# Operator algebras of higher rank numerical semigroups

## TL;DR

This paper explores the operator algebras associated with higher rank numerical semigroups, revealing their character space as the polydisc and using this to identify semigroups from their algebras, contributing to the understanding of dilation problems.

## Contribution

It demonstrates that the nonselfadjoint semigroup algebras of higher rank numerical semigroups have the polydisc as their character space and uses this to identify semigroups from their algebras.

## Key findings

- Character space of these algebras is the polydisc.
- These algebras provide examples related to Arveson's Dilation Problem.
- Method to identify semigroups from their operator algebras.

## Abstract

A higher rank numerical semigroup is a positive cone whose seminormalization is isomorphic to the free abelian semigroup. The corresponding nonselfadjoint semigroup algebras are known to provide examples that answer Arveson's Dilation Problem to the negative. Here we show that these algebras share the polydisc as the character space in a canonical way. We subsequently use this feature in order to identify higher rank numerical semigroups from the corresponding nonselfadjoint algebras.