von Neumann’s inequality for row contractive matrix tuples

Abstract

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