Maximum modulus principle for "holomorphic functions" on the quantum matrix ball

TL;DR

This paper characterizes the boundary behavior of quantum holomorphic functions on matrix balls, showing their algebraic structure relates to quantum unitary groups, thus advancing noncommutative function theory.

Contribution

It identifies the Shilov boundary ideal for a q-analog of holomorphic functions on matrix balls and establishes its $C^*$-envelope as the algebra of continuous functions on the quantum unitary group.

Findings

Shilov boundary ideal described for quantum matrix ball

The $C^*$-envelope is isomorphic to continuous functions on $U_q(n)$

Advances understanding of noncommutative function algebras

Abstract

We describe the Shilov boundary ideal for a q-analog of the algebra of holomorphic functions on the unit ball in the space of $n\times n$ matrices and show that its $C^*$-envelope is isomorphic to the $C^*$-algebra of continuous functions on the quantum unitary group $U_q(n)$.