Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · funct-an/9705003

Some universal constructions in abstract topological dynamics

Vladimir Pestov
View original source ↗Read source document ↗Export metadata ↓

Abstract

This small survey of basic universal constructions related to the actions of topological groups on compacta is centred around a new result --- an intrinsic description of extremely amenable topological groups (i.e., those having a fixed point in each compactum they act upon), solving a 1967 problem by Granirer. Another old problem whose solution (in the negative) is noted here, is a 1957 conjecture by Teleman on irreducible representations of general topological groups. Our exposition covers the greatest ambit and the universal minimal flow, as well as closely related constructions and results from representation theory. We present in a simplified fashion the known examples of extremely amenable groups and discuss their relationship with a 1969 problem by Ellis. Open questions, including those from bordering disciplines, are reviewed along the way.

Explore related subjects

funct-anmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vladimir Pestov. 1997-05-22. Some universal constructions in abstract topological dynamics. https://arxiv.org/abs/funct-an/9705003

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2020

Isomorphism classes for quantum Heisenberg manifolds

We embed the quantum Heisenberg manifold in a crossed product algebra. This enables us to show that, in the irrational case, all tracial states on $\dc$ induce the same homomorphism on the K_0-group. We conclude that two irrational quantum Heisenberg manifolds $\dc$ and $D^c_{μ' ν'}$ are isomorphic if and only if the parameters $(μ,ν)$ and $(μ',ν')$ belong to the same orbit under the usual action of $GL_2(\ZZ)$ on the torus.

funct-an↗
arXiv2007

Quantum Mechanics and Operator algebras on the Hilbert ball

Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C$^{*}$-algebra is represented as a function algebra on the set of pure states with a noncommutative $*$-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$. The space ${\cal P}({\cal H})$ is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature $c$. The Hilbert ball $B_{\cal H}$ is defined by the open unit ball in ${\cal H}$ and it is a Kähler manifold with $c<0$. We introduce the quantum mechanics on $B_{\cal H}$. As an application, we show the structure of the noncommutative function algebra on $B_{\cal H}$.

funct-an↗
arXiv2004

No More Than Mechanics. I

One can introduce so-called {\em Plain Mechanics} having an {\bf operator realization}. Then the set of one-dimension representations of this operator realization may be identified with the Classical Mechanics. Different irreducible infinite-dimension representations may be recognized as Quantum Mechanics for different $\hbar$ (the Planck constant). It can be done in the such manner that the following diagram will be commutative. Plain Mechanics / \ / \ / \ Quantum Mechanics --> Classical Mechanics h->0 Here the horizontal arrow is well known correspondence between Quantum and Classical Mechanics if Planck constant tensing to zero. A {\em realization} of this scheme for a particle in $n$-dimensional space by two-sided convolutions on the Heisenberg group is constructed. We also introduce the {\em motion equations} for observables in this realization. The left arrow of the given diagram carries this equation to the Heisenberg one and the right arrow maps it to the Hamilton equation.

funct-an↗