Let M be a von Neumann algebra and let (L_p(M), ||.||_p), 1<=p<infinity be Haagerup's L_p-space on M. It is proved that the differentiability properties of ||.||_p are precisely the same as those of classical (commutative) L_p-spaces. This resolves the problem suggested by G. Pisier and Q. Xu in their survey, 2003. The main instruments are the theories of multiple operator integrals and singular traces.
In this talk I will explain the commutative version of this result and give a general idea of the proof in the noncommutative case.
Joint work with: Denis Potapov, Fedor Sukochev and Dmitriy Zanin