Generalised noncommutative subsequence principles

Generalised subsequence principles extend almost everywhere convergence results for sequences of independent random variables, satisfying a moment condition, to subsequences of an arbitrary sequence of functions which satisfies the same moment condition.

Here we extend two such results to the noncommutative setting, where bilateral almost uniform convergence forms a natural substitute for almost everywhere convergence.

Our first result is a subsequence principle extending the law of large numbers for noncommutative $L^p$-spaces, with $p \gt 1$, in that the series is now permutation invariant, and the second result gives a subsequence principle in the quasi-Banach setting, providing a subsequence principle for $L^p$-spaces with $p \lt 1$. Our first result also corrects the proof presented in the classical case in Wojtaszczyk’s book (1991).