Bohr compactification

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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[edit]

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

 \mathbf{b}: G \rightarrow \mathbf{Bohr}(G).

The correspondence GBohr(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 if and only if the set of right translates gf where

 [{}_g f ] (x)  = f(g^{-1} \cdot x)

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 f1 on Bohr(G) (which is uniquely determined) such that

 f = f_1 \circ \mathbf{b}.

Maximally almost periodic groups[edit]

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 of finite dimension.