Let \({\mathcal M}\) be a semifinite von Neumann algebra with a faithful normal semifinite trace \(\tau\), \(L^0=L^0({\mathcal M},\tau)\) the \(*\)-algebra of \(\tau\)-measurable operators affiliated with \({\mathcal M}\), and let \(L^p=L^p({\mathcal M},\tau)\), \(1\leq p\leq \infty\), be the noncommutative \(L^p\) space associated with \(({\mathcal M},\tau)\). A linear map \(T:L^1+L^\infty \to L^1+ L^\infty\) such that \(\| T(x)\|_\infty \leq \| x\|_\infty\) for all \(x\in {\mathcal M}\) and \(\| T(x)\|_1 \leq \| x\|_1\) for all \(x\in L^1\) is called a Dunford-Schwartz operator.
Let \(\Phi\) be an Orlicz function and let \(L^\Phi =L^\Phi({\mathcal M},\tau)\) be the noncommutative Orlicz space associated with \(({\mathcal M},\tau)\). The Orlicz function satisfies a \(\Delta_2\)-condition (resp., \(\delta_2\)-condition) if there exists \(k>0\) and \(u_0\geq 0\) such that \(\Phi(2u)\leq k\Phi(u)\) for all \(u\geq u_0\), (resp., \(\Phi(2u)\leq k\Phi(u)\) for all \(u\in (0,u_0]\)).
The main result of the paper shows that, if \(T\) is a positive Dunford-Schwartz operator and \(\Phi\) is an Orlicz function satisfying the \((\delta_2, \Delta_2)\) condition, then, for all \(x\in L^\Phi\), the averages \(\frac{1}{n}\sum_{k=1}^n T^k(x)\) converge in the b.a.u.(bilateral almost uniform) topology to some \(\hat{x}\) in \(L^\Phi\). A sequence \((x_n)\subset L^0\) converges in the b.a.u.topology to \(x\) if, for all \(\varepsilon>0\), there exists a projection \(e\in{\mathcal M}\) such that \(\tau(e^\perp)<\varepsilon\) and \(\| e(x-x_n)e\|_\infty\to 0\).