Locally compact group

Last updated November 25, 2023

In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on G so that standard analysis notions such as the Fourier transform and $L^{p}$ spaces can be generalized.

Many of the results of finite group representation theory are proved by averaging over the group. For compact groups, modifications of these proofs yields similar results by averaging with respect to the normalized Haar integral. In the general locally compact setting, such techniques need not hold. The resulting theory is a central part of harmonic analysis. The representation theory for locally compact abelian groups is described by Pontryagin duality.

Examples and counterexamples

Any compact group is locally compact.
- In particular the circle group T of complex numbers of unit modulus under multiplication is compact, and therefore locally compact. The circle group historically served as the first topologically nontrivial group to also have the property of local compactness, and as such motivated the search for the more general theory, presented here.
Any discrete group is locally compact. The theory of locally compact groups therefore encompasses the theory of ordinary groups since any group can be given the discrete topology.
Lie groups, which are locally Euclidean, are all locally compact groups.
A Hausdorff topological vector space is locally compact if and only if it is finite-dimensional.
The additive group of rational numbers Q is not locally compact if given the relative topology as a subset of the real numbers. It is locally compact if given the discrete topology.
The additive group of p-adic numbers Q_p is locally compact for any prime number p.

Properties

By homogeneity, local compactness of the underlying space for a topological group need only be checked at the identity. That is, a group G is a locally compact space if and only if the identity element has a compact neighborhood. It follows that there is a local base of compact neighborhoods at every point.

A topological group is Hausdorff if and only if the trivial one-element subgroup is closed.

Every closed subgroup of a locally compact group is locally compact. (The closure condition is necessary as the group of rationals demonstrates.) Conversely, every locally compact subgroup of a Hausdorff group is closed. Every quotient of a locally compact group is locally compact. The product of a family of locally compact groups is locally compact if and only if all but a finite number of factors are actually compact.

Topological groups are always completely regular as topological spaces. Locally compact groups have the stronger property of being normal.

Every locally compact group which is first-countable is metrisable as a topological group (i.e. can be given a left-invariant metric compatible with the topology) and complete. If furthermore the space is second-countable, the metric can be chosen to be proper. (See the article on topological groups.)

In a Polish group G, the σ-algebra of Haar null sets satisfies the countable chain condition if and only if G is locally compact.^[1]

Locally compact abelian groups

For any locally compact abelian (LCA) group A, the group of continuous homomorphisms

Hom(A, S¹)

from A to the circle group is again locally compact. Pontryagin duality asserts that this functor induces an equivalence of categories

LCA^op→ LCA.

This functor exchanges several properties of topological groups. For example, finite groups correspond to finite groups, compact groups correspond to discrete groups, and metrisable groups correspond to countable unions of compact groups (and vice versa in all statements).

LCA groups form an exact category, with admissible monomorphisms being closed subgroups and admissible epimorphisms being topological quotient maps. It is therefore possible to consider the K-theory spectrum of this category. Clausen (2017) has shown that it measures the difference between the algebraic K-theory of Z and R, the integers and the reals, respectively, in the sense that there is a homotopy fiber sequence

K(Z) → K(R) → K(LCA).

Related Research Articles

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space $is said to be metrizable if there is a metric such that the topology induced by is Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.$

In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.

In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood.

In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexandroff. More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any topological space.

In mathematics, a topological ring is a ring $that is also a topological space such that both the addition and the multiplication are continuous as maps:$

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(g) is a unitary operator for every g ∈ G. The general theory is well-developed in the case that G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.

In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.

In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group, the finite abelian groups, and the additive group of the integers, the real numbers, and every finite-dimensional vector space over the reals or a $p$ -adic field.

In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in the 1930s in the course of his investigation of Boolean algebras, which culminated in his representation theorem for Boolean algebras.

In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers, or the real numbers or the circle are locally compact abelian groups.

In mathematics, a topological group G is called a discrete group if there is no limit point in it. Equivalently, the group G is discrete if and only if its identity is isolated.

In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group. That is, a TAG is both a group and a topological space, the group operations are continuous, and the group's binary operation is commutative.

In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".

In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In mathematics, a topological semigroup is a semigroup that is simultaneously a topological space, and whose semigroup operation is continuous.

In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields.

In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is usually taken to be the extension of the theory to all groups G that are locally compact. The case of compact groups is understood, qualitatively and after the Peter–Weyl theorem from the 1920s, as being generally analogous to that of finite groups and their character theory.

In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space. These kinds of fields were originally introduced in p-adic analysis since the fields $are locally compact topological spaces constructed from the norm on . The topology is essential because it allows one to construct analogues of algebraic number fields in the p-adic context.$

References

↑ Slawomir Solecki (1996) On Haar Null Sets, Fundamenta Mathematicae 149

Sources

Clausen, Dustin (2017), A K-theoretic approach to Artin maps, arXiv: 1703.07842v2