Von Neumann Type Inequalities for NC Operator Balls

Abstract

We establish finite-dimensional von Neumann type inequalities for commuting matrix tuples in noncommutative operator balls. Let $Ω=\mathbb{D}_Q$ be the NC operator ball of a linear pencil $Q$ with linearly independent coefficients. If $Ω(1)$ admits a transitive subgroup of automorphisms with NC lifts and $Ω(1)$ has the bounded $H^\infty$-Gleason property at the origin, then $\|f(T)\|\leq C_{Ω,n}\|f\|_{H^\infty(Ω(1))}$ for every commuting tuple $T\inΩ(n)$, with $C_{Ω,n}=(2ρ_ΩK_Ω)^{n-1}$; here $K_Ω$ is the Gleason constant and $ρ_Ω$ is the sum of the suprema of the coordinate functions on $Ω(1)$. The scalar coordinate bounds persist at every matrix level. The hypotheses hold for products of row and column balls, where $ρ_Ω=d$, yielding an unconditional estimate $(2dK_Ω)^{n-1}$. For a general type I Cartan domain $R_{p,q}$, we obtain only a conditional result: the same bound holds provided the scalar domain has the bounded $H^\infty$-Gleason property at the origin, which we do not establish when $\min(p,q)\geq 2$.