von Neumann's inequality for row contractive matrix tuples

Abstract

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 $\mathrm{Mult}({\mathcal{D}}_a({\mathbb{B}}_d))$ 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.

20 pages. v2: the constants C_{d,n} are shown to be uniformly bounded in d for fixed n. v3: small changes

References in corpus (1)

• Dilations of matricies

Cited by in corpus (4)

• On the classification of function algebras on subvarieties of noncommutative operator balls
• Dilations of commuting $C_0$-semigroups with bounded generators and the von Neumann polynomial inequality
• Some relations between Schwarz-Pick inequality and von Neumann's inequality
• The von Neumann inequality for $3\times 3$ matrices in the unit Euclidean ball