Sums of asymptotically midpoint uniformly convex spaces

Abstract

We study the property of asymptotic midpoint uniform convexity for infinite direct sums of Banach spaces, where the norm of the sum is defined by a Banach space $E$ with a 1-unconditional basis. We show that a sum $(\sum_{n=1}^\infty X_n)_E$ is asymptotically midpoint uniformly convex (AMUC) if and only if the spaces $X_n$ are uniformly AMUC and $E$ is uniformly monotone. We also show that $L_p(X)$ is AMUC if and only if $X$ is uniformly convex.