1
$\begingroup$

Is there any condition on the Hilbert space $\mathcal{H}$ (if $\mathcal{H}$ if infinite-dimensional) such that the C*-algebra of bounded operators on $\mathcal{H}$ - $\mathcal{L}(\mathcal{H})$ - is simple ?

$\endgroup$
5
$\begingroup$

A $C^{\star}$ - algebra is simple if it has no nontrivial closed ideals.

If $H$ is an infinite-dimensional Hilbert space then the ideal of compact operators is a nontrivial closed ideal of $L(H)$. So, $L(H)$ is not simple.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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