Let M be a von Neumann algebra with a faithful normal state \(\rho\) given by a bicyclic vector \(\xi_ 0\). A sequence of operators \(\{x_ n\}\) is bilaterally almost uniformly fundamental [convergent] if, given any \(\epsilon >0\), there is a projection \(p\in M\) with \(\rho (p^{\perp})<\epsilon\) such that \(\| p(x_ n-x_ m)p\| \to 0\) [resp. \(\| p(x_ n-x_ 0)p\| \to 0]\) as n,m\(\to \infty.\)
For each \(x\in M\) there is a corresponding linear form \(\omega_ x\in (M')_*\), where \((M')_*\) is the predual of the commutant of M. The norm of this form \(\omega_ x\) is denoted by \(\| x\|_ 1.\)
The author formulates two theorems on \(\|\) \(\|_ 1\) and bilateral uniform convergence of subadditive sequences and supermartingales, generalizing some resuls of R. Jajte [Strong limit theorems in noncommutative probability, Lecture Notes in Math. 1110 (1985; Zbl 0554.46033)].