⊥)q−p⊥τ4(pxq⊥)q. |
It is easy to verify that for each derivation d:G→G, we have
d(e)=−d(f)∈M⊕N, d(A)⊆A⊕M⊕N, d(M)⊆A⊕M⊕B. | (2.5) |
Lemma 2.3. eφ(e)e+fφ(e)f∈Z(G).
Proof. For any m∈M, there exists a Lie derivation φe of G such that
φe(m)=φe([e,m])=[φ(e),m]+[e,φe(m)]=φ(e)m−mφ(e)+eφe(m)f−fφe(m)e. |
Multiplying the above equality from the left by e and from the right by f, we arrive at
eφ(e)m=mφ(e)f. |
Similarly, for any n∈N, we have from φe(n)=φe([n,e])=[φe(n),e]+[n,φ(e)] that
fφ(e)n=nφ(e)e. |
Hence
eφ(e)e+fφ(e)f∈Z(G). |
In the sequel, we define ϕ:G→G by ϕ(x)=φ(x)−[x,eφ(e)f−fφ(e)e]. One can verify that ϕ is also a local Lie derivation. Moreover, by Lemma 2.3, we have ϕ(e)=eφ(e)e+fφ(e)f∈Z(G).
Lemma 2.4. ϕ(M)⊆M and ϕ(N)⊆N.
Proof. Let a∈A,m∈M and p1 be any idempotent in A. Taking p=p1, x=a and q=e+m in Lemma 2.2, it follows from the facts p⊥xq⊥ and pxq⊥ can be written as commutators that τ3(p⊥xq⊥)=τ4(pxq⊥)=0, hence
ϕ(p1a+p1am)=ϕ(p1a)(e+m)+p1ϕ(a+am)−p1ϕ(a)(e+m)+(1−p1)τ1(p1a+p1am)(f−m)−p1τ2(a+am−p1a−p1am)(f−m)=ϕ(p1a)e+ϕ(p1a)m+p1ϕ(a)f+p1ϕ(am)−p1ϕ(a)m+τ1(p1a)f−τ1(p1a)m+p1τ1(p1a)m+p1τ2(a−p1a)m. | (2.6) |
Multiplying (2.6) from the right by e, we arrive at
ϕ(p1am)e=p1ϕ(am)e. |
In particular,
ϕ(p1m)e=p1ϕ(m)e. |
By the above two equations, then
ϕ(p1p2⋯pnm)e=p1ϕ(p2⋯pnm)e=p1p2⋯pn−1ϕ(pnm)e=p1p2⋯pnϕ(m)e |
for any idempotents p1,…,pn∈A. It follows from A=J(A) that
ϕ(am)e=aϕ(m)e | (2.7) |
for all a∈A,m∈M. This implies that fϕ(M)e=0.
The hypothesis (2), (3) and Proposition 1.1 imply that there exist a derivation d:G→G and a linear map τ:G→Z(G) vanishing on each commutator such that
ϕ(e+m)=d(e+m)+τ(e+m)=d(e+m)+τ(e). | (2.8) |
It follows from (2.5), (2.8) and the fact fϕ(M)e=0 that
0=fϕ(e+m)e=fd(e)e |
and hence by (2.5) and (2.8) again,
eϕ(e)e+eϕ(m)e=ed(m)e+eτ(e)e=ed(mf)e+eτ(e)e=md(f)e+eτ(e)e=−md(e)e+eτ(e)e=eτ(e)e |
and
fϕ(e)f+fϕ(m)f=fd(m)f+fτ(e)f=fd(e)m+fτ(e)f=fτ(e)f. |
Then we have from the fact ϕ(e)=eϕ(e)e+fϕ(e)f∈Z(G) that
eϕ(m)e+fϕ(m)f=τ(e)−ϕ(e)∈Z(G). | (2.9) |
We assume without loss of generality that A does not contain nonzero central ideals. By (2.7) and (2.9) that eϕ(m)e in the central ideal of A. Thus eϕ(M)e=0. So, by (2.9), we get fϕ(M)f=0. Hence, ϕ(M)⊆M.
With the same argument, we can obtain that ϕ(N)⊆N.
Lemma 2.5. There exist a linear map h1 from A into Z(G) such that ϕ(a)−h1(a)∈A for all a∈A and a linear map h2 from B into Z(G) such that ϕ(b)−h2(b)∈B for all b∈B.
