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Let $H$ be a separable Hilbert space and consider the Calkin algebra $C(H)=\frac{B(H)}{K(H)}$.

Q) True or false: Any representation of $C(H)$ is a direct sum of irreducible representations.

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    $\begingroup$ I think the correct spelling is "Calkin". $\endgroup$
    – S. Carnahan ♦
    Commented Apr 5, 2016 at 8:18

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If you google "representations of the Calkin algebra", you find the 1967 paper of Sakai which answers your question in the negative: It says that the Calkin algebra has a type III factor representation. By the Lemma of Schur, however, every irrducible representation is a type I representation and a direct sum of more than one irreducible is not a factor.

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  • $\begingroup$ Thank a lot. Hence the second dual of Calkin algebra should be a very complicated von Neumann algebra. Is there any paper in which I find something concerning the second dual of Calkin algebra?! $\endgroup$
    – ABB
    Commented Apr 5, 2016 at 13:53

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