Bohr compactification
Encyclopedia
In mathematics
, the Bohr compactification of a topological group
G is a compact Hausdorff topological group H that may be canonical
ly associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous function
s on H. The concept is named after Harald Bohr
who pioneered the study of almost periodic function
s, on the real line
.
G, the Bohr compactification of G is a compact Hausdorff topological group Bohr(G) and a continuous homomorphism
which is universal
with respect to homomorphisms into compact Hausdorff groups; this means that if K is another compact Hausdorff topological group and
is a continuous homomorphism, then there is a unique continuous homomorphism
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
The correspondence G → Bohr(G) defines a covariant functor on the category of topological groups and continuous homomorphisms.
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.
The Bohr compactification also reduces many problems in the theory of almost periodic function
s on topological groups to that of functions on compact groups.
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
of finite dimension.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Bohr compactification of a topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
G is a compact Hausdorff topological group H that may be canonical
Canonical
Canonical is an adjective derived from canon. Canon comes from the greek word κανών kanon, "rule" or "measuring stick" , and is used in various meanings....
ly associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s on H. The concept is named after Harald Bohr
Harald Bohr
Harald August Bohr was a Danish mathematician and football player. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the Nobel Prize-winning physicist Niels Bohr...
who pioneered the study of almost periodic function
Almost periodic function
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov,...
s, on the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
.
Definitions and basic properties
Given a topological groupTopological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
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
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
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 is 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
The correspondence G → Bohr(G) defines a covariant functor on the category of topological groups and continuous homomorphisms.
The Bohr compactification is intimately connected to the finite-dimensional unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
theory of a topological group. The kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of b consists exactly of those elements of G which cannot be separated from the identity of G by finite-dimensional unitary representations.
The Bohr compactification also reduces many problems in the theory of almost periodic function
Almost periodic function
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov,...
s on topological groups to that of functions on compact groups.
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
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 groupsof finite dimension.