Frechet differentiability of the norm of non-commutative Lp spaces

Let $M$ be a von Neumann algebra and let $(L_p(M), \|.\|_p)$, $1 \leq p < \infty$ be Haagerup’s $L_p$-space on $M$. The main ideas of the proof that the differentiability properties of $\|.\|_p\|$ are precisely the same as those of classical (commutative) $L_p$-spaces are presented. Our main instruments are the theories of multiple operator integrals and singular traces.
Joint work with: Prof. F. Sukochev, Dr. D. Potapov and Dr. D. Zanin.