Complex uniform rotundity in symmetric spaces of measurable operators

Let M be a semifinite von Neumann algebra with a faithful, normal, semifinite trace τ and E be a symmetric Banach function space on [0,τ(1)). We show that E is complex uniformly rotund if and only if E(M,τ)+ 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,τ). Therefore if E has non-trivial convexity, complex uniform convexity of E is equivalent with complex uniform convexity of E(M,τ). We obtain an analogous result for the unitary matrix space CE and a symmetric Banach sequence space E. From the above we conclude that E(M,τ)+ is complex uniformly rotund if and only if its norm ‖⋅‖E(M,τ) is uniformly monotone.