Editing Bohr compactification

{{more footnotes|date=May 2020}}
In [[mathematics]], the  '''Bohr compactification''' of a [[topological group]] ''G'' is a [[compact Hausdorff space|compact Hausdorff]] topological group ''H'' that may be [[canonical form|canonically]] associated to ''G''. Its importance lies in the reduction of the theory of [[uniformly almost periodic function]]s on ''G'' to the theory of [[continuous function]]s on ''H''. The concept is named after [[Harald Bohr]] who pioneered the study of [[almost periodic function]]s, on the [[real line]].

==Definitions and basic properties==
Given a [[topological group]] ''G'', the '''Bohr compactification''' of ''G''  is a compact ''Hausdorff'' topological group '''Bohr'''(''G'') and a continuous homomorphism{{sfn|Zhu|2019|p=37 Definition 3.1.2}}

:'''b''': ''G'' → '''Bohr'''(''G'')

which is [[universal property|universal]] with respect to homomorphisms into compact Hausdorff groups;  this means that if ''K'' is another compact Hausdorff topological group and

:''f'': ''G'' → ''K''

is a continuous homomorphism, then there is a unique continuous homomorphism

:'''Bohr'''(''f''): '''Bohr'''(''G'') → ''K''

such that ''f'' = '''Bohr'''(''f'') <math>\circ</math> '''b'''.

'''Theorem'''.  The Bohr compactification exists{{sfn|Gismatullin|Jagiella|Krupiński|2023|p=3}}{{sfn|Zhu|2019|p=34 Theorem 3.1.1}} and is unique up to isomorphism.

We will denote the Bohr compactification of ''G'' by '''Bohr'''(''G'') and the canonical map by

:<math> \mathbf{b}: G \rightarrow \mathbf{Bohr}(G). </math>

The correspondence ''G'' ↦ '''Bohr'''(''G'') defines a covariant functor on the category of topological groups and continuous homomorphisms.

The Bohr compactification is intimately connected to the finite-dimensional [[unitary representation]] theory of a topological group.  The [[kernel (algebra)|kernel]] of '''b''' consists exactly of those elements of ''G'' which cannot be separated from the identity of ''G'' by finite-dimensional ''unitary'' representations.

The Bohr compactification also reduces many problems in the theory of [[almost periodic function]]s on topological groups to that of functions on compact groups.

A bounded continuous  complex-valued function ''f'' on a topological group ''G'' is '''uniformly almost periodic''' if and only if the set of right translates <sub>''g''</sub>''f'' where

:<math> [{}_g f ] (x)  = f(g^{-1} \cdot x) </math>

is relatively compact in the uniform topology as ''g'' varies through ''G''.

'''Theorem'''. A bounded continuous  complex-valued function ''f'' on ''G'' is uniformly almost periodic if and only if there is a continuous function ''f''<sub>1</sub> on '''Bohr'''(''G'') (which is uniquely determined) such that

:<math> f = f_1 \circ \mathbf{b}. </math>{{sfn|Zhu|2019|p=39 Theorem 3.1.4}}

==Maximally almost periodic groups==
Topological groups for which the Bohr compactification mapping is injective are called ''maximally almost periodic'' (or MAP groups).  For example all Abelian groups, all compact groups, and all free groups are MAP.{{sfn|Zhu|2019|p=39 Remark 3.6.3}} In the case ''G'' is a locally compact connected group, MAP groups are completely characterized: They are precisely products of compact groups with vector groups
of finite dimension.

== See also ==

* {{annotated link|Compact space}}
* {{annotated link|Compactification (mathematics)}}
* {{annotated link|Pointed set}}
* {{annotated link|Stone–Čech compactification}}
* {{annotated link|Wallman compactification}}

== References ==
===Notes===
{{reflist|30em}}

===Bibliography===
{{refbegin|colwidth=30em}}
*{{cite journal
 | last1 = Gismatullin | first1 = Jakub
 | last2 = Jagiella | first2 = Grzegorz
 | last3 = Krupiński | first3 = Krzysztof
 | arxiv = 2011.04822
 | doi = 10.1017/jsl.2022.10
 | issue = 3
 | journal = The Journal of Symbolic Logic
 | mr = 4636627
 | pages = 1103–1137
 | title = Bohr compactifications of groups and rings
 | url = https://math.uni.wroc.pl/~kkrup/bohr7.pdf
 | volume = 88
 | year = 2023}}
*{{cite book
|first=Yihan 
|last=Zhu
|title=Almost Periodic Functions on Topological Groups
|year=2019
|series=Theses, Dissertations, and Major Papers
|url=https://scholar.uwindsor.ca/cgi/viewcontent.cgi?article=8749&context=etd
}}
{{refend}}

==Further reading==
*{{Springer|title=Bohr compactification}}

{{DEFAULTSORT:Bohr Compactification}}
[[Category:Topological groups]]
[[Category:Harmonic analysis]]
[[Category:Compactification (mathematics)]]