Separable spaces of continuous functions as Calkin algebras

Abstract: For a Banach space $X$ denote $\mathcal{L}(X) = \{T:X\to X\text{ linear and bounded}\}$ and $\mathcal{K}(X) = \{T\in\mathcal{L}(X): T\text{ compact}\}$. The Calkin algebra of $X$ is the Banach algebra $\mathcal{C}al(X) = \mathcal{L}(X)/\mathcal{K}(X)$. A question that has gathered attention in recent years is what unital Banach algebras admit representations as Calkin algebras. We discuss developments in this topic as well as a recent contribution, namely that for every compact metric space $K$ there exists a Banach space $X$ so that $\mathcal{C}al(X)$ coincides isometrically with $C(K)$ as a Banach algebra.

functional analysisoperator algebras

Audience: advanced learners

( paper )

Functional analysis and operator algebras in Athens

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