Uniform equicontinuity of sequences of measurable operators and non-commutative ergodic theorems

Litvinov, Semyon

doi:10.1090/S0002-9939-2011-11483-7

Uniform equicontinuity of sequences of measurable operators and non-commutative ergodic theorems
HTML articles powered by AMS MathViewer

by Semyon Litvinov PDF

Proc. Amer. Math. Soc. 140 (2012), 2401-2409 Request permission

Abstract:

The notion of uniform equicontinuity in measure at zero for sequences of additive maps from a normed space into the space of measurable operators associated with a semifinite von Neumann algebra is discussed. It is shown that uniform equicontinuity in measure at zero on a dense subset implies the uniform equicontinuity in measure at zero on the entire space, which is then applied to derive some non-commutative ergodic theorems.

References

O. Bratelli, D.N. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer, Berlin, 1979.
V. I. Chilin and S. N. Litvinov, Uniform equicontinuity for sequences of homomorphisms into the ring of measurable operators, Methods Funct. Anal. Topology 12 (2006), no. 2, 124–130. MR 2238034
Vladimir Chilin, Semyon Litvinov, and Adam Skalski, A few remarks in non-commutative ergodic theory, J. Operator Theory 53 (2005), no. 2, 331–350. MR 2153152
Adriano M. Garsia, Topics in almost everywhere convergence, Lectures in Advanced Mathematics, No. 4, Markham Publishing Co., Chicago, Ill., 1970. MR 0261253
Michael Goldstein and Semyon Litvinov, Banach principle in the space of $\tau$-measurable operators, Studia Math. 143 (2000), no. 1, 33–41. MR 1814479, DOI 10.4064/sm-143-1-33-41
Semyon Litvinov, The Banach principle for topological groups, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 53 (2005), no. 2, 323–330. MR 2289403
Miguel de Guzmán, Real variable methods in Fourier analysis, Notas de Matemática [Mathematical Notes], vol. 75, North-Holland Publishing Co., Amsterdam-New York, 1981. MR 596037
Marius Junge and Quanhua Xu, Noncommutative maximal ergodic theorems, J. Amer. Math. Soc. 20 (2007), no. 2, 385–439. MR 2276775, DOI 10.1090/S0894-0347-06-00533-9
Richard V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2) 56 (1952), 494–503. MR 51442, DOI 10.2307/1969657
I. E. Segal, A non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401–457. MR 54864, DOI 10.2307/1969729
O. E. Tikhonov, Continuity of operator functions in topologies connected with a trace on a von Neumann algebra, Izv. Vyssh. Uchebn. Zaved. Mat. 1 (1987), 77–79 (Russian). MR 892008
F. J. Yeadon, Ergodic theorems for semifinite von Neumann algebras. I, J. London Math. Soc. (2) 16 (1977), no. 2, 326–332. MR 487482, DOI 10.1112/jlms/s2-16.2.326