Non Uniqueness of Invariant Means for Amenable Group Actions.
Poincaré’s classical recurrence theorem is generalised to the noncommutative setup where a measure space with a measure-preserving transformation is replaced by a von Neumann algebra with a weight and a Jordan morphism leaving the weight invariant. This is done by a suitable reformulation of the theorem in the language of -space rather than the original measure space, thus allowing the replacement of the commutative von Neumann algebra by a noncommutative one.
We study distance measures for lattice-generated sets in Rd, d>=3, with respect to non-isotropic distances l-l.K, induced by smooth symmetric convex bodies K. An effective Fourier-analytic approach is developed to get sharp upper bounds for the second moment of the weighted distance measure.
This paper is devoted to the following question. Suppose that a Polish group G has the property that some non-empty open subset U is covered by finitely many two-sided translates of every other non-empty open subset of G. Is then G necessarily locally compact? Polish groups which do not have the above property are called strongly non-locally compact. We characterize strongly non-locally compact Polish subgroups of in terms of group actions, and prove that certain natural classes of non-locally...
We present a theorem which generalizes some known theorems on the existence of nonmeasurable (in various senses) sets of the form X+Y. Some additional related questions concerning measure, category and the algebra of Borel sets are also studied.