Orthogonal almost locally minimal projections on ℓⁿ₁

A projection P on a Banach space X with ||P|| ≤ λ₀ is called almost locally minimal if, for every α > 0 small enough, the ball B(P, α) in the space of operators L(X) does not contain a projection Q with ||Q|| ≤ ||P||(1-Dα²), where D = D(λ₀) is a constant independent of ||P||. It is shown that, for every p ≥ 1 and every compact abelian group G, every translation invariant projection on L_p (G) is almost locally minimal. Orthogonal projections on ℓⁿ₁ are investigated with respect to some weaker local minimality properties.