Bohr compactification

The space $X$ of maximal ideals of the algebra of Bohr almost-periodic functions (cf. Banach algebra; Bohr almost-periodic functions). The Bohr almost-periodic functions form a commutative $C ^ {*}$- algebra $A$ over the reals $\mathbf R$. The algebra $A$ is isometrically isomorphic to the algebra $C(X)$ of all continuous functions on the compactum $X$. The real line $\mathbf R$ is naturally imbedded in $X$ as an everywhere-dense subset (however, this imbedding is not a homeomorphism). The compactum $X$ has the structure of a connected compact group which is identified with the group of characters of the real line if the latter is considered with the discrete topology. This isomorphism between the algebra of almost-periodic functions and the algebra of all continuous functions on the Bohr compactification makes it possible to simplify the proofs of a number of classical theorems. The concept of a Bohr compactification is also meaningful for the algebras of almost-periodic functions on other groups. In the case of the set of conditionally-periodic functions with $n$ independent fixed periods, the $n$- dimensional torus with these periods acts as the Bohr compactification.

References

 [1] L.H. Loomis, "An introduction to abstract harmonic analysis" , v. Nostrand (1953)

There are several other ways to characterize the Bohr compactification and to prove its existence; see e.g. [a1]. The most abstract one is the characterization of the Bohr compactification of a topological group $G$ as the reflection of $G$ in the category of all compact Hausdorff topological groups; its existence is then guaranteed by the adjoint functor theorem. This approach is also used for semi-groups and leads to several other types of compactifications (e.g. the weakly almost-periodic compactifications). See [a2].