  This paper is devoted to the study of noncommutative ergodic theorems for
connected amenable locally compact groups. For a dynamical system
$(\mathcal{M},\tau,G,\sigma)$, where $(\mathcal{M},\tau)$ is a von Neumann
algebra with a normal faithful finite trace and $(G,\sigma)$ is a connected
amenable locally compact group with a well defined representation on
$\mathcal{M}$, we try to find the largest noncommutative function spaces
constructed from $\mathcal{M}$ on which the individual ergodic theorems hold.
By using the Emerson-Greenleaf's structure theorem, we transfer the key
question to proving the ergodic theorems for $\mathbb{R}^d$ group actions.
Splitting the $\mathbb{R}^d$ actions problem in two cases according to
different multi-parameter convergence types---cube convergence and unrestricted
convergence, we can give maximal ergodic inequalities on $L_1(\mathcal{M})$ and
on noncommutative Orlicz space $L_1\log^{2(d-1)}L(\mathcal{M})$, each of which
is deduced from the result already known in discrete case. Finally we give the
individual ergodic theorems for $G$ acting on $L_1(\mathcal{M})$ and on
$L_1\log^{2(d-1)}L(\mathcal{M})$, where the ergodic averages are taken along
certain sequences of measurable subsets of $G$.
