Uniform equicontinuity of sequences of measurable operators and noncommutative ergodic theorems
Author:
Semyon Litvinov
Journal:
Proc. Amer. Math. Soc. 140 (2012), 24012409
MSC (2010):
Primary 46L51; Secondary 47A35
Published electronically:
December 7, 2011
MathSciNet review:
2898702
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
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 noncommutative ergodic theorems.
 1.
O. Bratelli, D.N. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer, Berlin, 1979.
 2.
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
(2007b:46112)
 3.
Vladimir
Chilin, Semyon
Litvinov, and Adam
Skalski, A few remarks in noncommutative ergodic theory, J.
Operator Theory 53 (2005), no. 2, 331–350. MR 2153152
(2006c:46053)
 4.
Adriano
M. Garsia, Topics in almost everywhere convergence, Lectures
in Advanced Mathematics, vol. 4, Markham Publishing Co., Chicago,
Ill., 1970. MR
0261253 (41 #5869)
 5.
Michael
Goldstein and Semyon
Litvinov, Banach principle in the space of 𝜏measurable
operators, Studia Math. 143 (2000), no. 1,
33–41. MR
1814479 (2002b:46107)
 6.
Semyon
Litvinov, The Banach principle for topological groups, Atti
Semin. Mat. Fis. Univ. Modena Reggio Emilia 53 (2005),
no. 2, 323–330. MR 2289403
(2007j:22004)
 7.
Miguel
de Guzmán, Real variable methods in Fourier analysis,
NorthHolland Mathematics Studies, vol. 46, NorthHolland Publishing
Co., AmsterdamNew York, 1981. Notas de Matemática [Mathematical
Notes], 75. MR
596037 (83j:42019)
 8.
Marius
Junge and Quanhua
Xu, Noncommutative maximal ergodic
theorems, J. Amer. Math. Soc.
20 (2007), no. 2,
385–439. MR 2276775
(2007k:46109), 10.1090/S0894034706005339
 9.
Richard
V. Kadison, A generalized Schwarz inequality and algebraic
invariants for operator algebras, Ann. of Math. (2)
56 (1952), 494–503. MR 0051442
(14,481c)
 10.
I.
E. Segal, A noncommutative extension of abstract integration,
Ann. of Math. (2) 57 (1953), 401–457. MR 0054864
(14,991f)
 11.
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
(88h:46120)
 12.
F.
J. Yeadon, Ergodic theorems for semifinite von Neumann algebras.
I, J. London Math. Soc. (2) 16 (1977), no. 2,
326–332. MR 0487482
(58 #7111)
 1.
 O. Bratelli, D.N. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer, Berlin, 1979.
 2.
 V. Chilin, S. Litvinov, Uniform equicontinuity for sequences of homomorphisms into the ring of measurable operators, Methods of Funct. Anal. Top., 12 (2)(2006), 124130. MR 2238034 (2007b:46112)
 3.
 V. Chilin, S. Litvinov, A. Skalski, A few remarks in noncommutative ergodic theory,
J. Operator Theory, 53 (2)(2005), 331350. MR 2153152 (2006c:46053)
 4.
 A. Garsia, Topics in Almost Everywhere Convergence, Markham: Lectures in Advanced Mathematics, 4, 1970. MR 0261253 (41:5869)
 5.
 M. Goldstein, S. Litvinov, Banach principle in the space of measurable operators, Studia Math., 143(2000), 3341. MR 1814479 (2002b:46107)
 6.
 S. Litvinov, The Banach principle for topological groups, Atti Sem. Mat. Fis. Univ. Modena e Reggio Emilia, 53 (2005), 323330. MR 2289403 (2007j:22004)
 7.
 M. de Guzman, Real Variable Methods in Fourier Analysis, NorthHolland: Math. Studies, 46, 1981. MR 596037 (83j:42019)
 8.
 M. Junge, Q. Xu, Noncommutative maximal ergodic theorems, J. Amer. Math. Soc., 20 (2)(2007), 385439. MR 2276775 (2007k:46109)
 9.
 R. V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2), 56(1952), 494503. MR 0051442 (14:481c)
 10.
 I. Segal, A noncommutative extension of abstract integration, Ann. of Math. (2), 57(1953), 401457. MR 0054864 (14:991f)
 11.
 O.E. Tichonov, Continuity of operator functions in topologies connected with a trace on a von Neumann algebra [Russian], Izv. Vyssh. Uchebn. Zaved. Mat., 1(1987), 7779. MR 892008 (88h:46120)
 12.
 F. J. Yeadon, Ergodic theorems for semifinite von Neumann algebras. I, J. London Math. Soc., 16 (2)(1977), 326332. MR 0487482 (58:7111)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
46L51,
47A35
Retrieve articles in all journals
with MSC (2010):
46L51,
47A35
Additional Information
Semyon Litvinov
Affiliation:
Department of Mathematics, Pennsylvania State University, 76 University Drive, Hazleton, Pennsylvania 18202
Email:
snl2@psu.edu
DOI:
http://dx.doi.org/10.1090/S000299392011114837
Keywords:
Semifinite von Neumann algebra,
uniform equicontinuity,
noncommu tative ergodic theorem
Received by editor(s):
February 20, 2011
Published electronically:
December 7, 2011
Communicated by:
Marius Junge
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
