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

Abstract

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