) Ever Be Amenable?

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