  In this paper, we investigate the boundedness of a square function from
ergodic theory on noncommutative $L_{p}$-spaces. The main result is a weak
$(1,1)$ type estimate of this square function. We also show the
$(L_{\infty},\mathrm{BMO})$ estimate, and thus strong $(L_{p},L_{p})$ estimate
by interpolation. The main novel difficulty lies in the fact that the kernel of
this square function does not enjoy any regularity, which is crucial in showing
such endpoint estimates for standard noncommutative Calder\'on-Zygmund singular
integral operators.
