Monotonicity and complex convexity in Banach lattices
It is well-known that every complex uniformly convex Banach space has finite cotype, and that the space \(L_1/H_1\) has cotype 2 but cannot be equivalently renormed to be complex uniformly convex. The author shows that in Banach lattices such examples are impossible: every Banach lattice of finite cotype possesses an equivalent complex uniformly convex norm. Unfortunately, by doing this the author just rediscovers a theorem obtained 24 years ago by \textit{E. V. Tokarev} [Funct. Anal. Appl. 15, 150--151 (1981); translation from Funkts. Anal. Prilozh. 15, No. 2, 90--91 (1981; Zbl 0466.46026)]. It is also shown that the complex strict (complex uniform) convexity of a Banach space \(X\) is inherited by vector-valued function spaces \(E(X)\) for those Köthe spaces \(E\) that are complex strictly convex or, respectively, complex uniformly convex.
scientific article; zbMATH DE number 2174926