Everything about Bohr Compactification totally explained
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.
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's 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
»
Maximally almost periodic groups
Topological groups for which the Bohr compactification mapping is injective are called
maximally almost periodic (or MAP groups). In the case
G is a locally compact connected group, MAP groups are completely characterized: They are precisely products of compact groups with vector groups
of finite dimension.
Further Information
Get more info on 'Bohr Compactification'.
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