Noncommutative functions are weakly algebraic on operatorial polynomial polyhedra

TL;DR

This paper demonstrates that noncommutative functions defined on operatorial polynomial polyhedra are weakly algebraic, meaning their values lie in the weak operator topology closure of the algebra generated by their inputs, without domain restrictions.

Contribution

It proves weak algebraicity of noncommutative functions on general operatorial polynomial polyhedra without assuming balanced domain conditions.

Findings

Noncommutative functions are weakly algebraic on all operatorial polynomial polyhedra.

The result holds without the balanced set assumption.

Values of functions lie in the weak operator topology closure of generated algebras.

Abstract

An operatorial polynomial polyhedron is a set of the form $$B_{\delta}(\mathcal{B}(\mathcal{H}))=\{X\in \mathcal{B}(\mathcal{H})^d : \Vert\delta(X)\Vert<1\}$$ where $\mathcal{B}(\mathcal{H})$ denotes the space of bounded operators on a separable Hilbert space $\mathcal{H}$, and $\delta$ is a matrix of polynomials in $d$ noncommuting variables. These sets appear throughout the literature on noncommutative function theory. While much of what has been written involves matricial polynomial polyhedra, there do exist $\delta$ such that the associated $B_{\delta}(\mathcal{B}(\mathcal{H}))$ is non-empty but contains no matrix points. Algebraicity of operatorial noncommutative functions has been established in the case that the domain $B_{\delta}(\mathcal{B}(\mathcal{H}))$ is a balanced set (hence contains the matrix point 0). In this paper, we dispense of such assumptions on the domain and prove that an operatorial noncommutative function on any $B_{\delta}(\mathcal{B}(\mathcal{H}))$ is weakly algebraic in the sense that its value at each operator tuple $Z$ lies in the weak operator topology closure of the unital algebra generated by the coordinates of $Z$.