Commutators and images of noncommutative polynomials

2020/01/28 by Brešar, Matej · 4 citations
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2001.10392

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 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.

Cited by

• Dynamics of Word Maps on Groups and Polynomial Maps on Algebras
• Upper triangular matrices with superinvolution: identities and images of multilinear polynomials
• A note on additive commutator groups in certain algebras
• A nonlinear analogue of additive commutators

Related

• On the image of a noncommutative polynomial
• A nonlinear analogue of additive commutators
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• The Waring Problem for Matrix Algebras