On local derivations in the Kadison sense (Q2759162)

A linear map \(d\) from an algebra \(A\) into itself is called a local derivation if for each \(a\in A\) there is a derivation \(d_a\) of \(A\) such that \(d(a)= d_a(a)\). These maps were first studied by \textit{R. V. Kadison} [J. Algebra 130, No. 2, 494-509 (1990; Zbl 0751.46041)]. The author calls an algebra \(A\) a Kadison algebra if every local derivation on \(A\) is a derivation. For various special algebras the question whether or not they are Kadison algebras is discussed. For example, the polynomial ring \(k[x_1,\dots, x_n]\) is a Kadison algebra if and only if \(k\) is an infinite field.