Strong Haagerup inequality with operator coefficients
arXiv:0903.0303 · doi:10.1016/j.jfa.2009.05.015
Abstract
We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, H_d denotes the subspace of the von Neumann algebra of a free group F_I 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 M_n(H_d), 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 is less than , where M_0,...,M_d are d+1 different block-matrices naturally constructed from the family (a_i)_{i \in I^d} for each decomposition of I^d = I^l \times I^{d-l} with l=0,...,d. It is also proved that the same inequality holds for the norms in the associated non-commutative L_p 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.
34 pages, 6 figures; changes according to the advices of the reviewer: typos corrected, introduction and proof of Corollary 2.4 rewritten. To appear in Journal of Functional Analysis