Noncommutative Maximal Ergodic Theorems

This paper is devoted to the study of various maximal ergodic theorems in noncommutative Lp-spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on Lp 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 theory of von Neumann algebras and in quantum probability.

[1]  O. Johnson.  Free Random Variables , 2004 .

[2]  Alexandre Nou.  Non injectivity of the q-deformed von Neumann algebra , 2004, math/0605789.

[3]  M. Junge,et al.  Maximal theorems of Menchoff–Rademacher type in non-commutative Lq-spaces , 2004 .

[4]  Éric Ricard.  Factoriality of q-Gaussian von Neumann Algebras , 2003, math/0311413.

[5]  Magdalena Musat.  Interpolation between non-commutative BMO and non-commutative Lp-spaces , 2003 .

[6]  V. Paulsen.  Completely Bounded Maps and Operator Algebras: Contents , 2003 .

[7]  M. Junge,et al.  Noncommutative Burkholder/Rosenthal inequalities , 2003 .

[8]  M. Junge.  Doob's inequality for non-commutative martingales , 2002, math/0206062.

[9]  N. Randrianantoanina.  Non-Commutative Martingale Transforms , 2001, math/0111264.

[10]  K. Dykema,et al.  Embeddings of reduced free products of operator algebras , 1999, math/9911012.

[11]  M. Bożejko.  ULTRACONTRACTIVITY AND STRONG SOBOLEV INEQUALITY FOR q-ORNSTEIN–UHLENBECK SEMIGROUP (-1 < q < 1) , 1999 .

[12]  G. Pisier,et al.  Non-Commutative Martingale Inequalities , 1997, math/9704209.

[13]  D. Shlyakhtenko.  FREE QUASI-FREE STATES , 1997 .

[14]  R. Speicher,et al.  Interpolations between bosonic and fermionic relations given by generalized brownian motions , 1996 .

[15]  R. Speicher,et al.  q-Gaussian Processes: Non-commutative and Classical Aspects , 1996, funct-an/9604010.

[16]  M. Choda.  Reduced free products of completely positive maps and entropy for free product of automorphisms , 1996 .

[17]  G. Pisier.  Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps , 1993, math/9306206.

[18]  K. Dykema.  Factoriality and Connes' invariant T(M) for free products of von Neumann algebras. , 1993, funct-an/9302005.

[19]  Eric Carlen,et al.  Optimal hypercontractivity for fermi fields and related non-commutative integration inequalities , 1992, hep-th/9209053.

[20]  Roland Speicher,et al.  An example of a generalized Brownian motion , 1991 .

[21]  Hideki Kosaki,et al.  Generalized s-numbers of τ-measurable operators , 1986 .

[22]  R. Jajte,et al.  Strong Limit Theorems in Non-Commutative Probability , 1985 .

[23]  H. Kosaki.  APPLICATIONS OF THE COMPLEX INTERPOLATION METHOD TO A VON NEUMANN ALGEBRA: NON-COMMUTATIVE LP-SPACES , 1984 .

[24]  N. Dang-Ngoc.  Pointwise convergence of martingales in von Neumann algebras , 1979 .

[25]  U. Haagerup.  An example of a non nuclearC*-algebra, which has the metric approximation property , 1978 .

[26]  B. Kümmerer.  A non-commutative individual ergodic theorem , 1978 .

[27]  J. Conze,et al.  Ergodic theorems for noncommutative dynamical systems , 1978 .

[28]  F. J. Yeadon.  Ergodic theorems for semifinite von Neumann algebras: II , 1977, Mathematical Proceedings of the Cambridge Philosophical Society.

[29]  Edward G. Effros,et al.  Injectivity and operator spaces , 1977 .

[30]  E. Christopher Lance,et al.  Ergodic theorems for convex sets and operator algebras , 1976 .

[31]  U. Haagerup.  Normal weights on W∗-algebras , 1975 .

[32]  I. Cuculescu.  Martingales on von Neumann algebras , 1971 .

[33]  Tord Holmstedt,et al.  Interpolation of Quasi-Normed Spaces. , 1970 .

[34]  E. Stein.  Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. , 1970 .

[35]  E. Stein,et al.  ON THE MAXIMAL ERGODIC THEOREM. , 1961, Proceedings of the National Academy of Sciences of the United States of America.

[36]  R. Kadison.  A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras , 1952 .

[37]  T. Mei.  Operator Valued Hardy Spaces , 2003 .

[38]  G. Pisier,et al.  Non-commutative L p spaces , 2003 .

[39]  Marius Junge,et al.  Théorèmes ergodiques maximaux dans les espaces Lp non commutatifsMaximal ergodic theorems in non-commutative Lp-spaces , 2002 .

[40]  K. Parthasarathy.  An Introduction to Quantum Stochastic Calculus , 1992 .

[41]  R. Jajte.  Strong Limit Theorems in Noncommutative L2-Spaces , 1991 .

[42]  M. Bożejko.  Positive-definite kernels, length functions on groups and a noncommutative von Neumann inequality , 1989 .

[43]  D. Voiculescu.  Symmetries of some reduced free product C*-algebras , 1985 .