105 (1–2) (2004), 145–150.
LOCAL DERIVATIONS OF SUBRINGS OF MATRIX RINGS
A. NOWICKI and I. NOWOSAD (Toru´n)
Abstract. We show that on finite incidence algebras every local derivation is a derivation.
1. Introduction
Let R be a commutative ring and let P be an R-algebra with unity. An R-linear mapping d : P → P is called a derivation of P if d(ab) = ad(b) + bd(a) for all a, b ∈ P . An R-linear mapping γ : P → P is called a local R- derivation of P if for each a ∈ P there exists an R-derivation da of P such that γ(a) = da(a).
Every derivation of P is of course a local derivation of P . We say (as in [7], [8]) that P is a Kadison algebra if every local R-derivation of P is an R-derivation.
R. Kadison [1] proved that polynomial rings over C are Kadison algebras.
It was proven in [7] that any polynomial ring over a field k is a Kadison algebra if and only if k is infinite. Properties and nontrivial examples of local R-derivations of P , in the case when the R-algebra P is commutative, are given in [12], [7], [13] and [8].
There are several papers on local derivations for non-commutative alge- bras. Larson and Sourour [2] proved that the algebra B(X ) of all bounded operators on a Banach space X is a Kadison algebra. Shul’man [9] showed that any C∗-algebra is a Kadison algebra. Recently Wiehl [11] proved that the Weyl algebra with one pair of generators is a Kadison algebra.
In the present paper we prove that all finite incidence algebras [10] over R are Kadison R-algebras. This means, in particular, that the ring Mn(R) of all n × n matrices over R is a Kadison R-algebra. The same is true for the ring of triangle matrices over R and for many other subrings of Mn(R).
Key words and phrases: derivation, local derivation, incidence algebra, Kadison algebra.
2000 Mathematics Subject Classification: primary 16W25; secondary 16S30, 15A99, 47B47.
The main role in the proof of this fact is played by Theorem 1 (from [3]) describing all R-derivations of finite incidence algebras.
2. Results
Throughout this article, R is a commutative ring with identity, n is a fixed natural number, ρ is a reflexive and transitive relation on the set In= {1, 2, . . . , n}, and Mn(R) is the ring of n × n matrices over R. If (i, j) ∈ In× In, then we denote by Aij the ij-coefficient of a matrix A, and by Eij the element of the standard basis of Mn(R).
If r ∈ R, then we denote by ¯r the diagonal matrix whose all coefficients on the diagonal are equal to r. The mapping r 7→ ¯r is an injective ring ho- momorphism from R to Mn(R) and, thanks to this homomorphism, Mn(R) is an R-algebra.
Consider the set
Mn(R)ρ:= A ∈ Mn(R); Aij = 0 for (i, j) 6∈ ρ .
This is an R-subalgebra of Mn(R), called the finite incidence algebra over R with respect to ρ [10]. Properties and applications of algebras of the form Mn(R)ρ can be found in many papers; see for example [10], [3], [4], [5], [6].
Note that Mn(R)ρis a free R-module on the basis Eij; (i, j) ∈ ρ .
A mapping f : ρ → R is called transitive ([3], [5]) if f (i, k) + f (k, j) = f (i, j) for all i, j, k ∈ In such that iρk and kρj. Every mapping σ : In→ R determines a transitive mapping (i, j) 7→ σ(i) − σ(j). Transitive mappings of this form are called trivial [3]. There exist nontrivial transitive mappings.
For example, if n = 4 and ρ = (1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (1, 4), (2, 3), (2, 4) , then f : ρ → R, defined by f (i, j) = 0 for all (i, j) 6= (1, 3) and f (1, 3)
= 1, is a nontrivial transitive mapping. Note that if f : ρ → R is a transitive mapping, then f (i, i) = 0 for all i ∈ In, and f (i, j) = −f (j, i) if iρj and jρi.
An R-linear mapping d of an R-algebra P into itself is called a deriva- tion of P if d(ab) = ad(b) + d(a)b for all a, b ∈ P . We say that d is an inner derivation of P if there exists an element c ∈ P such that d = [c, ], that is, d(x) = cx − xc for all x ∈ P .
