Algebras of diagonal operators of the form scalar-plus-compact are Calkin algebras

2017/11/03 by Motakis, Pavlos, Puglisi, Daniele, Tolias, Andreas
#46B03 #46B25 #46B28 #46B45 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1711.01340

Abstract

For every Banach space X with a Schauder basis consider the Banach algebra ℝ I\oplusK_diag(X) of all diagonal operators that are of the form λI + K. We prove that ℝ I\oplusK_diag(X) is a Calkin algbra i.e., there exists a Banach space Y_X so that the Calkin algebra of Y_X is isomorphic as a Banach algebra to ℝ I\oplusK_diag(X). Among other applications of this theorem we obtain that certain hereditarily indecomposable spaces and the James spaces J_p and their duals endowed with natural multiplications are Calkin algebras, that all non-reflexive 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.

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