Quantum Expanders and Geometry of Operator Spaces II
  In this appendix to our paper with the same title posted on arxiv we give a quick proof of an inequality that can be substituted to Hastings's result, quoted as Lemma 1.9 in our previous paper. Our inequality is less sharp but also appears to apply with more general (and even matricial) coefficients. It shows that up to a universal constant all moments of the norm of a linear combination of the form   $$S=\sum\nolimits_j a_j U_j \otimes \bar U_j (1-P)$$ are dominated by those of the corresponding Gaussian sum   $$S'=\sum\nolimits_j a_j Y_j \otimes \bar Y'_j .$$ The advantage is that $S'$ is now simply separately a Gaussian random variable with respect to the independent Gaussian random matrices $(Y_j)$ and $(Y'_j)$, and hence is much easier to majorize. Note we plan to incorporate this appendix into our future publication. 
Gilles Pisier
math.OA math.FA math.PR