## Abstract

We prove that for all n∈ N, there exists a constant C_{n} 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: ‖p(T)‖≤Cnsupz∈Bd|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^{∞}(B_{d}) for d≥ 2. Second, we prove that the multiplier algebra Mult(Da(Bd)) of the weighted Dirichlet space D_{a}(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(D_{a}(B_{d})) of Mult(Da(Bd)) 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.

Original language | English |
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Pages (from-to) | 3877-3894 |

Number of pages | 18 |

Journal | Mathematische Zeitschrift |

Volume | 301 |

Issue number | 4 |

DOIs | |

State | Published - 1 Aug 2022 |

Externally published | Yes |

## Keywords

- Gleason’s problem
- Noncommutative function theory
- Von Neumann type inequality

## ASJC Scopus subject areas

- General Mathematics