Strong Haagerup inequality with operator coefficients

TL;DR

This paper establishes a new, explicit Strong Haagerup inequality with operator coefficients for free group von Neumann algebras, extending previous results and applying to non-commutative L_p spaces and R-diagonal operators.

Contribution

It provides a generalized and explicit upper bound on operator norms in free group algebras, improving and extending prior inequalities to broader settings including non-holomorphic cases.

Findings

Derived explicit norm bounds for elements in free group von Neumann algebras.

Extended inequalities to non-commutative L_p spaces with p even and greater than d.

Applied results to free circular and R-diagonal operators.

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 $\sum_{i=(i_1,..., i_d)} a_i \otimes \lambda(g_{i_1} ... g_{i_d})$ is less than $4^5 \sqrt e (\|M_0\|^2+...+\|M_d\|^2)^{1/2}$, 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.