Functional identities in one variable. (Q397983)

Let \(A\) be an algebra over a field \(F\). The authors require \(F\) to be of characteristic 0 but several of their results, as they mention, hold under weaker assumptions on the field. A commuting map \(q\colon A\to A\) is a map such that \(q(a)a=aq(a)\) for all \(a\in A\). If \(A\) and \(B\) are two algebras then a map \(q\colon A\to B\) is the trace of a \(d\)-linear map if there is a \(d\)-linear map \(M\colon A^d\to B\) with \(q(a)=M(a,\ldots,a)\), \(a\in A\).NEWLINENEWLINE A major contribution of the paper under review is the following theorem. Let \(A\) be a centrally closed prime algebra. If \(q\) is a commuting trace of a \(d\)-linear map then \(q\) is of the so-called standard form. Recall that \(q\) is of standard form if \(q(a)=\sum_{i=0}^d\mu_i(a)a_i\) where \(\mu_i\) are traces of \((d-i)\)-linear maps \(A^{d-i}\to A\). This is obtained first by means of a (quite nontrivial) reduction to the case when \(A\) is a matrix algebra and then passing to the algebra of generic matrices which enables the use of methods from commutative algebra.NEWLINENEWLINE The authors also consider various applications of the above theorem in the context of functional identities. They obtain a description of the traces of \(d\)-linear maps \(q_i\) in centrally closed prime algebras such that \(\sum_{i=0}^ma^iq_i(a)a^{m-i}\in F\) for all \(a\in A\).