Unital Banach algebras not isomorphic to Calkin algebras of separable Banach spaces

TL;DR

This paper constructs examples of unital Banach algebras that meet necessary conditions to be Calkin algebras of separable Banach spaces but are proven not to be isomorphic to any such Calkin algebra, highlighting limitations in these representations.

Contribution

It provides explicit examples of unital Banach algebras that cannot be realized as Calkin algebras of separable Banach spaces, addressing a key open question.

Findings

Examples include algebras of the form C(X), ℓ₁(G), and simple unital AF C*-algebras.

These examples meet the density condition but are not isomorphic to any Calkin algebra of a separable Banach space.

Extensions to higher densities are also demonstrated.

Abstract

Recent developments in Banach space theory provided unexpected examples of unital Banach algebras that are isomorphic to Calkin algebras of Banach spaces, however no example of a unital Banach algebra that cannot be realised as a~Calkin algebra has been found so far. This naturally led to the question of possible limitations of such assignments. In the present note we provide examples of unital Banach algebras meeting the necessary density condition for being the Calkin algebra of a separable Banach space that are not isomorphic to Calkin algebras of such spaces, nonetheless. The examples may be found of the form $C(X)$ for a compact space $X$, $\ell_1(G)$ for some torsion-free Abelian group, and a~simple, unital AF $C^*$-algebra. Extensions to higher densities are also presented.