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: G → Bohr(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: G → K
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
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 if and only if the set of right translates gf where
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
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