ℬ(ℓ^{𝓅}) is never amenable

We show that if E E is a Banach space with a basis satisfying a certain condition, then the Banach algebra ℓ ∞ ( K ( ℓ 2 ⊕ E ) ) \ell ^\infty ({\mathcal K}(\ell ^2 \oplus E)) is not amenable; in particular, this is true for E = ℓ p E = \ell ^p with p ∈ ( 1 , ∞ ) p \in (1,\infty ) . As a consequence, ℓ ∞ ( K ( E ) ) \ell ^\infty ({\mathcal K}(E)) is not amenable for any infinite-dimensional L p {\mathcal L}^p -space. This, in turn, entails the non-amenability of B ( ℓ p ( E ) ) {\mathcal B}(\ell ^p(E)) for any L p {\mathcal L}^p -space E E , so that, in particular, B ( ℓ p ) {\mathcal B}(\ell ^p) and B ( L p [ 0 , 1 ] ) {\mathcal B}(L^p[0,1]) are not amenable.