  We prove that the von Neumann algebra generated by q-gaussians is not
injective as soon as the dimension of the underlying Hilbert space is greater
than 1. Our approach is based on a vector valued Khintchine type inequality for
Wick products. The same proof also works for the more general setting of a
Yang-Baxter deformation. Our techniques can also be extended to the so called
q-Araki-Woods von Neumann algebras recently introduced by Hiai. In this latter
case, we obtain the non injectivity under some asssumption on the spectral set
of the positive operator associated with the deformation.
