The uniqueness-of-norm problem for Calkin algebras

TL;DR

This paper investigates whether Calkin algebras have unique algebra norms, providing the first negative examples through recent Banach space constructions and extending the results to weak Calkin algebras.

Contribution

It offers the first known negative examples showing non-uniqueness of algebra norms in Calkin algebras, expanding understanding of their structural properties.

Findings

Calkin algebra does not always have a unique algebra norm

Recent Banach space constructions provide negative examples

Weak Calkin algebra also lacks a unique algebra norm

Abstract

We examine the question of whether the Calkin algebra of a Banach space must have a unique complete algebra norm. We present a survey of known results, and make the observation that a recent Banach space construction of Argyros and Motakis (preprint, 2015) provides the first negative answer. The parallel question for the weak Calkin algebra also has a negative answer; we demonstrate this using a Banach space of Read (J. London Math. Soc. 1989).