Bohr compactification

In mathematics, the Bohr compactification is a special way to study certain types of mathematical structures called topological groups. It takes a group G and creates a new, more organized group H that is both compact and follows special rules called being Hausdorff. This new group helps mathematicians understand complicated functions on G by turning them into simpler ones on H.

The idea behind Bohr compactification comes from the work of Harald Bohr, a mathematician who studied functions that repeat in a very regular way. These functions, called uniformly almost periodic functions, are easier to handle when we use the Bohr compactification. Instead of dealing with these tricky functions directly on G, we can look at regular continuous functions on the new group H.

This concept is important because it connects two areas of mathematics: the study of groups and the study of functions. By doing this, mathematicians can solve problems that would otherwise be very hard. The Bohr compactification shows how deep ideas in mathematics can simplify complex theories.

Definitions and basic properties

The Bohr compactification is a way to turn a topological group into a special kind of group that is both compact and Hausdorff. This helps simplify many problems, especially those involving almost periodic functions.

In simple terms, it connects the study of certain functions on a group to the study of continuous functions on a compact group. This makes it easier to understand and work with these functions. The Bohr compactification is named after Harald Bohr, who worked on almost periodic functions.

Maximally almost periodic groups

Some special groups of mathematical objects are called "maximally almost periodic" groups, or MAP groups. These groups have a special property related to their Bohr compactification. Examples of MAP groups include all Abelian groups, all compact groups, and all free groups. When dealing with locally compact connected groups, MAP groups are exactly those that can be described as combinations of compact groups and vector groups with a fixed number of dimensions.