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Noncommutative Poincaré recurrence theorem

Andrzej Łuczak (2001)

Colloquium Mathematicae

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 L -space rather than the original measure space, thus allowing the replacement of the commutative von Neumann algebra L by a noncommutative one.

Non-isotropic distance measures for lattice-generated sets.

Alexander Iosevich, Misha Rudnev (2005)

Publicacions Matemàtiques

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.

Non-locally compact Polish groups and two-sided translates of open sets

Maciej Malicki (2008)

Fundamenta Mathematicae

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 S in terms of group actions, and prove that certain natural classes of non-locally...

Nonmeasurable algebraic sums of sets of reals

Marcin Kysiak (2005)

Colloquium Mathematicae

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.

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