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. This new group is both compact and follows special rules.

The idea comes from the work of Harald Bohr. He studied functions that repeat in a regular way. These functions are called uniformly almost periodic functions. Using the Bohr compactification, mathematicians can turn hard problems into easier ones.

This concept connects two areas of mathematics: the study of groups and the study of functions. It helps mathematicians solve problems that would be very hard otherwise.

Definitions and basic properties

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

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

Bohr compactification