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

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.