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| 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]].
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| ==Definitions and basic properties==
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| Given a [[topological group]] ''G'', the '''Bohr compactification''' of ''G'' is a compact ''Hausdorff'' topological group '''Bohr'''(''G'') and a continuous homomorphism
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| :'''b''': ''G'' → '''Bohr'''(''G'')
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| 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
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| :''f'': ''G'' → ''K''
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| is a continuous homomorphism, then there is a unique continuous homomorphism
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| :'''Bohr'''(''f''): '''Bohr'''(''G'') → ''K''
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| such that ''f'' = '''Bohr'''(''f'') ∘ '''b'''.
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| '''Theorem'''. The Bohr compactification exists and is unique up to isomorphism.
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| This is a direct application of the [[Tychonoff theorem]].
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| We will denote the Bohr compactification of ''G'' by '''Bohr'''(''G'') and the canonical map by
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| :<math> \mathbf{b}: G \rightarrow \mathbf{Bohr}(G). </math>
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| The correspondence ''G'' ↦ '''Bohr'''(''G'') defines a covariant functor on the category of topological groups and continuous homomorphisms.
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| 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.
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| 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.
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| 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
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| :<math> [{}_g f ] (x) = f(g^{-1} \cdot x) </math> | |
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| is relatively compact in the uniform topology as ''g'' varies through ''G''.
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| '''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
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| :<math> f = f_1 \circ \mathbf{b}. </math>
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| ==Maximally almost periodic groups==
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| Topological groups for which the Bohr compactification mapping is injective are called ''maximally almost periodic'' (or MAP groups). 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
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| of finite dimension.
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| ==References==
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| *{{Springer|id=B/b016780}}
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| {{DEFAULTSORT:Bohr Compactification}}
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| [[Category:Topological groups]]
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| [[Category:Harmonic analysis]]
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| [[Category:Compactification]]
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My name is Reinaldo (42 years old) and my hobbies are Aircraft spotting and Machining.
Review my blog :: 4Inkjets Discount Coupons