von Neumann’s inequality for row contractive matrix tuples

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