Local Derivations of Finitary Incidence Algebras

TL;DR

This paper proves that every R-linear local derivation of finitary incidence algebras over a poset is actually a derivation, extending previous results in the algebraic structure of these algebras.

Contribution

It generalizes a known result by showing that local derivations are derivations in the context of finitary incidence algebras over a poset.

Findings

Every R-linear local derivation of FI(P,R) is a derivation

Generalizes previous results by Nowicki and Nowosad

Extends understanding of derivation structures in incidence algebras

Abstract

Let $P$ be a partially ordered set, $R$ a commutative ring with identity and $FI(P,R)$ the finitary incidence algebra of $P$ over $R$. In this note we prove that each $R$-linear local derivation of $FI(P,R)$ is a derivation, which partially generalizes a result by Nowicki and Nowosad.

Equations

$$d(rs)=d(r)s+rd(s)$$

$$d(e)=d_e(e)=d_e(e)e+e{d}_e(e)=d(e)e+ed(e).$$

$$\alpha =\sum _{x\le y}\alpha (x,y)e_{xy},$$

$$\alpha \beta =\sum _{x\le y}\left(\sum _{x\le z\le y}\alpha (x,z)\beta (z,y)\right)e_{xy}$$

$$e_{xy}{e}_{uv}={\delta }_{yu}{e}_{xv},$$

$$e_x\alpha e_y=\{\begin{array}{ll}\alpha (x,y)e_{xy},& x\le y,\\ 0,& x\nleqq y.\end{array}$$

$$\alpha {|}_x^y=\alpha (x,y)e_{xy}+\sum _{x\le v<y}\alpha (x,v)e_{xv}+\sum _{x<u\le y}\alpha (u,y)e_{uy}.$$

$$(\alpha {|}_x^y){|}_x^y$$

$$(e_X){|}_x^y$$

$$d(\alpha )(x,y)=d(\alpha {|}_x^y)(x,y).$$

$$d(\alpha (x,y)e_{xy})=d(e_x\alpha e_y)=d(e_x)\alpha e_y+e_{x}d(\alpha )e_y+e_x\alpha d(e_y),$$

$$d(\alpha )(x,y)$$

$$d((\alpha {|}_x^y)(x,y)e_{xy})(x,y)-(d(e_x)\alpha {|}_x^y)(x,y)-(\alpha {|}_x^{y}d(e_y))(x,y),$$

$$d(\alpha )(x,y)$$

$$=d_{\alpha -\alpha {|}_x^y}((\alpha -\alpha {|}_x^y){|}_x^y)(x,y)+d(\alpha {|}_x^y)(x,y)$$

$$=d_{\alpha -\alpha {|}_x^y}(\alpha {|}_x^y-(\alpha {|}_x^y){|}_x^y)(x,y)+d(\alpha {|}_x^y)(x,y)$$

$$=d(\alpha {|}_x^y)(x,y),$$

$$\displaystyle d(e_{X})(u,v)=\begin{cases}d(e_{u})(u,v),&\mbox{if $u\in X$ and $v\not\in X$},\\ d(e_{v})(u,v),&\mbox{if $u\not\in X$ and $v\in X$},\\ 0,&\mbox{otherwise}.\end{cases}$$

$$d(e_x)(x,y)=-d(e_y)(x,y).$$

$$\alpha =\sum _{x\le y}d(e_y)(x,y)e_{xy}\in I(P,R).$$

$$\alpha =-\sum _{x\le y}d(e_x)(x,y)e_{xy},$$

$$d(e_x)=d(e_x)e_x+e_{x}d(e_x)=\alpha e_x-e_x\alpha ={\mathrm{ad}}_{\alpha }(e_x).$$

