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 K₀-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 K₀(B(C([0,ω₁])))≅Z, where C([0,ω₁]) denotes the Banach space of scalar-valued, continuous functions defined on the compact Hausdorff space of ordinals not exceeding the first uncountable ordinal ω₁, endowed with the order topology.