TITLE: Characterization of ideals in the bounded operators
QUESTION [6 upvotes]: Let $\mathcal{B}(H)$ denote the C*-algebra of all bounded operators on a separable infinite dimensional complex Hilbert space $H$. It is a well-known fact that $\mathcal{B}_0(H)$, the ideal of compact operators, is the unique, nontrivial, proper closed ideal in $\mathcal{B}(H)$. Furthermore, any proper ideal in $\mathcal{B}(H)$ must be contained in $\mathcal{B}_0(H)$.
Is there any reasonable characterization of all (not necessarily closed) ideals in $\mathcal{B}(H)$ which contain the ideal of finite rank operators (possibly under some definability constraints, like being Borel in the strong operator topology, etc)?
Also, not being an operator theorist, are there important ideals of this type other than $\mathcal{B}_0(H)$ and the Schatten $p$-ideals (including the trace-class and Hilbert-Schmidt operators)?

REPLY [4 votes]: One of my ancient papers, joint work with Gary Weiss, concerns the question whether the ideal of compact operators is the sum of two properly smaller (and therefore not closed) ideals.  We proved that the answer is affirmative if one assumes the continuum hypothesis (or various weaker assumptions).  The paper contains a description of ideals that "reduces" this sort of question to set theory.  Later work showed that the use of assumptions (like the continuum hypothesis) beyond the usual axioms of set theory is needed; it's consistent with ZFC that $K(H)$ is not the sum of two properly smaller ideals.  
The joint paper with Gary Weiss is "A characterization and sum decomposition of operator ideals" [Trans. A. M. S. 246 (1978) 407-417].  For the later work, see my paper "Near Coherence of Filters II: Applications to operator ideals, the Stone-Cech remainder of a half-line, order-ideals of sequences, and slenderness of groups" [Trans. A. M. S. 300 (1987) 557-581] and the references there.