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DTC ultrafilters on groups

Pachl Jan, Stepr?ns Juris
16 Oct 2019 math.GR math.GN arxiv.org/abs/1910.07505

We say that an ultrafilter on an infinite group $G$ is DTC if it determines the topological centre of the semigroup $\beta G$. We prove that DTC ultrafilters do not exist for virtually BFC groups, and do exist for the countable groups that are not virtually FC. In particular, an infinite finitely generated group is virtually abelian if and only if it does not admit a DTC ultrafilter.

Approximate fixed points and B-amenable groups

Pachl Jan
06 Nov 2017 math.GR math.FA arxiv.org/abs/1711.02171

A topological group $G$ is B-amenable if and only if every continuous affine action of $G$ on a bounded convex subset of a locally convex space has an approximate fixed point. Similar results hold more generally for slightly uniformly continuous semigroup actions.

Semipullbacks of labelled Markov processes

Pachl Jan,
08 Jun 2017 math.PR cs.LO arxiv.org/abs/1706.02801

A labelled Markov process (LMP) consists of a measurable space $S$ together with an indexed family of Markov kernels from $S$ to itself. This structure has been used to model probabilistic computations in Computer Science, and one of the main problems in the area is to define and decide whether two LMP $S$ and $S'$ "behave the same". There are two natural categorical definitions of sameness of behavior: $S$ and $S'$ are bisimilar if there exist an LMP $T$ and measure preserving maps forming a diagram of the shape $ S\leftarrow T \rightarrow{S'}$; and they are behaviorally equivalent if there exist some $ U$ and maps forming a dual diagram $ S\rightarrow U \leftarrow{S'}$. These two notions differ for general measurable spaces but Doberkat (extending a result by Edalat) proved that they coincide for analytic Borel spaces, showing that from every diagram $ S\rightarrow U \leftarrow{S'}$ one can obtain a bisimilarity diagram as above. Moreover, the resulting square of measure preserving maps is commutative (a "semipullback"). In this paper, we extend the previous result to measurable spaces $S$ isomorphic to a universally measurable subset of a Polish space with the trace of the Borel $\sigma$-algebra, using a version of Strassen's theorem on common extensions of finitely additive measures.

Proof of the Ghahramani-Lau conjecture

Losert Viktor, Neufang Matthias, Pachl Jan, Stepr?ns Juris
19 Oct 2015 math.FA math.GR arxiv.org/abs/1510.05412

The Ghahramani-Lau conjecture is established; in other words, the measure algebra of every locally compact group is strongly Arens irregular. To this end, we introduce and study certain new classes of measures (called approximately invariant, respectively, strongly singular) which are of interest in their own right. Moreover, we show that the same result holds for the measure algebra of any (not necessarily locally compact) Polish group.

Continuity of convolution and SIN groups

Pachl Jan, Stepr?ns Juris
27 Jul 2015 math.FA math.GR arxiv.org/abs/1507.07506

Let the measure algebra of a topological group be equipped with the topology of uniform convergence on bounded right uniformly equicontinuous sets of functions. Convolution is separately continuous on the measure algebra, and it is jointly continuous if and only if the group has the SIN property.

Minimal sets determining the topological centre of the algebra LUC(G)*

Ferri Stefano, Neufang Matthias, Pachl Jan
29 Oct 2013 math.FA arxiv.org/abs/1310.7931

The Banach algebra LUC(G)* associated to a topological group G has been of interest in abstract harmonic analysis. A number of authors have studied the topological centre of LUC(G)*, which is defined as the set of elements in LUC(G)* for which the left multiplication is w--w-continuous on LUC(G)*. Several recent works show that for a locally compact group G it is sufficient to test the continuity of the left multiplication at just one specific point in order to determine whether an element of LUC(G)* belongs to the topological centre. In this work we extend some of these results to a much larger class of groups which includes many non-locally compact groups as well as all the locally compact ones. This answers a question raised by H.G. Dales. We also obtain a corollary about the topological centre of any subsemigroup of LUC(G)* containing the uniform compactification of G. In particular, we prove that there are sets of just one point determining the topological centre of the uniform compactification itself.

Cauchy filters from Pelant's games

Pachl Jan
07 Oct 2013 math.GN math.FA arxiv.org/abs/1310.1808

The language of finite games is used to rephrase Pelant's proof of his result: The separable modification of the complete metric space $C([0,\omega_1])$ is not complete.

Uniformly equicontinuous sets, right multiplier topology, and continuity of convolution

Neufang Matthias, Pachl Jan, Salmi Pekka
20 Feb 2012 math.FA arxiv.org/abs/1202.4350

The dual space of the C*-algebra of bounded uniformly continuous functions on a uniform space carries several natural topologies. One of these is the topology of uniform convergence on bounded uniformly equicontinuous sets, or the UEB topology for short. In the particular case of a topological group and its right uniformity, the UEB topology plays a significant role in the continuity of convolution. In this paper we derive a useful characterisation of bounded uniformly equicontinuous sets on locally compact groups. Then we demonstrate that for every locally compact group G the UEB topology on the space of finite Radon measures on G coincides with the right multiplier topology. In this sense the UEB topology is a generalisation to arbitrary topological groups of the multiplier topology for locally compact groups. In the final section we prove results about UEB continuity of convolution.

Measurable centres in convolution semigroups

Pachl Jan
19 Jul 2011 math.FA arxiv.org/abs/1107.3799

In a convolution semigroup over a locally compact group, measurability of the translation by a fixed element implies continuity. In other words, the measurable centre coincides with the topological centre.

Weak uniform structures on probability distributions

Pachl Jan
20 Jul 2010 math.PR math.FA arxiv.org/abs/1007.3433

In dealing with asymptotic approximation of possibly divergent nets of probability distributions, we are led to study uniform structures on the set of distributions. This paper identifies a class of such uniform structures that may be considered to be reasonable generalizations of the weak topology. It is shown that all structures in the class yield the same notion of asymptotic approximation for sequences (but not for general nets) of probability distributions.