Noncommutative Wiener–Wintner type ergodic theorems

Abstract

We obtain a version of the noncommutative Banach Principle suitable to prove Wiener-Wintner type results for weights in W1-space. This is used to obtain noncommutative Wiener-Wintner type ergodic theorems for various types of weights for certain types of positive Dunford-Schwartz operators. We also study the b.a.u. (a.u.) convergence of some subsequential averages and moving averages of such operators.

Noncommutative Modulated Individual Ergodic Theorems for General Weights

“…The commutative version of result is Theorem 3.5 in [15]; we note that our result will unfortunately not be as general when 2 < p < ∞ due to the mentioned technicality on q. Afterwards, we show that one drawback of the second main result mentioned of [18] can be partially fixed to allow certain W q Hartman sequences on other L p -spaces (though, unfortunately, not for as many weights as on the space the assumption holds). Finally, in Section 4 we show that the results can be further extended to include more general weighted averages using the approach of Cuny and Weber in [7], allowing certain number theoretic weights to be able to be considered.…”

“…In [18], this problem doesn't actually occur due to different assumptions being made on T . In fact, the methods and assumption in that article allow even more general weights to be used (in particular, one replaces W q with W 1 ).…”

“…In [18], if the assumptions on T (inspired by this decomposition) in Theorem 3.2 below held on some L r , where 1 < r < ∞, the author obtained the convergence of the averages for all sequences in W 1 + ∩ H. However, for any other p = r one could only guarantee the convergence on L p when the weights were in W ∞ ∩ H using the methods of that paper (unless the same assumptions also held on L p as well).…”

“…For the results mentioned above in the noncommutative setting, the corresponding weighted maximal inequalities, which were used to prove convergence, required that the weighting sequence be bounded. However, in [18], with an additional assumption to the other result of the paper already mentioned above (that is often already satisfied as a byproduct of the first assumption), the author showed that the ergodic averages, when weighted by W 1 sequences (see Section 2 below for the definition of W r -spaces), are bilaterally uniformly equicontinuous in measure at zero. From this, it was shown that the set of Hartman sequences in W 1 + satisfy a Wiener-Wintner type ergodic theorem for the operator as well.…”

“…In [9], Hong and Sun showed that the weights considered by Bellow and Losert satisfy a Wiener-Wintner type result in a multiparameter form for certain τ -preserving * -automorphisms of finite von Neumann algebras. In [18], under a strong assumption on the positive Dunford-Schwartz operator under consideration (frequently satisfied for operators satisfying certain spectral theoretic properties, like being self-adjoint on L 2 ), the author was able to prove that all bounded Hartman sequences satisfy a Wiener-Wintner type result.…”