K-theory for algebras of operators on Banach spaces

It is proved that, for each pair (m, n) of non-negative integers, there is a Banach space X for which K-0(B(X)) congruent to Z(m) and K-1(B(X)) congruent to Z(n). The K-groups of all closed ideals of operators contained in the ideal of strictly singular operators are computed, and some results about the existence of splittings of certain short exact sequences are derived.