Quenched central limit theorem for random walks with a spectral gap

Let G be a semi-group of measure preserving transformations of a probability space (X, B, m) and let mu be a probability measure on G. We prove a quenched central limit theorem for functions in L(0)(p)(m), p > 2, when the spectral gap condition holds for the diagonal action of G on (X x X, m circle times m). (C) 2011 Academie des sciences.