von Neumann's inequality for row contractive matrix tuples

We prove that for all $n\in \mathbb{N}$, 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: $$\|p(T)\| \le C_{n} \sup_{z \in \mathbb{B}_d} |p(z)| .$$ 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 $H^\infty(\mathbb{B}_d)$ for $d \ge 2$. Second, we prove that the multiplier algebra $\operatorname{Mult}(\mathcal{D}_a(\mathbb{B}_d))$ 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 $\operatorname{Mult}(\mathcal{D}_a(\mathbb{B}_d))$ 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.