Hubbry Logo
Locally compact groupLocally compact groupMain
Open search
Locally compact group
Community hub
Locally compact group
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Locally compact group
Locally compact group
Locally compact group
View on Wikipedia
from Wikipedia

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 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

[edit]
  • Any compact group is locally compact.
    • In particular the circle group S1 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 becomes a topological group when given the discrete topology.
  • The additive groups of the real numbers R and of the complex numbers C, if given their standard topologies, are locally compact, as are the multiplicative groups of non-zero numbers Rx and Cx.
  • Lie groups, which are locally Euclidean, are locally compact. Here we find many examples of non-abelian locally compact groups.
  • A normable topological vector space over a local field is locally compact if and only if it is finite-dimensional.
  • The additive group of rational numbers Q is not locally compact if given its standard topology, 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 Qp with its standard topology is locally compact for any prime number p.

Properties

[edit]

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.

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 T0 and 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

[edit]

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

LCAop → 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).

See also

[edit]

References

[edit]

Sources

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

Further reading

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Locally compact group

View on Grokipedia
from Grokipedia
In mathematics, a locally compact group is a topological group GG whose underlying topology is locally compact and Hausdorff, meaning that every point in GG has a compact neighborhood. This structure ensures that the group operations of multiplication and inversion are continuous, allowing for the application of both algebraic and analytic techniques. Locally compact groups form a fundamental class in abstract harmonic analysis and representation theory, as they admit a left-invariant Haar measure, which is a positive Radon measure unique up to positive scalar multiples and serves as the basis for integration on the group. Key properties include the closedness of compact subgroups and the local compactness of quotients by closed normal subgroups, facilitating the study of group actions and extensions. Prominent examples encompass finite groups, which are compact, and more generally discrete groups, Lie groups such as the general linear group GLn(R)\mathrm{GL}_n(\mathbb{R}), the circle group U(1)U(1), and non-abelian groups like the Heisenberg group over R\mathbb{R}. For the subclass of locally compact abelian groups, Pontryagin duality provides a powerful tool, establishing an isomorphism between the group and the dual group of continuous homomorphisms to the circle, which interchanges compactness and discreteness. These groups underpin applications in , , and , with their representations decomposing into irreducible components via tools like the Peter-Weyl theorem for compact cases and more general Plancherel formulas for non-compact ones. Non-unimodular examples, such as the ax+b group of affine transformations of the line, highlight the role of the modular function in adjusting left and right Haar measures.

Fundamentals

Definition

A is a group GG together with a on GG such that the multiplication map G×G→GG \times G \to G, (g,h)↦gh(g, h) \mapsto gh, and the inversion map G→GG \to G, g↦g−1g \mapsto g^{-1}, are continuous, where G×GG \times G is equipped with the . This structure ensures that the algebraic operations respect the topological properties, allowing for the study of both group-theoretic and analytic behaviors within the same framework. A locally compact group is a GG whose underlying is Hausdorff and locally compact. The Hausdorff condition requires that for any two distinct points in GG, there exist disjoint open neighborhoods separating them, which guarantees the uniqueness of limits and prevents pathological behaviors in arguments. Local compactness means that every point x∈Gx \in G has a compact neighborhood, i.e., a neighborhood UU of xx such that every open cover of UU has a finite subcover. This property is fundamental, as it enables the existence of a —a left-invariant measure defined up to scalar multiples—on GG. In standard notation, such groups are denoted by GG, and the topology is often assumed to be second countable when additional is needed. Under this assumption, GG is σ\sigma-compact, meaning it can be expressed as a countable union of compact subsets, which facilitates many analytical constructions.

