 
Summary: Pointwise Ergodic Theorems for Group Representations in L
by Arkady Tempelman
Abstract
Pointwise, maximal and dominated ergodic theorem with "weighted" averages
for Lamperti representations of amenable compact locally compact groups in L
,
(1 < < ), over a finite measure space (, F, m) will be considered; in particular,
these theorems imply ergodic theorems for positive group representations and group
actions; Cesaro averages represent a special case when the "weights" are nonzero
constants on integrable sets. We discuss various conditions on the "weights" and
their supports, under which these theorems hold. It is wellknown that the pointwise
ergodic theorem is not valid, in general, for Cesaro averages of powers of invertible
positive operators in L1
so our restriction to the case > 1 is quite natural (in the
case when m() < , the pointwise ergodic theorem for "tempered" Cesaro averages
of representations in L1
(, F, m), induced by measure preserving group actions, has
been proved by a different method by E. Lindenstrauss).
The results generalize, in several directions, some of the results contained in the
Ph.D. thesis by A. Shulman (Vilnius, 1988), in Shulman (1988) and in Tempelman