Proof. Taking m=0 in (2.6), we have
eϕ(p1a)f=p1ϕ(a)f and fϕ(p1a)f=τp1a(p1a)f∈πB(Z(G)). | (2.10) |
In particular,
eϕ(p1)f=p1ϕ(e)f=0. |
By the two equations above, we obtain
eϕ(p1p2⋯pn)f=p1ϕ(p2⋯pn)f=p1p2⋯pn−1ϕ(pn)f=0 |
for all idempotents pi in A. It follows from A=J(A) that eϕ(a)f=0. Similarly, by taking p=e, x=a and q=p1 in Lemma 2.2, we get
fϕ(ap1)e=fϕ(a)p1. |
This implies that fϕ(a)e=0. So ϕ(a)∈A⊕B.
By the hypothesis (2) of Theorem 2.1, there exists a algebra isomorphism η:Z(B)→Z(A) such that η(b)⊕b∈Z(G) for any b∈Z(B).
It follows from (2.10) that fϕ(a)f∈πB(Z(G))=Z(B). We define h1:A→Z(G) by h1(a)=η(fϕ(a)f)⊕fϕ(a)f. It is clear that h1 is linear and
ϕ(a)−h1(a)=eϕ(a)e+fϕ(a)f−η(fϕ(a)f)−fϕ(a)f=eϕ(a)e−η(fϕ(a)f)∈A. |
With the similar argument, we can define a linear map h2:B→Z(G) such that ϕ(b)−h2(b)∈B for all b∈B.
Now for any x∈G, we define two linear maps h:G→Z(G) and δ:G→G by
h(x)=h1(exe)+h2(fxf) and δ(x)=ϕ(x)−h(x). |
It is easy to verify that δ(e)=0. Moreover, we have
δ(A)⊆A, δ(B)⊆B, δ(M)=ϕ(M)⊆M, δ(N)=ϕ(N)⊆N. |
Lemma 2.6. δ is a derivation.
Proof. We divide the proof into the following three steps.
Step 1. We first prove that
δ(p1p2…pnm)=δ(p1p2…pn)m+p1p2…pnδ(m) | (2.11) |
for all idempotents pi in A and m∈M.
Let a∈A, m∈M and p1 be any idempotent in A. Taking p=p1, x=a and q=e+m in (2.2), we have
ϕ(a+am−p1a−p1am)=d2(a+am−p1a−p1am)+τ2(a+am−p1a−p1am)=d2(a+am−p1a−p1am)+τ2(a−p1a). | (2.12) |
It follows from (2.5) and (2.12) that
0=fd2(a−p1a)e=fd2(e(a−p1a))e=fd2(e)(a−p1a) |
and hence by (2.5) and (2.12) again,
fϕ(a−p1a)f=fd2(am−p1am)f+fτ2(a−p1a)f=fd2(e)(a−p1a)m+fτ2(a−p1a)f=fτ2(a−p1a)f. | (2.13) |
Multiplying (2.6) by f from both sides, we arrive at
fϕ(p1a)f=fτ1(p1a)f. | (2.14) |
By (2.13) and (2.14), then mτ1(p1a)=mϕ(p1a) and
p1mτ2(a−p1a)=p1mϕ(a−p1a)=p1mϕ(a)−p1mϕ(p1a)=p1mϕ(a)−p1mτ1(p1a). |
Hence (2.6) implies that
δ(p1am)=ϕ(p1am)=ϕ(p1a)m+p1ϕ(am)−p1ϕ(a)m−mϕ(p1a)+p1mϕ(a)=(δ(p1a)+h(p1a))m+p1δ(am)−p1(δ(a)+h(a))m−m(δ(p1a)+h(p1a))+p1m(δ(a)+h(a))=δ(p1a)m+p1δ(am)−p1δ(a)m. | (2.15) |
Taking a=e in (2.15), we have from δ(e)=0 that
δ(p1m)=δ(p1)m+p1δ(m). |
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=J(A) that δ(am)=δ(a)m+aδ(m).
Similarly, we can get δ(mb)=δ(m)b+mδ(b), δ(mb)=δ(m)b+mδ(b) and δ(na)=δ(n)a+nδ(a).
