On a linearization trick

TL;DR

This paper presents a linearization technique for matrix polynomials within the open unit ball, particularly in the context of self-adjoint unitary generators of a $C^*$-algebra, simplifying their structure.

Contribution

It introduces a method to express matrix polynomials in generators as products of degree-1 polynomials within the open unit ball, enhancing understanding of their algebraic structure.

Findings

Matrix polynomials in certain $C^*$-algebra generators can be factorized into degree-1 polynomials.

The factorization preserves the property of being in the open unit ball.

This simplifies the analysis of such polynomials in operator algebra contexts.

Abstract

In several situations, mainly involving a self-adjoint set of unitary generators of a $C^*$-algebra, we show that any matrix polynomial in the generators and the unit that is in the open unit ball can be written as a product of matrix polynomials of degree 1 also in the open unit ball.