Noncommutative maximal ergodic theorems
By Marius Junge, Quanhua Xu. 2005
223 citations recorded by the source corpus.
This paper is devoted to the study of various maximal ergodic theorems in noncommutative -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on and the analogue of Stein's maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.
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Citations in the corpus
- Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori (2015, background): “[17] was made possible by the recent developments on noncommutative martingale/ergodic inequalities and the Littlewood-Paley-Stein theory for quantum Markovian semigroups, which had been achieved thanks to the efforts of many researchers; see, for instance, [56, 31, 36, 37, 61, 62, 52], and [32, 44,”
- Noncommutative Riesz transforms -- a probabilistic approach (2008, unknown): “It has been shown in [JX07] that Ps is a positive average of the Mt and hence (1.”
- A reduction method for noncommutative L_p-spaces and applications (2008, methodology): “The preceding theorem is proved in [JX4].”
- Pseudo-Localization of Singular Integrals and Noncommutative Littlewood–Paley Inequalities (2008, methodology): “It is known [10, 17] that square and maximal functions are bounded maps on Lp.”
- On ergodic theorems for free group actions on noncommutative spaces (2004, background): “Junge and Xu have generalized in [20] both the individual Dunford-Schwartz ergodic theorem and the classical Stein ergodic theorem proved in [40].”
- Noncommutative martingale deviation and Poincaré type inequalities with applications (2012, background): “Then it was shown in [35] that Fix is a von Neumann subalgebra and Fix⊥ = L∞(N ).”
- BMO spaces associated with semigroups of operators (2011, methodology): “∞ 0 λisΓ(1−is)(1 +is)−1ds = e−λ− Z1 0 e−uλdu, we deduce by functional calculus that 1 π Z+∞ 0 (tL)isΓ(1 −is)(1 +is)−1ds = e−tL− Z1 0 e−utLdu = T t− 1 t Zt 0 (5.5) Tvdv. Theorem 4.1 and Theorem 4.5 of [JX07] imply that ksup t 1 t Zt 0 TvfdvkL p(N) ≤ cmax{p, 1 (p −1)2 }kfkL (5.6) p(N). On the other ha”
- Noncommutative Boyd interpolation theorems (2012, background): “If T moreover satisfies (d) τ(T (y)x) = τ(yT (x)) for all x, y ∈ L(2)(M) ∩M, then, as was observed in [19], Theorem 5.”
- Noncommutative maximal ergodic inequalities associated with doubling conditions (2017, methodology): “] obtained a maximal ergodic theorem in the preduals of semifinite von Neumann algebras. But the corresponding maximal inequalities in Lp-spaces remained open until the celebrated work of Junge and Xu [JX07], which established the noncommutative analogue of the Dunford-Schwartz maximal ergodic theore”
- Smooth Fourier multipliers on group von Neumann algebras (2010, background): “Reciprocally, kerA∞ is a ∗-subalgebra of M by [37].”
- $H^{\infty }$ functional calculus and square functions on noncommutative $L^p$-spaces (2006, background): “The noncommutative maximal ergodic theorem in [36] is our starting point for the study of noncommutative diffusion semigroups.”
- Noncommutative Burkholder/Rosenthal inequalities II: Applications (2007, methodology): “We refer to [J1] for more information (see also [JX3]).”