Bohr Compactification Research Articles

ON MODEL-THEORETIC CONNECTED GROUPS

AbstractWe introduce and study the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with the trivial Bohr compactification, that is, the class of minimally almost periodic groups.

On Bohr compactifications and profinite completions of group extensions

AbstractLet $G= N\rtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H. The Bohr compactification ${\rm Bohr}(G)$ and the profinite completion ${\rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 \rtimes {\rm Bohr}(H)$ and $Q_2 \rtimes {\rm Prof}(H)$ for appropriate quotients $Q_1$ of ${\rm Bohr}(N)$ and $Q_2$ of ${\rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of H on appropriate subsets of the dual space of N. In the case where N is abelian, we have ${\rm Bohr}(G)\cong A \rtimes {\rm Bohr}(H)$ and ${\rm Prof}(G)\cong B \rtimes {\rm Prof}(H),$ where A (respectively B) is the dual group of the group of unitary characters of N with finite H-orbits (respectively with finite image). Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where $G= \Lambda\wr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${\rm Bohr}(\Lambda\wr H)$ is isomorphic to ${\rm Bohr}(\Lambda^{\rm Ab}\wr H)$ and ${\rm Prof}(\Lambda\wr H)$ is isomorphic to ${\rm Prof}(\Lambda^{\rm Ab} \wr H),$ where $\Lambda^{\rm Ab}=\Lambda/ [\Lambda, \Lambda]$ is the abelianisation of $\Lambda.$ As examples, we compute ${\rm Bohr}(G)$ and ${\rm Prof}(G)$ when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring.

Star product approach for loop quantum cosmology

Guided by recent developments toward the implementation of the deformation quantization program within the loop quantum cosmology (LQC) formalism, in this paper we address the introduction of both the integral and differential representation of the star product for LQC. To this end, we consider the Weyl quantization map for cylindrical functions defined on the Bohr compactification of the reals. The integral representation contains all of the common properties that characterize a star product which, in the case under study here, stands for a deformation of the usual pointwise product of cylindrical functions. Our construction also admits a direct comparison with the integral representation of the Moyal product which may be reproduced from our formulation by judiciously substituting the appropriate characters that identify such representation. Further, we introduce a suitable star commutator that correctly reproduces both the quantum representation of the holonomy-flux algebra for LQC and, in the proper limit, the holonomy-flux classical Poisson algebra emerging in the cosmological setup. Finally, we propose a natural way to obtain the quantum dynamical evolution in LQC in terms of this star commutator for cylindrical functions as well as a differential representation of the star product using discrete finite differences. We expect that our findings may contribute to a better understanding of certain issues arising within the LQC program, in particular, those related to the semiclassical limit and the dynamical evolution of quantum states.

A note on Pontryagin duality and continuous logic

We exhibit Pontryagin duality as a special case of Stone duality in a continuous logic setting. More specifically, given an abelian topological group A, and F the family (group) of continuous homomorphisms from A to the circle group T, then, viewing (A,+) equipped with the collection F as a continuous logic structure M, we show that the local type space SF(M) is precisely the Pontryagin dual of the group F where the latter is considered as a discrete group.We conclude, using Pontryagin duality (between compact and discrete abelian groups), that SF(M) is the Bohr compactification of the topological group A.

Generating ideals by additive subgroups of rings

We obtain several fundamental results on finite index ideals and additive subgroups of rings as well as on model-theoretic connected components of rings, which concern generating in finitely many steps inside additive groups of rings.Let R be any ring equipped with an arbitrary additional first order structure, and A a set of parameters. We show that whenever H is an A-definable, finite index subgroup of (R,+), then H+R⋅H contains an A-definable, two-sided ideal of finite index. As a corollary, we obtain a positive answer to Question 4.9 of [4]: if R is unital, then (R¯,+)A00+R¯⋅(R¯,+)A00+R¯⋅(R¯,+)A00=R¯A00, which also implies that R¯A00=R¯A000, where R¯≻R is a sufficiently saturated elementary extension of R, (R¯,+)A00 [resp. R¯A00] denotes the smallest A-type-definable, bounded index additive subgroup [resp. ideal] of R¯, and R¯A000 is the smallest invariant over A, bounded index ideal of R¯. If R is of positive characteristic (not necessarily unital), we get a sharper result: (R¯,+)A00+R¯⋅(R¯,+)A00=R¯A00. We obtain a similar result (but with more steps required) for finitely generated (not necessarily unital) rings. We obtain analogous results for topological rings. The above result for unital rings implies that the simplified descriptions of the definable (and so also classical) Bohr compactifications of triangular groups over unital rings obtained in Corollary 4.5 of [4] are valid for all unital rings.We also analyze many concrete examples, where we compute the number of steps needed to generate a group by (R¯∪{1})⋅(R¯,+)A00 and study related aspects, showing “optimality” of some of our main results and yielding answers to some natural questions.

