Projections in the space of bounded linear operators

Thorp has shown that for X and Y certain Banach spaces of sequences there is no continuous linear projection of the bounded linear operators from X to Y onto the compact linear operators from X to Y.In this paper, this result, as well as related results for the weakly compact linear operators, is demonstrated for cases including (a)X an infinite dimensional abstract L-space and Y an infinite dimensional space whose conjugate contains a countable total set and (b)X a separable £>-space and Y = C(S) with S either a metric space containing an infinite number of points or S a compact space which contains a one-to-one convergent sequence.We recall that a subspace of a Banach space X is said to be complemented (in X) if there is a continuous linear projection of X onto that subspace,, In [14] it is shown that for X and Y certain Banach spaces of sequences the subspace K(X, Y) of compact linear operators from X to Y is not complemented in B(X, Y), the space of bounded linear operators from X to Y o Here, we will prove similar results for either X an abstract L-space or 7 a space of type C(S) and will also consider projections on the subspace W(X, Y) of weakly compact linear operators mapping X to Y.All maps will be linear and X and Y will be Banach spaces.Abstract L-spaces are defined in [7, page 394]; C(S) shall be the space of bounded continuous functions on a topological space S and we use the sup norm.We recall that a set in X', the conjugate of the Banach space X, is total if the only vector mapped into zero by that set is the zero vector.Our main results are Theorems 1 and 2 below.