$$d(e_X)(x_i,y_i)$$

$$=d(e_{X\setminus \{x_i\}})(x_i,y_i)+d(e_{x_i})(x_i,y_i)$$

$$=d(e_{x_i})(x_i,y_i)=-\alpha (x_i,y_i)\ne 0.$$

$$d(e_x)=0$$

$$d(\alpha )(x,y)=\sigma (x,y)\alpha (x,y)$$

$$d(e_{xy})(u,v)=0\text{mbox}for(u,v)\ne (x,y).$$

$$d(e_{xy})(u,v)=d((e_{xy}){|}_u^v)(u,v).$$

$$d(e_y+e_{xy})(u,v)$$

$$=d(e_y+e_{xy})(y,v)=d(e_{xy})(y,v)=0.$$

$$d(e_x+e_{xy})(u,v)$$

$$=d(e_x+e_{xy})(u,x)=d(e_{xy})(u,x)=0,$$

$$\sigma =\sum _{x\le y}d(e_{xy})(x,y)e_{xy}\in I(P,R).$$

Full text

Introduction

Local derivations appeared in the early 90’s in the works by Kadison [13] and Larson-Sourour [18]. Kadison proved in [13, Theorem A] that each local derivation of a von Neumann algebra with values in its dual bimodule is a derivation. Brešar showed in [6] that Theorem A by Kadison remains valid for any normed bimodule. The main result of Larson and Sourour [18] says that the algebra of all bounded operators on a complex infinite-dimensional Banach space has no proper local derivations. An alternative proof of this fact (which also works in the real case) was given in [7]. In the case of $2$-local derivations one can even drop the linearity and continuity as was shown by Šemrl in [23].

The incidence algebra $I(P,R)$ of a locally finite preordered set $P$ over a commutative ring $R$ is a classical object in the area of derivations and their generalizations. When $|P|=n<\infty$, the algebra $I(P,R)$ can be seen as a subalgebra of the full matrix algebra $M_{n}(R)$, and by this reason $I(P,R)$ is sometimes called a structural matrix algebra. We would like to note that $M_{n}(R)$, as well as its subalgebra $T_{n}(R)$ of upper triangular matrices over $R$, are particular cases of $I(P,R)$. On the other hand, if $P$ is finite and connected with $|P|\geq 2$, then $I(P,R)$ is a triangular algebra [25]111In [2] such an algebra is called the idealization of a bimodule. (when $P$ is finite, but not necessarily connected, one has $I(P,R)=\bigoplus_{j=1}^{k}I(P_{j},R)$, where $P_{1},\dots,P_{k}$ are the connected components of $P$, so if each $P_{j}$ has at least $2$ elements, then $I(P,R)$ is a direct sum of triangular algebras). The case of finite $P$ is easier to deal with, since $I(P,R)$ possesses the natural basis formed by matrix units, and it only suffices to study the behavior of a derivation on the elements of the basis (see [19, 20, 9, 12, 21, 4, 5, 11, 8, 27, 1]). In the infinite case the latter does not work (unless one imposes some extra restrictions as in [24]), and some other technique is needed (see [3, 22, 17, 14, 16, 26]).

Based on an earlier work by Nowicki [19], Nowicki and Nowosad proved in [21, Theorem 3] that each $R$-linear local derivation of $I(P,R)$ is a derivation, provided that $P$ is a finite preordered set and $R$ is a commutative ring. Alizadeh and Bitarafan improved a particular case of [21, Theorem 3] by showing in [1, Theorem 3.7] that $M_{n}(R)$ has no proper (additive, but not necessarily $R$-linear) local derivations with values in $M_{n}(\mathcal{M})$, where $\mathcal{M}$ is $2$-torsion free central $R$-bimodule and $n\geq 3$. Applying arguments similar to those used by Nowicki and Nowosad [21], Zhao, Yao and Wang proved in [27, Theorem 2.1] that each local Jordan derivation of $T_{n}(R)$ is a derivation.

In this short note, which was inspired by the recent preprint [10] by Courtemanche, Dugas and Herden, we adapt the ideas from [21] to the infinite case using the technique elaborated in [14, 16]. More precisely, we show that each $R$-linear local derivation of the finitary incidence algebra $FI(P,R)$ of an arbitrary poset $P$ over a commutative unital ring $R$ is a derivation, giving thus another partial generalization of [21, Theorem 3].

