Local derivations of finitary incidence algebras (Q1705190)

A local derivation on a ring \(R\) is an additive map \(d:\, R\to R\) such that for all \(r\in R\) there is a derivaton \(d_r\) of \(R\) with \(d(r)= d_r(r)\). Several major results in operator theory dating back to the 1990s deal with the problem of finding criteria for a local derivation to be a derivation on \(R\). Let \(R\) be a commutative ring and \(P\) a poset. Let \[ A=\{(x,y):\,x,\,y\in P,\;x\leq y \text{ and}[x,y]\text{ is finite}\}. \] For each \((x,y)\in A\), let \(e_{xy}\) be an indeterminate. The finitary incidence algebra \(FI(P,R)\) of poset \(P\) over \(R\) is the \(R\)-module of all formal sums \(\alpha=\sum_{(x,y)\in A} \alpha(x,y)e_{xy}\) where \(\alpha(x,y)\in R\). With multiplication defined by convolution, \(FI(P,R)\) is an \(R\)-algebra. The main result of this paper is that every local derivation on \(FI(P,R)\) is a derivation.