von Neumann's inequality for row contractive matrix tuples

Abstract

We prove that for all n ∈ N, there exists a constant Cn 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: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(B d )) of the weighted Dirichlet space Da(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(Da(B d )) of Mult(Da(B d )) 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.

Multiplier tests and subhomogeneity of multiplier algebras

“…The recent preprint [34] shows that Mult(D a (B d )) is not topologically subhomogeneous for 0 ≤ a < d, thus answering Question 10.3.…”

Free outer functions in complete Pick spaces

“…turns out that the 2-point multiplier norm equals the H ∞ -norm, hence the lemma cannot be used to construct examples of subinner functions in the Drury-Arveson space, see [32,Lemma 3.3].…”

An invitation to the Drury-Arveson space

“…We will give a proof of the dilation theorem in the non-pure case in Subsection 3.5. holds. In fact, one can even take C d,n to be independent of d; see [93].…”