Hyperrigidity I: singly generated commutative $C^*$-algebras

TL;DR

This paper investigates hyperrigidity of monomial sets in commutative $C^*$-algebras, establishing new characterizations and connections to functional analysis and physics.

Contribution

It identifies which monomial sets are hyperrigid in commutative $C^*$-algebras and introduces a topological approach for their characterization.

Findings

Hyperrigid sets of monomials are characterized using topological limits.

Connections between hyperrigidity and areas like functional analysis and physics are established.

Examples demonstrate the optimality of the developed criteria.

Abstract

Although Arveson's hyperrigidity conjecture was recently resolved negatively by B. Bilich and A. Dor-On, the problem remains open for commutative $C^*$-algebras. Relatively few examples of hyperrigid sets are known in the commutative case. The main goal of this paper is to determine which sets of monomials in $t$ and $t^*$, where $t$ is a generator of a commutative unital $C^*$-algebra, are hyperrigid. We show that this class of hyperrigid sets has significant connections to other areas of functional analysis and mathematical physics. Moreover, we develop a topological approach based on weak and strong limits of normal (or subnormal) operators to characterize hyperrigidity tracing back to ideas of C. Kleski and L. G. Brown. Employing Choquet boundary techniques, we present examples that discuss the optimality of our results.