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Encyclopedia > Bohr compactification

In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H. The concept is named after Harald Bohr who pioneered the study of almost periodic functions, 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

b: GBohr(G)

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

f: GK

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

Bohr(f): Bohr(G) → K

such that f = Bohr(f) b.


Theorem. The Bohr compactification exists and is unique up to isomorphism.


This is a direct application of the Tychonoff theorem.


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

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 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 functions 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 iff the set of right translates gf where

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


Theorem. A bounded continous complex_valued function f on G is uniformly almost periodic iff there is a continuous function f1 on Bohr(G) (which is uniquely determined) such that



  Results from FactBites:
 
Bohr compactification - Wikipedia, the free encyclopedia (336 words)
In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G.
The Bohr compactification is intimately connected to the finite-dimensional unitary representation theory of a topological group.
The Bohr compactification also reduces many problems in the theory of almost periodic functions on topological groups to that of functions on compact groups.
Compactification (mathematics) - Wikipedia, the free encyclopedia (1102 words)
The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".
An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. Embeddings into compact Hausdorff spaces may be of particular interest.
The Bohr compactification of a topological group arises from the consideration of almost periodic functions.
  More results at FactBites »

 
 

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