Noncommutative functions are weakly algebraic on operatorial polynomial polyhedra

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 $Z$.