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It is well-known (and easy to prove) that the only closed ideals of $B(\ell_2)$ are $\{0\}$, $B(\ell_2)$ and $K(\ell_2)$, the ideal of compact operators on $\ell_2$. I am curious whether we know what is the lattice of closed, two-sided ideals of $B(\ell_2)^{**}$.

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    $\begingroup$ The second dual of $B(H)$? Are you mad?? $\endgroup$
    – Nik Weaver
    May 22, 2014 at 22:37
  • $\begingroup$ You could find something in Matthew Daws Ph.D. thesis $\endgroup$
    – Norbert
    Jul 28, 2014 at 11:40
  • $\begingroup$ @userNaN Do you have a specific reference? I'm reasonably familiar with Daws's thesis and he looks at the lattice of closed 2-sided ideals in $B(\ell_p(I))$ for higher cardinals $I$, but IIRC nothing on ideals in $B(H)^{**}$ $\endgroup$
    – Yemon Choi
    Dec 23, 2014 at 4:08
  • $\begingroup$ My bad, Daws discussed only semisimplicity of $\mathcal{B}(E)^{**}$ $\endgroup$
    – Norbert
    Dec 23, 2014 at 7:43

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