A Note on Values of Noncommutative Polynomials

TL;DR

This paper identifies a class of algebras where every nontrivial noncommutative polynomial's values span the entire algebra, including bounded and compact operators on infinite-dimensional Hilbert spaces.

Contribution

It introduces a class of algebras with the property that polynomial values are dense, expanding understanding of polynomial identities in operator algebras.

Findings

Algebras where polynomial values span the entire algebra

Includes all bounded and compact operators on infinite-dimensional Hilbert spaces

Provides insight into polynomial identities in operator algebras

Abstract

We find a class of algebras A satisfying the following property: for every nontrivial noncommutative polynomial, the linear span of all its values in A equals A. This class includes the algebras of all bounded and all compact operators on an infinite dimensional Hilbert space.