For every Banach space with a Schauder basis, consider the Banach algebra of all diagonal operators that are of the form . We prove that is a Calkin algebra, that is, there exists a Banach space such that the Calkin algebra of is isomorphic as a Banach algebra to . Among other applications of this theorem, we obtain that certain hereditarily indecomposable spaces and the James spaces 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.
"Algebras of Diagonal Operators of the Form Scalar-Plus-Compact Are Calkin Algebras." Michigan Math. J. 69 (1) 97 - 152, March 2020. https://doi.org/10.1307/mmj/1574845272