Von Neumann’s inequality for row contractive matrix tuples

Abstract

We prove that for all nin {mathbb {N}}, there exists a constant C_{n} such that for all d in {mathbb {N}}, for every row contraction T consisting of d commuting n times n matrices and every polynomial p, the following inequality holds: ‖p(T)‖≤Cnsupz∈Bd|p(z)|.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert p(T)\\Vert \\le C_{n} \\sup _{z \\in {\\mathbb {B}}_d} |p(z)| . \\end{aligned}$$\\end{document}We apply this result and the considerations involved in the proof to several open problems from the pertinent literature. First, we show that Gleason’s problem cannot be solved contractively in H^infty ({mathbb {B}}_d) for d ge 2. Second, we prove that the multiplier algebra {{,mathrm{Mult},}}({mathcal {D}}_a({mathbb {B}}_d)) of the weighted Dirichlet space {mathcal {D}}_a({mathbb {B}}_d) on the ball is not topologically subhomogeneous when d ge 2 and a in (0,d). In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra A({mathcal {D}}_a({mathbb {B}}_d)) of {{,mathrm{Mult},}}({mathcal {D}}_a({mathbb {B}}_d)) generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball mathfrak {C}mathfrak {B}_d that is levelwise uniformly continuous but not globally uniformly continuous.