1. Preliminaries

Let $R$ be a ring. An additive map $d:R\to R$ is called a derivation of $R$, if it satisfies

[TABLE]

for all $r,s\in R$. Each $a\in R$ defines the derivation $\mathrm{ad}_{a}$, given by $\mathrm{ad}_{a}(r)=ar-ra$. A derivation of such a form is called inner. A local derivation [13, 18] of $R$ is an additive map $d:R\to R$, such that for any $r\in R$ there is a derivation $d_{r}$ of $R$ with $d(r)=d_{r}(r)$. Obviously, each derivation of $R$ is a local derivation of $R$. Observe also that for any local derivation $d$ of $R$ and any idempotent $e$ of $R$ one has

[TABLE]

Let $(P,\leq)$ be a partially ordered set and $R$ a commutative ring with identity. With any pair of $x\leq y$ from $P$ associate a symbol $e_{xy}$ and denote by $I(P,R)$ the $R$-module of formal sums

[TABLE]

where $\alpha(x,y)\in R$. If $x$ and $y$ run through a subset $X$ of the ordered pairs $x\leq y$ in the sum 2, then it is meant that $\alpha(x,y)=0$ for any pair $x\leq y$ which does not belong to $X$.

The sum 2 is called a finitary series [15], whenever for any pair of $x,y\in P$ with $x<y$ there exists only a finite number of $u,v\in P$, such that $x\leq u<v\leq y$ and $\alpha(u,v)\neq 0$. The set of finitary series, denoted by $FI(P,R)$, is an $R$-submodule of $I(P,R)$ which is closed under the convolution of the series:

[TABLE]

for $\alpha,\beta\in FI(P,R)$. Thus, $FI(P,R)$ is an $R$-algebra, called the finitary incidence algebra of $P$ over $R$. Moreover, $I(P,R)$ is a bimodule over $FI(P,R)$ under 3.

2. Local derivations of $FI(P,R)$

Given $x\leq y$, we identify $e_{xy}$ with the series $1_{R}e_{xy}\in FI(P,R)$. Note that

[TABLE]

where $\delta$ is the Kronecker delta. In particular, the elements $e_{x}:=e_{xx}$ are orthogonal idempotents of $FI(P,R)$, and for any $\alpha\in FI(P,R)$ one has

[TABLE]

We shall also consider the idempotents $e_{X}:=\sum_{x\in X}1_{R}e_{xx}\in FI(P,R)$, where $X\subseteq P$.

For any $\alpha\in FI(P,R)$ and $x\leq y$ we define

[TABLE]

Observe that the sums in 6 are finite, so $\alpha\mapsto\alpha|_{x}^{y}$ is a well-defined map $FI(P,R)\to FI(P,R)$. Moreover, it is $R$-linear and satisfies

[TABLE]

The next result is a partial generalization of [14, Lemma 8].

Lemma 2.1**.**

For each $R$-linear local derivation $d$ of $FI(P,R)$ and $x\leq y$ one has

[TABLE]

Proof.

We first assume that $d$ is an $R$-linear derivation of $FI(P,R)$. By 5

[TABLE]

whence

[TABLE]

In view of 3 and 6 the right-hand side of 10 is

[TABLE]

which is $d(\alpha|_{x}^{y})(x,y)$ by the same 10, whence 9.

Now let $d$ be an $R$-linear local derivation of $FI(P,R)$. Then using the result of the previous case and 7

[TABLE]

which proves 9. ∎

We shall also need the following lemma which partially generalizes Lemma 1 from [14].

Lemma 2.2**.**

Let $d$ be an $R$-linear local derivation of $FI(P,R)$ and $X\subseteq P$. Then for all $u\leq v$ one has

[TABLE]

Proof.

The first two cases of 11, as well as the case $u,v\not\in X$, are immediate consequences of 8 and 2.1. Now let $u,v\in X$. Then $d(e_{X})(u,v)=d_{e_{X}}(e_{X})(u,v)$, the latter being zero by [14, Lemma 1]. ∎

Corollary 2.3**.**

Let $d$ be an $R$-linear local derivation of $FI(P,R)$ and $x\leq y$. Then

[TABLE]

Indeed, if $x=y$, then $d(e_{x})(x,x)=0$ thanks to Lemma 2.2, and if $x<y$, then $d(e_{x}+e_{y})(x,y)=d(e_{\{x,y\}})(x,y)=0$ by the same reason.

The following fact is a partial generalization of [14, Lemma 2].