Step 2. Let a,a′∈A. For any m∈M, on one hand, by Step 1, we have
δ(aa′m)=δ(a)a′m+aδ(a′m)=δ(a)a′m+aδ(a′)m+aa′δ(m). |
On the other hand,
δ(aa′m)=δ(aa′)m+aa′δ(m). |
Comparing these two equalities, we have
(δ(aa′)−δ(a)a′−aδ(a′))m=0 |
for any m∈M. Since M is a faithful left A-module, we get
δ(aa′)=δ(a)a′+aδ(a′). |
Similarly, by considering δ(mbb′), we can get
δ(bb′)=δ(b)b′+bδ(b′). |
Step 3. Let m,m′∈M and n∈N. Taking p=e−m′, x=n+m′n and q=e−m′ in Lemma 2.2, we have from pxq=pxq⊥=0 that
0=(e−m′)ϕ(m′n−m′nm′+n−nm′)−(e−m′)ϕ(m′n+n)(e−m′)−(e−m′)τ2(m′n−nm′)(f+m′)+(e−m′)τ3(nm′)(e−m′)=−ϕ(m′nm′)−eϕ(nm′)−m′ϕ(m′n)+m′ϕ(nm′)+ϕ(m′n)m′−m′ϕ(n)m′+eτ3(nm′)e−τ3(nm′)m′. | (2.16) |
This implies that
eϕ(nm′)=eτ3(nm′)e. |
Then eϕ(nm′)m′=τ3(nm′)m′ and hence by (2.16),
δ(m′nm′)=ϕ(m′nm′)=−m′ϕ(m′n)+m′ϕ(nm′)+ϕ(m′n)m′−m′ϕ(n)m′−ϕ(nm′)m′=−m′h(m′n)+m′δ(nm′)+m′h(nm′)+δ(m′n)m′+h(m′n)m′−m′δ(n)m′−h(nm′)m′=m′δ(nm′)+δ(m′n)m′−m′δ(n)m′. |
Replacing m′ with m+m′, we arrive at
δ(m′nm+mnm′)=δ(m′n)m+m′δ(nm)−m′δ(n)m+δ(mn)m′+mδ(nm′)−mδ(n)m′. |
On the other hand, by Steps 1 and 2, we have
δ(m′nm+mnm′)=δ(m′n)m+m′nδ(m)+δ(m)nm′+mδ(nm′). |
Comparing these two equalities, we have
(δ(mn)−δ(m)n−mδ(n))m′=−m′(δ(nm)−nδ(m)−δ(n)m). | (2.17) |
Set
f(m,n):=δ(mn)−δ(m)n−mδ(n) |
and
g(m,n):=δ(nm)−nδ(m)−δ(n)m. |
We assume without loss of generality that A does not contain nonzero central ideals. For any a∈A, by (2.17),
f(m,n)am′=−am′g(m,n)=af(m,n)m′. |
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
f(m,n)∈Z(A). |
By Steps 1 and 2, we have
f(am,n)=δ(amn)−δ(am)n−amδ(n)=δ(a)mn+aδ(mn)−δ(a)mn−aδ(m)n−amδ(n)=af(m,n). |
The above two equalities show that f(m,n) in the central ideal of A and hence
f(m,n)=0, | (2.18) |
that is
δ(mn)=δ(m)n+mδ(n) |
for all m∈M,n∈N. Since M is a faithful right B-module, it follows from (2.17) that
δ(nm)=nδ(m)+δ(n)m |
for all m∈M,n∈N.
Lemma 2.7. The map h:G→Z(G) vanishes on each commutator.
Proof. Step 1. Let a∈A, m∈M, n∈N and b∈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′∈A, we have ϕ([a,a′])=eϕ([a,a′])e+fϕ([a,a′])f∈A⊕B. On the other hand, Proposition 1.1 implies that ϕ([a,a′])=d([a,a′])∈A⊕M⊕N, where d is a derivation. Thus, fϕ([a,a′])f=0. This implies that h([a,a′])=h1([a,a′])=η(fϕ([a,a′])f)+fϕ([a,a′])f=0.
Similarly, we can get h([b,b′])=0, for all b,b′∈B.
