Complex uniform rotundity in symmetric spaces of measurable operators

Let M be a semifinite von Neumann algebra with a faithful, normal, semifinite trace tau and E be a symmetric Banach function space on [0, tau(1)). We show that E is complex uniformly rotund if and only if E(M, tau)(+) is complex uniformly rotund. Moreover, under the assumption that E is p-convex for some p > 1, complex uniform rotundity of E implies complex uniform rotundity of E(M, tau). Therefore if E has non-trivial convexity, complex uniform convexity of E is equivalent with complex uniform convexity of E(M, tau). We obtain an analogous result for the unitary matrix space C-E and a symmetric Banach sequence space E. From the above we conclude that E(M, tau)(+) is complex uniformly rotund if and only if its norm parallel to.parallel to(E(M.tau)) is uniformly monotone. (C) 2012 Elsevier Inc. All rights reserved.