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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 GA_googleFillSlot("encyclopedia_square");

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 &mapsto; 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.
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