  In this paper, we establish a multi-parameter version of Bellow and Losert's
Wiener-Wintner type ergodic theorem for dynamical systems not necessarily being
commutative. More precisely, we introduce a weight class $\mathcal{D}$, which
is shown to strictly include the multi-parameter bounded Besicovitch weight
class, thus including the set
$$\Lambda_d=\left\{\{\lambda^{k_1}_1\dotsm\lambda^{k_d}_d\}_{(k_1,\dots,k_d)\in
\mathbb{N}^d}:\quad (\lambda_1,\dots,\lambda_d) \in \mathbb{T}^d\right\};$$
then prove a multi-parameter Bellow and Losert's Wiener-Wintner type ergodic
theorem for the class $\mathcal{D}$ and for noncommutative trace preserving
dynamical system $(\mathcal{M},\tau,\mathbf{T})$. Restricted to consider the
set $\Lambda_d$, we also prove a noncommutative multi-parameter analogue of
Bourgain's uniform Wiener-Wintner ergodic theorem.
  The noncommutativity and the multi-parameter induce some difficulties in the
proof. For instance, our arguments in proving the uniform convergence for a
dense subset turn out to be quite different since the "pointwise" argument does
not work in the noncommutative setting; to obtain the uniform convergence in
the largest spaces, we show maximal inequality between the Orlicz spaces, which
can not be deduced easily using classical extrapolation argument. Junge and
Xu's noncommutative maximal inequalities with optimal order, together with the
atomic decomposition of Orlicz spaces, play an essential role in overcoming the
second difficulty.

Operator Algebras;Dynamical Systems;Functional Analysis