Another ambitable group
It is an open question whether every topological group is precompact or ambitable. This note presents a topological group that is ambitable but does not satisfy previously known sufficient conditions for being so.
math.GR↗
arXiv subjects
Jan Pachl
Publications and source records attributed to Jan Pachl.
An embedding theorem for uniform measures on topological groups
The embedding theorem of Roelcke and Dierolf for the completions of four standard uniform structures on topological groups and their quotients holds more generally for spaces of uniform measures. The natural mappings between the four spaces of uniform measures on a topological group are injective and their restrictions to positive cones are topological embeddings. The same holds for spaces of uniform measures on the quotient of a topological group by a neutral subgroup.
math.GR↗
DTC ultrafilters on groups
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.
math.GR↗
Approximate fixed points and B-amenable groups
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.
math.GR↗
Semipullbacks of labelled Markov processes
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.
math.PR↗
Proof of the Ghahramani-Lau conjecture
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.
math.FA↗
Continuity of convolution and SIN groups
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.
math.FA↗
Minimal sets determining the topological centre of the algebra LUC(G)*
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.
math.FA↗
Cauchy filters from Pelant's games
The language of finite games is used to rephrase Pelant's proof of his result: The separable modification of the complete metric space $C([0,ω_1])$ is not complete.
math.GN↗
Reachability problems for communicating finite state machines
The paper deals with the verification of reachability properties in a commonly used state transition model of communication protocols, which consists of finite state machines connected by potentially unbounded FIFO channels. Although simple reachability problems are undecidable for general protocols with unbounded channels, they are decidable for the protocols with the recognizable channel property. The decidability question is open for the protocols with the rational channel property.
cs.LO↗
Uniformly equicontinuous sets, right multiplier topology, and continuity of convolution
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.
math.FA↗
Measurable centres in convolution semigroups
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.
math.FA↗
Weak uniform structures on probability distributions
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.
math.PR↗
Uniform measures and convolution on topological groups
Uniform measures are the functionals on the space of bounded uniformly continuous functions that are continuous on every bounded uniformly equicontinuous set. This paper describes the role of uniform measures in the study of convolution on an arbitrary topological group.
math.FA↗
Ambitable topological groups
A topological group is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the group with its right uniformity is contained in an ambit. For n=0,1,2,..., every locally aleph_n bounded topological group is either precompact or ambitable. In the familiar semigroups constructed over ambitable groups, topological centres have an effective characterization.
math.FA↗
Semiuniform semigroups and convolution
Semiuniform semigroups provide a natural setting for the convolution of generalized finite measures on semigroups. A semiuniform semigroup is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the semigroup is contained in an ambit. In the convolution algebras constructed over ambitable semigroups, topological centres have a tractable characterization.
math.FA↗
Uniform measures and countably additive measures
Uniform measures are defined as the functionals on the space of bounded uniformly continuous functions that are continuous on bounded uniformly equicontinuous sets. If every cardinal has measure zero then every countably additive measure is a uniform measure. The functionals sequentially continuous on bounded uniformly equicontinuous sets are exactly uniform measures on the separable modification of the underlying uniform space.
math.FA↗
The sum of two measurable functions
The paper describes two Borel-measurable functions from a measure space into a locally convex space such that the image measure for each function is Radon but their sum is not Borel-measurable.
math.FA↗