Functional identities in one variable

Let A be a centrally closed prime algebra over a characteristic 0 field k, and let q:A→A be the trace of a d-linear map (i.e., q(x)=M(x,…,x) where M:Ad→A is a d-linear map). If [q(x),x]=0 for every x∈A, then q is of the form q(x)=∑i=0dμi(x)xi where each μi is the trace of a (d−i)-linear map from A into k. For infinite dimensional algebras and algebras of dimension >d2 this was proved by Lee, Lin, Wang, and Wong in 1997. In this paper we cover the remaining case where the dimension is ⩽d2. Using this result we are able to handle general functional identities in one variable on A; more specifically, we describe the traces of d-linear maps qi:A→A that satisfy ∑i=0mxiqi(x)xm−i∈k for every x∈A.