Assume now that P = Mn(R)ρ. If f : ρ → R is a transitive mapping, then we denote by ∆f the mapping from P to P defined by
∆f(B)ij = f (i, j)Bij,
for all B ∈ P and (i, j) ∈ ρ. In particular, ∆f(Eij) = f (i, j)Eij for iρj. It is known [3] that ∆f is an R-derivation of P , and that ∆f is inner if and only if f is trivial.
The following theorem describes all R-derivations of Mn(R)ρ.
Theorem 1 [3]. Let P = Mn(R)ρ. Every R-derivation of P has a unique representation
d = [A, ] + ∆f,
where f : ρ → R is a transitive mapping, and A is a matrix from P such that Aii= 0 for all i ∈ In. Moreover, the derivation d is inner if and only if f is trivial.
If i ∈ In, then we denote by Li and Ri the subsets of In defined by Li := {p ∈ In; pρi} and Ri= {q ∈ In; iρq}, resp. Observe that if iρj, then Li j Lj and Rj j Ri. It is easy to check that if A ∈ Mn(R)ρ, then [A, Eij] = P
p∈LiApiEpj−P
q∈RjAjqEiq, for all iρj. This equality and Theorem 1 im- ply the following useful lemma.
Lemma 2. If d is an R-derivation of Mn(R)ρ, then there exist a unique matrix A ∈ Mn(R)ρ and a unique transitive mapping f : ρ → R such that Aii= 0 for all i ∈ In and
d(Eij) = f (i, j)Eij +X
p∈Li
ApiEpj− X
q∈Rj
AjqEiq,
for all iρj.
Let us recall that an R-linear mapping γ : P → P , where P = Mn(R)ρ, is a local R-derivation of P if for any A ∈ P there exists an R-derivation d of P such that γ(A) = d(A). We say that P is a Kadison algebra if every local R-derivation of P is an R-derivation.
Theorem 3. Every R-algebra of the form Mn(R)ρ is a Kadison algebra.
Proof. Let γ : P → P be a local R-derivation, where P = Mn(R)ρ. Then, for every i ∈ In, there exists an R-derivation di : P → P such that γ(Eii) = di(Eii). By Lemma 2 there exists a matrix A(i) ∈ P , for every i ∈ In, such that A(i)pp = 0 for p = 1, . . . , n, and γ(Eii) =P
p∈LiA(i)piEpi− P
q∈RiA(i)iqEiq.
Denote by E the identity matrix of P . It is obvious that d(E) = 0 for every derivation d of P . This implies that γ(E) = 0. But E = E11+ E22 + · · · + Enn. So, we have
0 =
n
X
i=1
X
p∈Li
A(i)piEpi− X
q∈Ri
A(i)iqEiq
.
Fix a pair (i, j) ∈ ρ. Comparing in the above equality the coefficients with respect to Eij, we obtain the equality 0 = A(j)ij − A(i)ij . Hence, A(i)ij = A(j)ij for all iρj. Consider the matrix A ∈ P defined by Aij = A(i)ij = A(j)ij , and let δ = [A, ]. Then δ is an R-derivation of P and γ(Eii) = δ(Eii) for all i ∈ In. Let γ1 = γ − δ. The mapping γ1 is a local R-derivation of P such that γ1(Eii) = 0 for all i ∈ In. We will show that γ1 is an R-derivation. This is clear in the case when ρ = (1, 1), (2, 2), . . . , (n, n) because then γ1 = 0.
Assume that (i, j) ∈ ρ and i 6= j. There exists an R-derivation dij of P such that γ1(Eij) = dij(Eij). Hence, by Lemma 2, there exist a matrix A(i,j)
∈ P and an element f (i, j) ∈ R such that A(i,j)pp = 0 for p ∈ In, and (∗) γ1(Eij) = f (i, j)Eij + X
p∈Li
A(i,j)pi Epj− X
q∈Rj
A(i,j)jq Eiq.
