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.
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.
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 Topological groups for which the Bohr compactification mapping is injective are called maximally almost periodic (or MAP groups).