Nontrivial closed ideal of $\mathbb{B(H)}$, $\mathbb{H}$ is a non-separable Hilbert space.

Question

$\mathbb{H}$ is a non-separable Hilbert space. Give an example of nontrivial closed ideal $I$of $\mathbb{B(H)}$, that is different from $\mathbb{B_0(H)}$ which is the ideal of compact operators.

Any disscussion would me welcomed! Any hints would be appreciated!

Answer 1

You can see the ideal of compact operators as the ideal generated by the finite-rank operators. You can play the same game and consider the ideal generated by those operators which have separable range (or any other cardinality less than the dimension of $\mathbb H$).

Answer 2

You will find complete descriptions of all closed ideals in the papers

M. Daws, Closed ideals in the Banach algebra of operators on classical non-separable spaces, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 2, 317–332.

W.B. Johnson, T. Kania, G. Schechtman, Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support, arXiv (2015).