Step 3. It follows from (2.18) that
(ϕ(mn)−η(fϕ(mn)f)−ϕ(m)n−mϕ(n))m′=−m′(ϕ(nm)−η−1(eϕ(nm)e)−nϕ(m)−ϕ(n)m). | (2.19) |
Since fϕ(a)f∈πB(Z(G)), eϕ(b)e∈πA(Z(G)), we get that
m′fϕ(mn)f=η(fϕ(mn)f)m′,eϕ(nm)em′=m′η−1(eϕ(nm)e). |
It further follows from (2.19) that
ϕ(mn)m′−m′fϕ(mn)f−ϕ(m)nm′−mϕ(n)m′=−m′ϕ(nm)+eϕ(nm)m′+m′nϕ(m)+m′ϕ(n)m. |
Hence
(ϕ(mn)−eϕ(nm)−ϕ(m)n−mϕ(n))m′=m′(−ϕ(nm)+fϕ(mn)f+nϕ(m)+ϕ(n)m). |
Using an argument similar to that in the proof of (2.18), we arrive that
eϕ(mn)e−eϕ(nm)−ϕ(m)n−mϕ(n)=0, | (2.20) |
and
−fϕ(nm)f+fϕ(mn)f+nϕ(m)+ϕ(n)m=0. |
By (2.19) and (2.20), we get that eϕ(nm)e=η(fϕ(mn)f). Note that h([m,n])=h1(mn)−h2(nm)=η(fϕ(mn)f)+fϕ(mn)f−eϕ(nm)e−η−1(eϕ(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 δ, we have φ(x)=δ(x)+[x,eφ(e)f−fφ(e)e]+h(x) for all x∈A, where δ 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 Mk×m(A) be the set of all k×m matrices over A. For n≥2 and each 2≤l<n−1, the full matrix algebra Mn(A) can be represented as a generalized matrix algebra of the form
(Ml×l(A)Ml×(n−l)(A)M(n−l)×l(A)M(n−l)×(n−l)(A)). |
Corollary 2.8. Let Mn(A) be a full matrix algebra with n≥4. Then each local Lie derivation φ on Mn(A) is of the form φ=d+τ, where d is a derivation of Mn(A) and τ is a linear map from Mn(A) into its center Z(A)⋅In vanishing on each commutator.
Proof. It follows from the example (C) of [2] that the matrix algebras Ml(A) and Mn−l(A) are generated by their idempotents for 2≤l<n−1. Since Z(Mn(A))=Z(A)⋅In, Z(Ml(A))=Z(A)⋅Il and Z(Mn−l(A))=Z(A)⋅In−l, the condition (2) of Theorem 2.1 is satisfied. By [5,Lemma 1], Mk(A) does not contain nonzero central ideals for k≥2. Hence by Theorem 2.1, every local Lie derivation of Mn(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 φ:R→R is a local Lie derivation, then there exit a derivation d and a linear central map τ vanishing on each commutator, such that φ=d+τ.
Proof. Let R be an unital simple algebra with a nontrivial idempotent e0 and let f0 denote the idempotent 1−e0. Then R can be represented in the so-called Peirce decomposition form
R=e0Re0+e0Rf0+f0Re0+f0Rf0, |
where e0Re0 and f0Rf0 are subalgebras with unitary element e0 and f0, respectively, e0Rf0 is an (e0Re0,f0Rf0)-bimodule.
Next, we will show that
e0xe0⋅e0Rf0={0} implies e0xe0=0 |
and
e0Rf0⋅f0xf0={0} implies f0xf0=0. |
That is e0Rf0 is faithful as an (e0Re0,f0Rf0)-bimodule. Let e=f0+e0Rf0, then e2=e and [e,R]⊆eR(1−e)+(1−e)Re. Note that
(1−e)Re=(e0−e0Rf0)R(f0+e0Rf0)⊆e0Rf0. |
Furthermore, the assumption e0xe0⋅e0Rf0={0} implies
e0xe0eR(1−e)=e0xe0(f0+e0Rf0)R(e0+e0Rf0)={0} |
and then
e0xe0[e,R]={0}. |
Let r=[e,y] and z,w∈R. It follows from
zrw=[e,z[e,r]w]−[e,z][e,rw]−[e,zr][e,w]+2[e,z]r[e,w] |
that e0xe0zrw=0. Then
e0xe0R[e,R]R=0. | (2.21) |
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), e0xe0R=0. Since 1∈R, we get e0xe0=0. Similarly, we can show that e0Rf0⋅f0xf0={0} implies f0xf0=0. Now, we can conclude that R can be represented as a generalized matrix algebra of the form R=e0Re0+e0Rf0+f0Re0+f0Rf0.
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 e0Re0 and f0Rf0 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∗-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.
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.
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.
All authors declare no conflicts of interest in this paper.
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