Strong Haagerup inequalities for free R-diagonal elements

In this paper, we generalize Haagerup's inequality [U. Haagerup, An example of a nonnuclear C∗-algebra, which has the metric approximation property, Invent. Math. 50 (1978/1979) 279–293] (on convolution norm in the free group) to a very general context of R-diagonal elements in a tracial von Neumann algebra; moreover, we show that in this “holomorphic” setting, the inequality is greatly improved from its original form. We give combinatorial proofs of two important special cases of our main result, and then generalize these techniques. En route, we prove a number of moment and cumulant estimates for R-diagonal elements that are of independent interest. Finally, we use our strong Haagerup inequality to prove a strong ultracontractivity theorem, generalizing and improving the one in [P. Biane, Free hypercontractivity, Comm. Math. Phys. 184 (1997) 457–474].