Observe that γ1(Eij)ij = f (i, j). Thus we have a set of all elements from R of the form A(i,j)pq , where iρj, i 6= j, pρq, with A(i,j)pp = 0 for p ∈ In. We also have a function f from ρ\ (1, 1), . . . , (n, n) to R. Let us extend this function to ρ, putting f (p, p) = 0 for all p ∈ In. Then we have the function f : ρ → R (not necessarily transitive). Moreover, we have the equalities of the form (∗).
Now we show that A(i,j)pi = 0 for all p ∈ Li. Fix a pair (i, j) ∈ ρ and con- sider the matrix U = Eii+ Eij. Note that γ1(U ) = γ1(Eij). Let d : P → P be an R-derivation such that d(U ) = γ1(U ) = γ1(Eij). Then, by Lemma 2, there exist a matrix B ∈ P and a transitive mapping g : ρ → R such that Bpp= 0 for p ∈ In, and
g(i, i)Eii+ g(i, j)Eij+ X
p∈Li
BpiEpi− X
q∈Ri
BiqEiq+ X
p∈Li
BpiEpj
− X
q∈Rj
BjqEiq = f (i, j)Eij+ X
p∈Li
A(i,j)pi Epj− X
q∈Rj
A(i,j)jq Eiq.
Let p ∈ Li. Comparing in this equality the coefficients with respect to Epi and Epj we obtain, respectively, that 0 = Bpi and 0 = Bpi− A(i,j)pi . So, A(i,j)pi
= Bpi= 0.
By a similar way, using the matrix U = Ejj+ Eij, we observe that A(i,j)jq
= 0 for all q ∈ Rj. Therefore, the equalities (∗) have the following simpler form:
(∗∗) γ1(Eij) = f (i, j)Eij, for all iρj with i 6= j, where f : ρ → R is a mapping.
Now we show that f is a transitive mapping. Let iρk and kρj. If i = k or k = j, then of course f (i, k) + f (k, j) = f (i, j). Assume that k 6= i and k 6= j and consider the matrix U = Eik+ Ekj− Eij− Ekk. Let d be an R- derivation of P such that γ1(U ) = d(U ). Then, by Lemma 2, there exist a matrix B ∈ P and a transitive mapping g : ρ → R such that Bpp= 0 for all p ∈ In and
f (i, k)Eik+ f (k, j)Ekj− f (i, j)Eij = g(i, k)Eik+ g(k, j)Ekj − g(i, j)Eij
+ X
p∈Li
BpiEpk− X
q∈Rk
BkqEiq+ X
p∈Lk
BpkEpj− X
q∈Rj
BjqEkq
− X
p∈Li
BpiEpj+ X
q∈Rj
BjqEiq− X
p∈Lk
BpkEpk+ X
q∈Rk
BkqEkq.
Compare in this equality the coefficients with respect to Eik, Ekj, Eij and Ekk. Then we have
f (i, k) = g(i, k) + Bjk− Bik, f (k, j) = g(k, j) − Bki+ Bkj, f (i, j) = g(i, j) + Bkj− Bik, 0 = Bki− Bjk.
Note that the quantities Bjk and Bkiappear only if jρk and kρi. If (j, k) 6∈ ρ, then of course Bjk = 0. Similarly, if (k, i) 6∈ ρ, then Bki = 0.
Since g is transitive, the above equalities imply that f (i, k) + f (k, j) − f (i, j) = Bjk− Bki = 0.
So, the mapping f is transitive.
This fact and the equalities (∗∗) imply that γ1= ∆f. Hence, γ1 is an R-derivation of P . But γ = γ1+ δ, so γ is an R-derivation of P .
Using the above theorem for some specific relations ρ we obtain, for ex- ample, the following corollaries.
Corollary 4. If R is a commutative ring and n = 1, then the matrix ring Mn(R) is a Kadison R-algebra.
Corollary 5. If R is a commutative ring and n = 1, then the ring of all n × n triangle matrices over R is a Kadison R-algebra.
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(Received April 23, 2003)
FACULTY OF MATHEMATICS AND COMPUTER SCIENCE N. COPERNICUS UNIVERSITY
87–100 TORU ´N POLAND
E-MAIL: ANOW@MAT.UNI.TORUN.PL ILON@MAT.UNI.TORUN.PL
