Commutators and images of noncommutative polynomials

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 span$f(A)$, the linear span of the image of $f$ in $A$. In particular, we show that $[A,A]=A$ implies span$f(A)=A$. 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 $M_n(C)$, and the polynomial $f$ is neither an identity nor a central polynomial of $M_n(F)$, 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 $H$.

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