# Calcular Algebras

## TL;DR

This paper explores the structure and representation of calcular algebras, which are subalgebras of bounded holomorphic functions characterized by operator norm supremums over specific operator classes.

## Contribution

It characterizes which algebras can be realized as calcular algebras and investigates their possible representations.

## Key findings

- Identification of conditions for algebras to be calcular algebras
- Representation methods for calcular algebras
- Characterization of algebras arising from operator classes

## Abstract

A calcular algebra is a subalgebra of $H^\infty(\Omega)$ with norm given by $\| \phi \| = \sup \| \phi(T) \|$ as $T$ ranges over a given class of commutative $d$-tuples of operators with Taylor spectrum in $\O$. We discuss what algebras arise this way, and how they can be represented.