BOHR COMPACTIFICATIONS OF GROUPS AND RINGS

AbstractWe introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group ${\mathrm {UT}}_3({\mathbb {Z}})$ , the continuous Heisenberg group ${\mathrm {UT}}_3({\mathbb {R}})$ , and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.

Amenability, connected components, and definable actions

We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version (Massicot and Wagner in J Ec Polytech Math 2:55–63, 2015) of the stabilizer theorem (Hrushovski in J Am Math Soc 25:189–243, 2012), and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain “weak Bohr compactification” introduced in Krupiński and Pillay (Adv Math 345:1253–1299, 2019). In other words, the conclusion says that certain connected components of G coincide: G^{00}_{{{,mathrm{{top}},}}} = G^{000}_{{{,mathrm{{top}},}}}. We also prove wide generalizations of this result, implying in particular its extension to a “definable-topological” context, confirming the main conjectures from Krupiński and Pillay (2019). We also introduce bigvee -definable group topologies on a given emptyset -definable group G (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of G implies (under some assumption) that {{,mathrm{{cl}},}}(G^{00}_M) = {{,mathrm{{cl}},}}(G^{000}_M) for any model M. Secondly, we study the relationship between (separate) definability of an action of a definable group on a compact space [in the sense of Gismatullin et al. (Ann Pure Appl Log 165:552–562, 2014)], weakly almost periodic (wap) actions of G [in the sense of Ellis and Nerurkar (Trans Am Math Soc 313:103–119, 1989)], and stability. We conclude that any group G definable in a sufficiently saturated structure is “weakly definably amenable” in the sense of Krupiński and Pillay (2019), namely any definable action of G on a compact space supports a G-invariant probability measure. This gives negative solutions to some questions and conjectures raised in Krupiński (J Symb Log 82:1080–1105, 2017) and Krupiński and Pillay (2019). Stability in continuous logic will play a role in some proofs in this part of the paper. Thirdly, we give an example of a emptyset -definable approximate subgroup X in a saturated extension of the group {{mathbb {F}}}_2 times {{mathbb {Z}}} in a suitable language (where {{mathbb {F}}}_2 is the free group in 2-generators) for which the bigvee -definable group H:=langle X rangle contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) “model” exists for each approximate subgroup does not work in general (they proved in (Massicot and Wagner 2015) that it works for definably amenable approximate subgroups).

Additive conjugacy and the Bohr compactification of orthogonal representations

We say that two unitary or orthogonal representations of a finitely generated group G are additive conjugates if they are intertwined by an additive map, which need not be continuous. We associate to each representation of G a topological action that is a complete additive conjugacy invariant: the action of G by group automorphisms on the Bohr compactification of the underlying Hilbert space. Using this construction we show that the property of having almost invariant vectors is an additive conjugacy invariant. As an application we show that G is amenable if and only if there is a nonzero homomorphism from \(L^2(G)\) into \(\mathbb {R}/\mathbb {Z}\) that is invariant to the G-action.

Topological groups without infinite precompact continuous homomorphic images

Let P be a property of topological groups. We say that a topological group G is MinAP moduloP if its image f[G] under each continuous homomorphism f:G→K from G to a compact group K has property P. When P is the property of being the trivial group, MinAP modulo P groups are precisely the minimally almost periodic groups of von Neumann and Wigner. We give a characterization of these new classes of groups for five properties P: finite, bounded, torsion, compact and connected. We show that these five classes are distinct and differ from the standard class of minimally almost periodic groups. We equip every Abelian group with a Hausdorff MinAP modulo finite group topology, thereby showing that, unlike the classical minimal almost periodicity, the property MinAP modulo finite imposes no restrictions whatsoever on the algebraic structure of the underlying group. At last but not least, we prove that the quotient group G/n(G) of every topological group G with respect to its von Neumann kernel n(G) is the categorical reflection of G in the class of maximally almost periodic (MAP) groups, and the natural quotient map plays the role of reflection homomorphism. As a consequence, this quotient group (G/n(G))+ equipped with its Bohr topology is topologically isomorphic to the image of G under the canonical homomorphism to its Bohr compactification.

