# Hyperrigid generators in C*-algebras

## TL;DR

This paper investigates conditions under which certain sets of operators generate $C^*$-algebras in a hyperrigid manner, providing new criteria and examples for hyperrigidity in operator algebras.

## Contribution

It establishes new hyperrigidity results for generators of $C^*$-algebras, including irreducible essential unitaries and normal operators, and discusses non-hyperrigidity of minimal sets for isometries.

## Key findings

- $ ext{S, SS*}$ is hyperrigid for the algebra generated by an irreducible essential unitary $S$.
- $ ext{T, T*T, TT*}$ form a hyperrigid set for the algebra generated by any operator $T$.
- Normal operators $T$ have $ ext{T, TT*}$ as hyperrigid generators; unitaries $ ext{T}$ are generated by $ ext{T}$ alone.

## Abstract

In this article, we show that, if $S\in \mathcal{B}(H)$ is irreducible and essential unitary, then $\{S,SS^*\}$ is a hyperrigid generator for the unital $C^*$-algebra $\mathcal{T}$ generated by $\{S,SS^*\}$. We prove that, if $T$ is an operator in $\mathcal{B}(H)$ that generates an unital $C^*$-algebra $\mathcal{A}$ then $\{T,T^*T,TT^*\}$ is a hyperrigid generator for $\mathcal{A}$. As a corollary it follows that, if $T\in \mathcal{B}(H)$ is normal then $\{T,TT^*\}$ is hyperrigid generator for the unital $C^*$-algebra generated by $T$ and if $T\in \mathcal{B}(H)$ is unitary then $\{T\}$ is hyperrigid generator for the $C^*$-algebra generated by $T$. We show that if $V\in \mathcal{B}(H)$ is an isometry (not unitary) that generates the $C^*$-algebra $\mathcal{A}$ then the minimal generating set $\{V\}$ is not hyperrigid for $\mathcal{A}$.