Topological and Algebraic Prerequisites

A is a group equipped with a such that the group and inversion are continuous. In such groups, the algebraic operations induce continuous maps, ensuring that the interacts compatibly with the group . Specifically, for any fixed element gg in the group GG, the left Lg:x↦gxL_g: x \mapsto gx and the right Rg:x↦xgR_g: x \mapsto xg are homeomorphisms, preserving and closedness across the . This homogeneity allows properties at arbitrary points to be reduced to those near the , facilitating analysis of the group's . The of a is determined by a neighborhood basis at the ee, consisting of open sets UU containing ee. These neighborhoods generate the entire via : a set WW is open if and only if for every g∈Wg \in W, there exists an identity neighborhood UU such that gU⊆WgU \subseteq W. In the context of local compactness, the existence of a compact neighborhood of the identity provides a local basis of compact sets, which underpins the around each point through homeomorphisms. This basis simplifies the study of convergence and continuity by localizing global properties to the identity. Topological groups naturally carry a uniform structure, induced by the neighborhoods of the identity, which equips the group with a notion of stronger than mere topological continuity. The entourages of this are sets of the form (U×U)(U \times U) where UU is a symmetric neighborhood of ee, ensuring that multiplication becomes with respect to the right (or left, analogously). This structure captures "nearness" in a translation-invariant way, allowing the definition of Cauchy sequences and completeness relative to the group operations. Many results in the theory of locally compact groups assume second-countability, meaning the topology has a countable basis of open sets, which often implies metrizability and simplifies proofs involving covers and limits. Under second-countability, locally compact groups are σ-compact, expressible as a countable union of compact subsets, facilitating the construction of invariant measures and structural decompositions. These assumptions ensure that the group admits a left-invariant proper metric compatible with the topology, aiding in the analysis of homogeneity and subgroup properties in standard settings.

Examples

Positive Examples

The additive group of the Rn\mathbb{R}^n, equipped with the standard , is a prototypical example of a locally compact group, as every point has a compact neighborhood given by a closed ball of finite radius. Finite groups, endowed with the discrete , are locally compact since every singleton subset is compact and serves as a compact neighborhood for each element. Lie groups, such as the general linear group GL(n,R)GL(n, \mathbb{R}) or the special orthogonal group SO(n)SO(n), are locally compact due to their structure as smooth manifolds, which provide compact neighborhoods around the via coordinate charts. The additive group of the pp-adic numbers Qp\mathbb{Q}_p, for a prime pp, forms a locally compact group under the pp-adic topology, where closed balls are compact subsets serving as local bases of neighborhoods. Infinite discrete groups, such as with the discrete topology, are locally compact but not compact, as singletons provide compact neighborhoods while the whole group is unbounded.

Counterexamples

The additive group of rational numbers Q\mathbb{Q}, endowed with the subspace topology induced from the real line R\mathbb{R}, provides a fundamental example of a metrizable topological group that fails to be locally compact. In this topology, every non-empty open set is unbounded and contains sequences converging to irrational numbers outside Q\mathbb{Q}, preventing any compact subset from containing an open neighborhood of any point; specifically, if a compact set K⊆QK \subseteq \mathbb{Q} contained an open interval intersected with Q\mathbb{Q}, its closure in R\mathbb{R} would be compact but unable to cover the dense irrationals without contradiction, as sequential compactness fails in Q\mathbb{Q}. Another illustrative counterexample is the (or restricted product) of countably infinitely many copies of the integers, denoted ⨁n=1∞Z\bigoplus_{n=1}^\infty \mathbb{Z} or equivalently the group of integer sequences with finitely many non-zero terms under componentwise addition, equipped with the where each Z\mathbb{Z} carries the discrete ; however, the full product ∏n=1∞Z\prod_{n=1}^\infty \mathbb{Z} with the same serves as the completion, and neither is locally compact at the identity. A basic open neighborhood of the zero element depends on only finitely many coordinates being restricted to finite subsets, but any such neighborhood projects onto an entire copy of Z\mathbb{Z} in later coordinates, whose image under the continuous projection map would be compact if the neighborhood were contained in a compact set, contradicting the non-compactness of Z\mathbb{Z}. The space C[0,1]C[0,1] of continuous real-valued functions on the compact interval [0,1][0,1], viewed as an additive topological group under the supremum norm ∥f∥∞=sup⁡x∈[0,1]∣f(x)∣\|f\|_\infty = \sup_{x \in [0,1]} |f(x)|, is a further example of a complete metric topological group that is not locally compact. This failure stems from its infinite-dimensional nature as a : the closed unit ball is not compact, and by the Riesz lemma, no neighborhood of the zero function is relatively compact, as one can construct sequences of functions with disjoint supports that remain at positive distance from each other and from zero. These examples highlight the limitations of topological groups without local compactness, notably the non-existence of a , which requires the space to admit a regular invariant under left translations and finite on compact sets.