Quasi-probability distributions in loop quantum cosmology

In this paper, we introduce a complete family of parametrized quasi-probability distributions in phase space and their corresponding Weyl quantization maps with the aim to generalize the recently developed Wigner–Weyl formalism within the loop quantum cosmology (LQC) program. In particular, we intend to define those quasi-distributions for states valued on the Bohr compactification of the real line in such a way that they are labeled by a parameter that accounts for the ordering ambiguity corresponding to non-commutative quantum operators. Hence, we notice that the projections of the parametrized quasi-probability distributions result in marginal probability densities which are invariant under any ordering prescription. We also note that, in opposition to the standard Schrödinger representation, for an arbitrary character the quasi-distributions determine a positive function independently of the ordering. Further, by judiciously implementing a parametric-ordered Weyl quantization map for LQC, we are able to recover in a simple manner the relevant cases of the standard, anti-standard, and Weyl symmetric orderings, respectively. We expect that our results may serve to analyze several fundamental aspects within the LQC program, in special those related to coherence, squeezed states, and the convergence of operators, as extensively analyzed in the quantum optics and in the quantum information frameworks.

Mapping ideals of quantum group multipliers

We study the dual relationship between quantum group convolution maps L1(G)→L∞(G) and completely bounded multipliers of Gˆ. For a large class of locally compact quantum groups G, we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with Mcb(L1(Gˆ)), yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter coinciding with ℓ1(bGˆ), where bG is the quantum Bohr compactification of G. For quantum groups whose dual has bounded degree, we show that the completely compact convolution maps coincide with C(bG). Our techniques comprise a mixture of operator space theory and abstract harmonic analysis, including Fubini tensor products, the non-commutative Grothendieck inequality, quantum Eberlein compactifications, and a suitable notion of inner co-amenable quantum group, that we introduce and for which we exhibit examples from the bicrossed product construction. Our main results are new even in the setting of group von Neumann algebras VN(G) for quasi-SIN locally compact groups G.

The Bohr compactification of an abelian group as a quotient of its Stone–Čech compactification

We will prove that, for any abelian group G, the canonical (surjective and continuous) mapping $$\varvec{\beta }G\rightarrow \mathfrak {b}G$$ from the Stone–Cech compactification $$\varvec{\beta }G$$ of G to its Bohr compactification $$\mathfrak {b}G$$ is a homomorphism with respect to the semigroup operation on $$\varvec{\beta }G$$ , extending the multiplication on G, and the group operation on $$\mathfrak {b}G$$ . Moreover, the Bohr compactification $$\mathfrak {b}G$$ is canonically isomorphic (both in algebraic and topological sense) to the quotient of $$\varvec{\beta }G$$ with respect to the least closed congruence relation on $$\varvec{\beta }G$$ merging all the Schur ultrafilters on G into the unit of G.

The universal minimal one parameter system with quasi-discrete spectrum

The universal minimal one parameter system will be characterized as the space \(\Gamma ^{\infty }\), in which \(\Gamma\) is the Bohr compactification of the additive group \({\mathbb {R}}\) of real numbers. In this way, we need to show that \(\Gamma ^\infty\) is isomorphic to the spectrum of \(W({\mathbb {R}})\), the norm closure of the invariant algebra generated by the maps \(\exp q(t)\), where q(t) is a real polynomial on \({\mathbb {R}}\).

$$\pmb {{\mathcal {H}}_{p}}$$-Theory of General Dirichlet Series

Inspired by results of Bayart on ordinary Dirichlet series $$\sum a_n n^{-s}$$, the main purpose of this article is to start an $${\mathcal {H}}_p$$-theory of general Dirichlet series $$\sum a_n e^{-\lambda _{n}s}$$. Whereas the $${\mathcal {H}}_p$$-theory of ordinary Dirichlet series, in view of an ingenious identification of Bohr, may be seen as a sub-theory of Fourier analysis on the infinite dimensional torus $${\mathbb {T}}^\infty $$, the $${\mathcal {H}}_p$$-theory of general Dirichlet series is build as a sub-theory of Fourier analysis on certain compact abelian groups, including the Bohr compactification $${\overline{{\mathbb {R}}}}$$ of the reals. Our approach allows to extend various important facts on Hardy spaces of ordinary Dirichlet series to the much wider setting of $${\mathcal {H}}_p$$-spaces of general Dirichlet series.

