Norm of convolution by operator-valued functions on free groups

Abstract:

We present a connection between the Leinert sets and the non-crossing two-partitions and we use this connection to give a simple proof of the free Khintchine inequality in discrete non-commutative $L_p$-spaces. Moreover we extend the inequality of Haagerup-Pisier, \[ \left \| \sum _{g\in S} \lambda (g)\otimes a_g\right \|_{C_\lambda ^*(F_n)\otimes A} \le 2\max \left \{\left \| \sum _{g\in S} a_g^*a_g\right \|^{\frac 12}, \left \|\sum _{g\in S} a_g a_g^*\right \|^{\frac 12}\right \}, \] where $\lambda$ is the left regular representation of the group $F_n$, $a_g$ are elements of the $C^*$-algebra $A$, and $S$ is the set of the words with length one, to the set $S$ of the words with arbitrary fixed length.