Strong Haagerup inequalities with operator coefficients

We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, Hd denotes the subspace of the von Neumann algebra of a free group FI spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on Mn(Hd), which improves and generalizes previous results by Kemp–Speicher (in the scalar case) and Buchholz and Parcet–Pisier (in the non-holomorphic setting). Namely the norm of an element of the form ∑i=(i1,…,id)ai⊗λ(gi1⋯gid) is less than , where M0,…,Md are d+1 different block-matrices naturally constructed from the family (ai)i∈Id for each decomposition of Id≃Il×Id−l with l=0,…,d. It is also proved that the same inequality holds for the norms in the associated non-commutative Lp spaces when p is an even integer, p⩾d and when the generators of the free group are more generally replaced by ∗-free R-diagonal operators. In particular it applies to the case of free circular operators. We also get inequalities for the non-holomorphic case, with a rate of growth of order d+1 as for the classical Haagerup inequality. The proof is of combinatorial nature and is based on the definition and study of a symmetrization process for partitions.