Generalised noncommutative subsequence principles (Q6580004)

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.\N\NThe authors extend two such results to the noncommutative setting, where bilateral almost uniform convergence forms a natural substitute for almost everywhere convergence.\N\NLet \(\mathcal M\) be a semifinite von Neumann algebra with a faithful normal semifinite trace \(\tau\). \(L^p(\mathcal M,\tau)\) is the noncommutative \(L^p\) space of \(\mathcal M\) associated with \(\tau\). If \(\mathcal M\) is hyperfinite and \(1<p<\infty\), then for any \(L^p\)-norm bounded sequence \(\{f_n\}_{n=1}^{\infty}\subseteq L^p(\mathcal M,\tau)\), there exists an element \(f\in L^p(\mathcal M,\tau)\) and a subsequence\(\{g_n\}_{n=1}^{\infty}\subseteq \{f_n\}_{n=1}^{\infty}\) such that for every further subsequence \(\{h_n\}_{n=1}^{\infty}\subseteq\{g_n\}_{n=1}^{\infty}\), and any permutation \(\sigma\) of the positive integers, the sequence \[\left\{\frac1n \sum\limits_{k=1}^{n}h_{\sigma(k)}\right\}_{n=1}^{\infty}\] converges bilaterally almost uniformly to \(f\). This result may be regarded as a subsequence principle extending the law of large numbers for noncommutative \(L^p\)-spaces with \(p>1\) in that the series is now permutation invariant. If \(1<p<1\), then \[ n^{-\frac1p} \sum\limits_{k=1}^{n}h_{ k} \to 0\] bilaterally almost uniformly, which gives a subsequence principle for \(L^p\)-spaces with \(p<1\).