Algebras of Diagonal Operators of the Form Scalar-Plus-Compact Are Calkin Algebras

Abstract

For every Banach space X with a Schauder basis, consider the Banach algebra RI⊕Kdiag(X) of all diagonal operators that are of the form λI+K. We prove that RI⊕Kdiag(X) is a Calkin algebra, that is, there exists a Banach space YX such that the Calkin algebra of YX is isomorphic as a Banach algebra to RI⊕Kdiag(X). Among other applications of this theorem, we obtain that certain hereditarily indecomposable spaces and the James spaces Jp and their duals endowed with natural multiplications are Calkin algebras; that all nonreflexive Banach spaces with unconditional bases are isomorphic as Banach spaces to Calkin algebras; and that sums of reflexive spaces with unconditional bases with certain James–Tsirelson type spaces are isomorphic as Banach spaces to Calkin algebras.