Non-commutative subsequence principles (Q1434514)

This paper generalizes several well-known results concerning convergence of subsequences to the non-commutative setting. Suppose that \(\tau\) is a semifinite normal faithful trace on a von Neumann subalgebra \(M\) of \(B(H)\). A (not necessarily bounded) operator \(x\) is called affiliated with \(M\) if \(xu = ux\) for every \(u \in M^\prime\). Then \(\tilde{M}\) denotes the set of \(\tau\)-measurable operators -- that is, of (unbounded) operators \(x\) s.t. for every \(\varepsilon > 0\) there exists an orthogonal projection \(p \in M\) s.t. \(p(H) \subset {\text{dom}} x\), \(\tau({\mathbf{1}} - p) < \varepsilon\), and \(xp \in M\). The predual of \(M\) can be identified with \(L_1(M,\tau)\) and viewed as a subset of \(\tilde{M}\). We say that a sequence \((x_n) \subset \tilde{M}\) converges to \(x \in \tilde{M}\) bilaterally if for every \(\varepsilon > 0\) there exists an orthogonal projection \(p \in M\) s.t. \(\tau({\mathbf{1}} - p) < \varepsilon\), and \(\lim_n \| p (x_n - x) p\| _M = 0\). The main result of the present paper provides a noncommutative version of the well-known subsequence theorem of \textit{J.~Komlós} [Acta Math. Acad. Sci. Hung. 18, 217--229 (1967; Zbl 0228.60012)]. The author shows that, for every sequence \((f_n) \subset M_*\) (here, \(M\) is a hyperfinite von Neumann algebra, equipped with a semifinite normal trace \(\tau\)) with \(\sup_n \| f_n\| _1 < \infty\), there exists \(f \in M_*\) and a subsequence \((g_n)\) such that \((m^{-1} \sum_{k=1}^m g_{n_k})_{m=1}^\infty\) converges to \(f\) bilaterally whenever \(n_1 < n_2 < \ldots\). In the non-hyperfinite case, a similar result holds, except that one has to assume that \(f_n \in L_p(M,\tau)\), and \(\sup_n \| f_n\| _p < \infty\). If \((M,\tau)\) is finite (not necessarily hyperfinite), it is shown that, whenever \(\sup_n \| f_n\| _1 < \infty\), the subsequence \((g_n)\) can be chosen in such a way that \((m^{-1} \sum_{k=1}^m g_{n_k})_{m=1}^\infty\) converges to \(f\) in measure (recall that a sequence \((x_n) \subset \tilde{M}\) converges to \(x\) in measure if for every \(\varepsilon > 0\) there exists \(N\) s.t. for every \(n \geq N\) there exists an orthogonal projection \(p\) satisfying \(\tau({\mathbf{1}} - p) < \varepsilon\), and \(\| (x_n - x)p\| _M < \varepsilon\)). Alternatively, the subsequence \((g_n)\) can be selected so that for every sequence \(n_1 < n_2 < \ldots\), the sequence of the Cesàro means \((m^{-1} \sum_{k=1}^m (1 - (k-1)/m) g_{n_k})_{m=1}^\infty\) converges to \(f\) bilaterally. The last section gives some applications of the results listed above. It is shown that, for any orthonormal system \((\phi_n)\) in \(L_2(M)\) (here, \(M\) is the hyperfinite \(II_1\) factor \(M\), equipped with its canonical trace \(\tau\)), there exists an increasing sequence \((m_k)\) such that \((\sum_{n=1}^{m_k} c_n \phi_n)_{k=1}^\infty\) converges bilaterally to \(\sum_{n=1}^\infty c_n \phi_n\) whenever \(\sum | c_n| ^2 < \infty\). The proof relies heavily on the properties of the noncommutative Walsh system, introduced by \textit{Sh.~Ayupov}, \textit{S.~Ferleger} and \textit{F.~Sukochev} [Math. Scand. 78, 271--285 (1996; Zbl 0865.46048]).