Can B(lp) ever be amenable?

It is known that B(l(p)) is not amenable for p = 1, 2, infinity, but whether or not B(l(p)) is amenable for p is an element of (1, infinity) \ {2} is an open problem. We show that, if B(l(p)) is amenable for p is an element of (1, infinity), then so are l(infinity) (B(l(P))) and l(infinity) (K(l(P))). Moreover, if l(infinity) (K(l(p))) is amenable so is l(infinity) (I, K(E)) for any index set I and for any infinite-dimensional L-P-space E; in particular, if l(infinity) (K(l(P))) is amenable for p is an element of (1, infinity), then so is l infinity (K(l(p) circle plus l(2) )). We show that l(infinity)(K(l(p) circle plus l(2))) is not amenable for p = 1, infinity, but also that our methods fail us if p is an element of (1, infinity). Finally, for p is an element of (1, 2) and a free ultrafilter U over N, we exhibit a closed left ideal of (K(l(p)))u lacking a right approximate identity, but enjoying a certain very weak complementation property.