Let X and Y be Banach spaces such that the ideal of operators which factor through Y has
codimension one in the Banach algebra B(X) of all bounded operators on X,
and suppose that Y contains a complemented subspace which is isomorphic to Y+Y and that X is isomorphic to X+Z
for every complemented subspace Z of Y. Then the K

_{0}-group of B(X) is isomorphic to the additive group Z of integers.
A number of Banach spaces which satisfy the above conditions are identified.
Notably, it follows that K

_{0}(B(C([0,w

_{1}]))) is Z, where C([0,w

_{1}]) denotes the Banach space of scalar-valued,
continuous functions defined on the compact Hausdorff space of ordinals not exceeding the first uncountable ordinal w

_{1}, endowed with the order topology.