General Properties

Inner Structure and Subgroups

Closed subgroups of a locally compact group GG inherit the local compactness property from GG in the . Specifically, a H≤GH \leq G is locally compact if and only if it is closed in GG. Open subgroups in a locally compact group GG are necessarily closed, since GG is assumed Hausdorff, and thus they are also locally compact. The cosets of an open subgroup HH are open sets homeomorphic to HH, so they share the topological properties of HH, including compactness if HH is compact. For quotient groups, if HH is a closed of a locally compact group GG, then the space G/HG/H is locally compact with the . Conversely, if the G/HG/H is locally compact, then HH must be closed in GG. This ensures that the algebraic structure of quotients aligns well with the topological framework of local compactness. In totally disconnected locally compact groups, compact open subgroups play a central role in the inner structure. By van Dantzig's theorem, every totally disconnected locally compact group admits a compact open subgroup, and in fact, the compact open subgroups form a base of neighborhoods at the identity. This property facilitates the study of such groups through their actions on discrete quotients and tidy subgroups. A prominent example arises in pp-adic groups, such as the additive group of pp-adic numbers Qp\mathbb{Q}_p, where the pp-adic integers Zp\mathbb{Z}_p form a compact open . Similarly, for the general linear group GLn(Qp)\mathrm{GL}_n(\mathbb{Q}_p), the GLn(Zp)\mathrm{GL}_n(\mathbb{Z}_p) is compact and open, illustrating how these subgroups capture the "integral" structure in non-Archimedean local fields. The homogeneity of locally compact groups stems from the fact that left (or right) translations by elements of the group are homeomorphisms, preserving compactness of subsets. Thus, if K⊆GK \subseteq G is compact, then gKgK is compact for any g∈Gg \in G. This invariance under group actions underscores the uniform topological behavior across the group.

Measure Theory Integration

A left Haar measure on a locally compact Hausdorff GG is a nonzero μ\mu on GG that is finite on every compact set, positive on every nonempty , and left-invariant, meaning μ(gA)=μ(A)\mu(gA) = \mu(A) for every A⊆GA \subseteq G and every g∈Gg \in G, where gA={ga∣a∈A}gA = \{ga \mid a \in A\}. The existence of such a measure on second countable locally compact groups was established by Alfred Haar in 1933. This result was generalized to arbitrary locally compact Hausdorff groups by in 1940 using the . The uniqueness of the left Haar measure, up to multiplication by a positive scalar, was proved by in 1936 for second countable cases and extends to the general setting. One standard construction of the Haar measure proceeds via the applied to the space of continuous complex-valued functions with compact support Cc(G)C_c(G), equipped with the inductive limit topology. Specifically, a left-invariant positive linear functional on Cc(G)C_c(G) is constructed using approximations by convolutions with approximate identities supported in a fixed relatively compact open neighborhood of the identity, yielding a regular that satisfies the required invariance. If GG is second countable, then any is σ\sigma-finite, as GG admits a countable basis of relatively compact open sets. Right Haar measures exist analogously but are invariant under right translations A↦AgA \mapsto Ag. For a left Haar measure μ\mu, the corresponding right Haar measure μr\mu_r satisfies dμr(g)=Δ(g−1) dμ(g)d\mu_r(g) = \Delta(g^{-1}) \, d\mu(g), where Δ:G→(0,∞)\Delta: G \to (0, \infty) is the modular function (or Haar modulus), a continuous uniquely determined by Δ(g)=dμgdμ\Delta(g) = \frac{d\mu_g}{d\mu} with μg(A)=μ(Ag)\mu_g(A) = \mu(Ag) the right translate of μ\mu by gg, where Ag={ag∣a∈A}Ag = \{a g \mid a \in A\}. The modular function adjusts for the lack of right invariance of the left Haar measure, via the change-of-variables formula ∫Gf(gh) dμ(g)=Δ(h)−1∫Gf(g) dμ(g)\int_G f(gh) \, d\mu(g) = \Delta(h)^{-1} \int_G f(g) \, d\mu(g) for all integrable f:G→Cf: G \to \mathbb{C} and h∈Gh \in G. Groups for which Δ≡1\Delta \equiv 1 are called unimodular; all compact and discrete groups are unimodular. The space L1(G)L^1(G) of μ\mu-integrable functions on GG, equipped with the product (f∗ϕ)(x)=∫Gf(y)ϕ(y−1x) dμ(y)(f * \phi)(x) = \int_G f(y) \phi(y^{-1} x) \, d\mu(y) for f,ϕ∈L1(G)f, \phi \in L^1(G) and x∈Gx \in G, forms a under the L1L^1-norm ∥f∥1=∫G∣f(g)∣ dμ(g)\|f\|_1 = \int_G |f(g)| \, d\mu(g), as ∥f∗ϕ∥1≤∥f∥1∥ϕ∥1\|f * \phi\|_1 \leq \|f\|_1 \|\phi\|_1. This structure underpins much of on groups, enabling the study of representations and Fourier transforms.