Topological dynamics and the complexity of strong types

We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by Glasner) and various Galois groups of the theory in question, obtaining essentially new information about them, e.g., we present the closure of the identity in the Lascar Galois group of the theory as the quotient of a compact, Hausdorff group by a dense subgroup. We apply this to describe the complexity of bounded, invariant equivalence relations, obtaining comprehensive results, subsuming and extending the existing results and answering some open questions from earlier papers. We show that, in a countable theory, any such relation restricted to the set of realizations of a complete type over O is type-definable if and only if it is smooth. Then we show a counterpart of this result for theories in an arbitrary (not necessarily countable) language, obtaining also new information involving relative definability of the relation in question. As a final conclusion we get the following trichotomy. Let $$\mathfrak{C}$$ be a monster model of a countable theory, p ∈ S(O), and E be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on p( $$\mathfrak{C}$$ ). Then, exactly one of the following holds: (1) E is relatively definable (on p( $$\mathfrak{C}$$ )), smooth, and has finitely many classes, (2) E is not relatively definable, but it is type-definable, smooth, and has 2ℵ0 classes, (3) E is not type definable and not smooth, and has 2ℵ0 classes. All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups.

Lattice of compactifications of a topological group

We show that the lattice of compactifications of a topological group $G$ is a complete lattice which is isomorphic to the lattice of all closed normal subgroups of the Bohr compactification $bG$ of $G$. The correspondence defines a contravariant functor from the category of topological groups to the category of complete lattices. Some properties of the compactification lattice of a topological group are obtained.

Loop Quantization and Symmetry: Configuration Spaces

Given two sets S1, S2 and unital C*-algebras $${\mathfrak{A}_1}$$ , $${\mathfrak{A}_2}$$ of functions thereon, we show that a map $${\sigma : {\bf S}_1 \longrightarrow {\bf S}_2}$$ can be lifted to a continuous map $${\bar\sigma : {\rm spec}\, \mathfrak{A}_1 \longrightarrow {\rm spec}\, \mathfrak{A}_2}$$ iff $${\sigma^\ast \mathfrak{A}_2 := \{\sigma^\ast f\, |\, f \in \mathfrak{A}_2\} \subseteq \mathfrak{A}_1}$$ . Moreover, $${\bar \sigma}$$ is unique if existing, and injective iff $${\sigma^\ast \mathfrak{A}_2}$$ is dense. Then, we apply these results to loop quantum gravity and loop quantum cosmology. For all usual technical conventions, we decide whether the cosmological quantum configuration space is embedded into the gravitational one; indeed, both are spectra of some C*-algebras, say, $${\mathfrak{A}_{\rm cosm}}$$ and $${\mathfrak{A}_{\rm grav}}$$ , respectively. Typically, there is no embedding, but one can always get an embedding by the defining $${\mathfrak{A}_{\rm cosm} := C^\ast(\sigma^\ast \mathfrak{A}_{\rm grav})}$$ , where $${\sigma}$$ denotes the embedding between the classical configuration spaces. Finally, we explicitly determine $${C^\ast(\sigma^\ast \mathfrak{A}_{\rm grav})}$$ in the homogeneous isotropic case for $${\mathfrak{A}_{\rm grav}}$$ generated by the matrix functions of parallel transports along analytic paths. The cosmological quantum configuration space so equals the disjoint union of $${\mathbb{R}}$$ and the Bohr compactification of $${\mathbb{R}}$$ , appropriately glued together.

On a Roelcke-precompact Polish group that 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 $\mathrm{Aut}^*(\mu)$ and $\mathrm{Homeo}^+[0,1]$, 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.

DEFINABLE TOPOLOGICAL DYNAMICS

AbstractFor a group G definable in a first order structure M we develop basic topological dynamics in the category of definable G-flows. In particular, we give a description of the universal definable G-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of G, that is to the quotient ${G^{\rm{*}}}/G_M^{{\rm{*}}00}$ (where G* is the interpretation of G in a monster model). More generally, we obtain these results locally, i.e., in the category of Δ-definable G-flows for any fixed set Δ of formulas of an appropriate form. In particular, we define local connected components $G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ and $G_{{\rm{\Delta }},M}^{{\rm{*}}000}$, and show that ${G^{\rm{*}}}/G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ is the Δ-definable Bohr compactification of G. We also note that some deeper arguments from [14] can be adapted to our context, showing for example that our epimorphism from the Ellis group to the Δ-definable Bohr compactification factors naturally yielding a continuous epimorphism from the Δ-definable generalized Bohr compactification to the Δ-definable Bohr compactification of G. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.

Limits of Regularizations for Generalized Function Solutions to the Schr\xf6dinger Equation with \u2018Square Root of Delta\u2019 Initial Value

We briefly review results on generalized solutions to the Cauchy problem for linear Schrödinger-type equations with non-smooth principal part and their compatibility with classical and distributional solutions. In the main part, we study convergence properties of regularized solutions to the standard Schrödinger equation with initial values corresponding to ‘square roots’ of Dirac measures in various duals of classical subspaces of the space of continuous functions. In particular, the main result establishes as limit the invariant mean on the space of almost periodic functions as the restriction of the Haar measure on the Bohr compactification of mathbb {R}^n.