Functional identities of one variable

Abstract

Let $A$ be a centrally closed prime algebra over a characteristic 0 field $k$, and let $q:A\to A$ be the trace of a $d$-linear map (i.e., $q(x)=M(x,...,x)$ where $M:A^d\to A$ is a $d$-linear map). If $[q(x),x]=0$ for every $x\in A$, then $q$ is of the form $q(x)={\sum }_{i=0}^d{\mu }_i(x)x^i$ where each ${\mu }_i$ is the trace of a $(d-i)$-linear map from $A$ into $k$. For infinite dimensional algebras and algebras of dimension $>d^2$ this was proved by Lee, Lin, Wang, and Wong in 1997. In this paper we cover the remaining case where the dimension is $\le d^2$. Using this result we are able to handle general functional identities of one variable on $A$; more specifically, we describe the traces of $d$-linear maps $q_i:A\to A$ that satisfy ${\sum }_{i=0}^m{x}^i{q}_i(x)x^{m-i}\in k$ for every $x\in A$.

10 pages