1,041 research outputs found
The dynamical hierarchy for Roelcke precompact Polish groups
We study several distinguished function algebras on a Polish group , under
the assumption that is Roelcke precompact. We do this by means of the
model-theoretic translation initiated by Ben Yaacov and Tsankov: we investigate
the dynamics of -categorical metric structures under the action of
their automorphism group. We show that, in this context, every strongly
uniformly continuous function (in particular, every Asplund function) is weakly
almost periodic. We also point out the correspondence between tame functions
and NIP formulas, deducing that the isometry group of the Urysohn sphere is
\Tame\cap\UC-trivial.Comment: 25 page
Logic and operator algebras
The most recent wave of applications of logic to operator algebras is a young
and rapidly developing field. This is a snapshot of the current state of the
art.Comment: A minor chang
Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups
We investigate the automorphism groups of -categorical structures
and prove that they are exactly the Roelcke precompact Polish groups. We show
that the theory of a structure is stable if and only if every Roelcke uniformly
continuous function on the automorphism group is weakly almost periodic.
Analysing the semigroup structure on the weakly almost periodic
compactification, we show that continuous surjective homomorphisms from
automorphism groups of stable -categorical structures to Hausdorff
topological groups are open. We also produce some new WAP-trivial groups and
calculate the WAP compactification in a number of examples
Canonical Effective Subalgebras of Classical Algebras as Constructive Metric Completions
We prove general theorems about unique existence of effective subalgebras of classical algebras. The theorems are consequences of standard facts about completions of metric spaces within the framework of constructive mathematics, suitably interpreted in realizability models. We work with general realizability models rather than with a particular model of computation. Consequently, all the results are applicable in various established schools of computability, such as type 1 and type 2 effectivity, domain representations, equilogical spaces, and others
On Roeckle-precompact Polish group which cannot act transitively on a complete metric space
We study when a continuous isometric action of a Polish group on a complete
metric space is, or can be, transitive. Our main results consist of showing
that certain Polish groups, namely and
, such an action can never be transitive (unless the
space acted upon is a singleton). We also point out "circumstantial evidence"
that this pathology could be related to that of Polish groups which are not
closed permutation groups and yet have discrete uniform distance, and give a
general characterisation of continuous isometric action of a Roeckle-precompact
Polish group on a complete metric space is transitive. It follows that the
morphism from a Roeckle-precompact Polish group to its Bohr compactification is
surjective
Automorphism groups of randomized structures
We study automorphism groups of randomizations of separable structures, with
focus on the -categorical case. We give a description of the
automorphism group of the Borel randomization in terms of the group of the
original structure. In the -categorical context, this provides a new
source of Roelcke precompact Polish groups, and we describe the associated
Roelcke compactifications. This allows us also to recover and generalize
preservation results of stable and NIP formulas previously established in the
literature, via a Banach-theoretic translation. Finally, we study and classify
the separable models of the theory of beautiful pairs of randomizations,
showing in particular that this theory is never -categorical (except
in basic cases).Comment: 28 page
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