214 research outputs found
Additive structure of difference sets and a theorem of Følner
A theorem of Folner asserts that for any set A subset of Z of positive upper density there is a Bohr neigbourhood B of 0 such that B \ (Lambda - Lambda) has zero density. We use this result to deduce some consequences about the structure of difference sets of sets of integers having a positive upper density
Existence of equilibria in countable games: an algebraic approach
Although mixed extensions of finite games always admit equilibria, this is
not the case for countable games, the best-known example being Wald's
pick-the-larger-integer game. Several authors have provided conditions for the
existence of equilibria in infinite games. These conditions are typically of
topological nature and are rarely applicable to countable games. Here we
establish an existence result for the equilibrium of countable games when the
strategy sets are a countable group and the payoffs are functions of the group
operation. In order to obtain the existence of equilibria, finitely additive
mixed strategies have to be allowed. This creates a problem of selection of a
product measure of mixed strategies. We propose a family of such selections and
prove existence of an equilibrium that does not depend on the selection. As a
byproduct we show that if finitely additive mixed strategies are allowed, then
Wald's game admits an equilibrium. We also prove existence of equilibria for
nontrivial extensions of matching-pennies and rock-scissors-paper. Finally we
extend the main results to uncountable games
Sub-additive ergodic theorems for countable amenable groups
In this paper we generalize Kingman's sub-additive ergodic theorem to a large
class of infinite countable discrete amenable group actions.Comment: Journal of Functional Analysi
Amenability and paradoxicality in semigroups and C*-algebras
We analyze the dichotomy amenable/paradoxical in the context of (discrete,
countable, unital) semigroups and corresponding semigroup rings. We consider
also F{\o}lner's type characterizations of amenability and give an example of a
semigroup whose semigroup ring is algebraically amenable but has no F{\o}lner
sequence.
In the context of inverse semigroups we give a characterization of
invariant measures on (in the sense of Day) in terms of two notions:
and . Given a unital representation of
in terms of partial bijections on some set we define a natural
generalization of the uniform Roe algebra of a group, which we denote by
. We show that the following notions are then equivalent: (1)
is domain measurable; (2) is not paradoxical; (3) satisfies the
domain F{\o}lner condition; (4) there is an algebraically amenable dense
*-subalgebra of ; (5) has an amenable trace; (6)
is not properly infinite and (7) in the
-group of . We also show that any tracial state on
is amenable. Moreover, taking into account the localization
condition, we give several C*-algebraic characterizations of the amenability of
. Finally, we show that for a certain class of inverse semigroups, the
quasidiagonality of implies the amenability of . The
converse implication is false.Comment: 29 pages, minor corrections. Mistake in the statement of Proposition
4.19 from previous version corrected. Final version to appear in Journal of
Functional Analysi
Pl\"unnecke inequalities for measure graphs with applications
We generalize Petridis's new proof of Pl\"unnecke's graph inequality to
graphs whose vertex set is a measure space. Consequently, this gives new
Pl\"unnecke inequalities for measure preserving actions which enable us to
deduce, via a Furstenberg correspondence principle, Banach density estimates in
countable abelian groups that improve on those given by Jin.Comment: 24 pages, 1 figur
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