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 C E 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.