3
$\begingroup$

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.

$\endgroup$
1
  • 2
    $\begingroup$ I think the correct spelling is "Calkin". $\endgroup$
    – S. Carnahan
    Apr 5, 2016 at 8:18

1 Answer 1

6
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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
    Apr 5, 2016 at 13:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.