Commutators and images of noncommutative polynomials

Abstract

Let A be an algebra and let f be a nonconstant noncommutative polynomial. In the first part of the paper, we consider the relationship between [A,A], the linear span of commutators in A, and spanf(A), the linear span of the image of f in A. In particular, we show that [A,A]=A implies spanf(A)=A. In the second part, we establish some Waring type results for images of polynomials. For example, we show that if C is a commutative unital algebra over a field F of characteristic 0, A is the matrix algebra Mn(C), and the polynomial f is neither an identity nor a central polynomial of Mn(F), then every commutator in A can be written as a difference of two elements, each of which is a sum of 7788 elements from f(A) (if C=F is an algebraically closed field, then 4 elements suffice). Similar results are obtained for some other algebras, in particular for the algebra B(H) of all bounded linear operators on a Hilbert space H.