Cesaro summability of subsequence of the partial sums of Fourier series in operator-valued setting

Let f is an element of L-1(L-infinity(T) (circle times) over barM), where T is the classical torus, M is a semi-finite von Neumann algebra. We prove the noncommutative weak type maximal inequality || (gamma(n)(f)) (n)||(Lambda 1,infinity)(N, l(infinity)) <= C ||f || L-1(N), where gamma(n)(f) is the Cesaro means of the subsequence of the partial sums sequence (S-n(f))(n >= 0). As a consequence, the Cesaro means.n( f) converges bilaterally almost uniformly to f whenever n -> infinity.