von Neumann’s inequality for row contractive matrix tuples

We prove that for all n∈ N, there exists a constant C_n such that for all d∈ N, for every row contraction T consisting of d commuting n× n matrices and every polynomial p, the following inequality holds: ‖p(T)‖≤Cnsupz∈Bd|p(z)|.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^∞(B_d) for d≥ 2. Second, we prove that the multiplier algebra Mult(Da(Bd)) of the weighted Dirichlet space D_a(B_d) on the ball is not topologically subhomogeneous when d≥ 2 and a∈ (0 , d). In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra A(D_a(B_d)) of Mult(Da(Bd)) generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball CB_d that is levelwise uniformly continuous but not globally uniformly continuous.