The compact operators on c₀ as a Calkin algebra

2024/03/07 by Motakis, Pavlos, Puglisi, Daniele
#46B07 #46B25 #46B28 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2403.04137

Abstract

For a Banach space X, let L(X) denote the algebra of all bounded linear operators on X and let K(X) denote the compact operator ideal in L(X). The quotient algebra L(X)/K(X) is called the Calkin algebra of X, and it is denoted Cal(X). We prove that the unitization of K(c₀) is isomorphic as a Banach algebra to the Calkin algebra of some Banach space Z_K(c0). This Banach space is an Argyros-Haydon sum (⊕_n=1^∞ X_n)_AH of a sequence of copies X_n of a single Argyros-Haydon space \mathfrakX_AH, and the external versus the internal Argyros-Haydon construction parameters are chosen from disjoint sets.

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