Lemma 2.4**.**

Let $d$ be an $R$-linear local derivation of $FI(P,R)$. Then there exists $\alpha\in FI(P,R)$ such that $d(e_{x})=\mathrm{ad}_{\alpha}(e_{x})$ for all $x\in P$.

Proof.

Define

[TABLE]

Then $\alpha e_{x}=d(e_{x})e_{x}$, and since by 12

[TABLE]

one similarly has $e_{x}\alpha=-e_{x}d(e_{x})$. So, by 1

[TABLE]

It remains to prove that $\alpha\in FI(P,R)$. Suppose that there is an infinite set $S$ of pairs $(x_{i},y_{i})$, such that $x\leq x_{i}<y_{i}\leq y$ and $\alpha(x_{i},y_{i})\neq 0$. For each fixed $u$ there is only a finite number of $i$ such that $x_{i}=u$, as $d(e_{u})(u,y_{i})=-\alpha(x_{i},y_{i})\neq 0$ for such $u$ and $d(e_{u})$ is a finitary series. Similarly for each $v$ there is only a finite number of $j$ such that $y_{j}=v$. Using this observation, similarly to what was done in the proof of [14, Lemma 2], we may construct an infinite $S^{\prime}\subseteq S$, such that for any two pairs $(x_{i},y_{i})$ and $(x_{j},y_{j})$ from $S^{\prime}$ one has $x_{i}\neq y_{j}$. Let $X=\{x_{i}\mid(x_{i},y_{i})\in S^{\prime}\}$. Note that $y_{i}\not\in X$ for any $(x_{i},y_{i})\in S^{\prime}$. So, using Lemma 2.2, we have for all $(x_{i},y_{i})\in S^{\prime}$

[TABLE]

This contradicts the fact that $d(e_{X})\in FI(P,R)$. ∎

It follows from Lemma 2.4 that it suffices to describe the local derivations of $FI(P,R)$ which satisfy

[TABLE]

for all $x\in P$.

Lemma 2.5**.**

Let $d$ be an $R$-linear local derivation of $FI(P,R)$ satisfying 13 for all $x\in P$. Then there exists $\sigma\in I(P,R)$, such that

[TABLE]

for all $\alpha\in FI(P,R)$ and $x\leq y$.

Proof.

We first show that

[TABLE]

In view of 13, equality 15 is trivial, when $x=y$. For $x<y$ observe by Lemma 2.1 that

[TABLE]

The latter may be non-zero in the following two cases:

1. (i)

$u=x<y\leq v$; 2. (ii)

$u\leq x<y=v$.

(i) Let $u=x<y<v$. Notice from 4 that $e_{y}+e_{xy}$ is an idempotent of $FI(P,R)$, so by 1, 13 and 16

[TABLE]

(ii) Let $u<x<y=v$. Considering the idempotent $e_{x}+e_{xy}\in FI(P,R)$, as above we get

[TABLE]

completing the proof of 15.

Define

[TABLE]

Using Lemmas 2.1, 6 and 15 and linearity of $d$ we conclude that

[TABLE]

∎

Lemma 2.6**.**

Let $d$ be as in Lemma 2.5. Then the corresponding element $\sigma\in I(P,R)$ given by 17 satisfies

[TABLE]

for all $x\leq y\leq z$.

Proof.

Clearly, 18 holds, when $x=y$ or $y=z$, thanks to 13 and 17. Suppose that $x<y<z$ and take

[TABLE]

Then by 14, 19, 2.1 and 6 we have

[TABLE]

Adding these equalities, we get

[TABLE]

Observe that the right-hand side of 20 is zero by [16, Lemma 4]. To show that 21 and 22 are also zero, write

[TABLE]

∎

Theorem 2.7**.**

Each $R$-linear local derivation of $FI(P,R)$ is a derivation.

Proof.

By Lemmas 2.4, 2.5 and 2.6 each $R$-linear local derivation of $FI(P,R)$ is a sum of an inner derivation and a map of the form 14 with $\sigma$ satisfying 18. It is readily checked by a direct application of 3 that such a map 14 is a derivation (see also [14, Lemma 3] for a similar construction). ∎