Positivity (2013) 17:27–46
Ergodic averages with vector-valued Besicovitch weights
Do˘gan Çömez · Semyon Litvinov
Received: 4 May 2011 / Accepted: 16 September 2011 / Published online: 14 October 2011
© Springer Basel AG 2011
Abstract For a von Neumann algebra M, we introduce M-valued Besicovitch
sequences and study the norm and individual convergences of the corresponding
weighted ergodic averages. The limits of the averages are examined under the condi-
tion that the contraction in question is weakly mixing.
Keywords Semiﬁnite von Neumann algebra · Non-commutative ergodic theorem ·
Operator-valued Besicovitch weights
Mathematics Subject Classiﬁcation (2000) Primary 46L50; Secondary 47A35
Non-commutative ergodic averages weighted by means of numerical Besicovitch
sequences were ﬁrst studied in , where the quasi-uniform convergence of such
averages was established for a von Neumann algebra with a faithful normal state.
In , vector-valued Besicovitch sequences were introduced and the quasi-uniform
convergence of the corresponding averages was proved.
Note that in  and in , the convergence was established for operators belonging
to the von Neumann algebra itself; nothing was said about convergence for integrable
operators. The point is that, in the commutative case, in order to prove convergence
North Dakota State University, Fargo, ND 58105, USA
S. Litvinov (
Pennsylvania State University, Hazleton, PA 18202, USA