Among the proper two-sided ideals in the ring of bounded operators ${\mathcal B}(H)$ on a separable Hilbert space, the algebraic $K$-theory of the largest ideal $\mathcal K$ of compact operators was determined by Suslin and Wodzicki in terms of topological $K$-theory. Similar results were obtained by Wodzicki for a large of ideals in terms of topological $K$-theory and cyclic homology, see the main index sequence (2.8).