Lifting of rotundity properties from $E$ to $L^p(\mu E)$
Giovanni Emmanuele, Alfonso Villani · Rocky Mountain Journal of Mathematics · 1987
We consider some rotundity properties which axe extensions of the uniform rotundity and show that these properties lift from the Banach space E (or from the conjugate Banach space E*) to the Lebesgue-Bochner function space L p (/x,£) (or to (LP(/x, £))*), 1 a finite measure space, and E, a Banach space, be given.Assume that E*, the conjugate space of E, satisfies the Radon-Nikodym property.Then L p (ß,E), 1 < p < oo, is weakly uniformly rotund if and only if E is.One of the purposes of this paper is to prove the above result without any assumption on E*. (By the way, it is unknown up to now whether the weak uniform rotundity of a Banach space E implies that E* has the Radon-Nikodym property).Moreover, we consider three other geometric properties, namely weak local uniform rotundity weak* uniform rotundity (in a conjugate space) and weak* local uniform rotundity, and we show that they lift from E (or E*) to L p {ß,E) (or (L p (/j,, E))*).Also for these properties we will make no assumption on E*.It is worth noting that assuming the Radon-Nikodym property for E* would be an effective restriction in this case (see Remark 4 at the end of the paper).Work performed under the auspices of G.N. A.F. A. of Italian C.N.