Abelian Case

Pontryagin Duality

For a locally compact abelian group GG, the Pontryagin dual group G^\hat{G} is defined as the set of all continuous group homomorphisms from GG to the circle group T=R/Z\mathbb{T} = \mathbb{R}/\mathbb{Z}, equipped with pointwise multiplication as the group operation. The circle group T\mathbb{T} serves as the codomain because it is a divisible compact abelian group, making homomorphisms to it capture the essential character data of GG. The dual group G^\hat{G} is endowed with the compact-open topology, defined by subbasis sets of the form {χ∈G^∣sup⁡x∈K∣χ(x)−η(x)∣<ϵ}\{ \chi \in \hat{G} \mid \sup_{x \in K} |\chi(x) - \eta(x)| < \epsilon \} for compact subsets K⊆GK \subseteq G, continuous characters η:G→T\eta: G \to \mathbb{T}, and ϵ>0\epsilon > 0. This topology ensures that G^\hat{G} is a Hausdorff topological group, and if GG is locally compact, then G^\hat{G} is also locally compact. Consequently, for abelian GG, the dual G^\hat{G} inherits the structure of a locally compact abelian group, preserving the class under duality. The Pontryagin duality theorem asserts that there exists a topological group isomorphism G≅G^^G \cong \hat{\hat{G}} between GG and its double dual, given by the evaluation map δ:G→G^^\delta: G \to \hat{\hat{G}} defined by δ(g)(χ)=χ(g)\delta(g)(\chi) = \chi(g) for g∈Gg \in G and χ∈G^\chi \in \hat{G}. This map is continuous, bijective, and bicontinuous, establishing a natural duality that generalizes classical to arbitrary locally compact abelian groups. Representative examples illustrate this: the dual of the additive group R\mathbb{R} of real numbers is isomorphic to R\mathbb{R} itself, with characters χt(x)=e2πitx\chi_t(x) = e^{2\pi i t x} for t∈Rt \in \mathbb{R}; similarly, the dual of the discrete additive group Z\mathbb{Z} of integers is the circle group T\mathbb{T}, with characters n↦znn \mapsto z^n for z∈Tz \in \mathbb{T}. A proof sketch relies on the existence of a on GG, which allows the definition of the Ff(χ)=∫Gf(g)χ(g)‾ dg\mathcal{F}f(\chi) = \int_G f(g) \overline{\chi(g)} \, dg for integrable functions f∈L1(G)f \in L^1(G). The invertibility of this transform, via the inversion formula f(g)=∫G^Ff(χ)χ(g−1) dμ^(χ)f(g) = \int_{\hat{G}} \mathcal{F}f(\chi) \chi(g^{-1}) \, d\hat{\mu}(\chi) (where μ^\hat{\mu} is the dual Haar measure), implies that δ\delta is injective by showing that if δ(g)=1\delta(g) = 1, then g=eg = e. Density arguments using continuous functions with compact support, combined with the for L2(G)L^2(G), establish surjectivity and continuity of the inverse, confirming the isomorphism.

Structure Theorems

The principal structure theorem for locally compact abelian (LCA) groups provides a canonical decomposition that reveals their inner and . Every LCA group GG contains an open HH topologically isomorphic to Rn×Zm×C\mathbb{R}^n \times \mathbb{Z}^m \times C, where nn and mm are non-negative integers, and CC is a compact ; the G/HG/H is discrete. The real part captures the continuous, divisible elements, the Zm\mathbb{Z}^m part handles the discrete torsion-free finite-rank elements, the compact part accounts for the bounded , and the discrete incorporates additional discrete components, which may include torsion and infinite rank. This decomposition is unique up to in its invariants. The connected component of the identity G0G_0 in any LCA group GG plays a central role in this ; it is compactly generated and topologically isomorphic to Rn×K\mathbb{R}^n \times K, while the G/G0G / G_0 is a discrete . This separation highlights how the connected structure is "tamed" by compact generation, ensuring the overall group remains locally compact. interacts seamlessly with this structure: the dual of a direct product of LCA groups is the direct product of their duals, so the dual of such a involves Rn×Zm×C×D^≅Rn×Tm×C^×D^\widehat{\mathbb{R}^n \times \mathbb{Z}^m \times C \times D} \cong \mathbb{R}^n \times \mathbb{T}^m \times \hat{C} \times \hat{D}
Add your contribution
Related Hubs
User Avatar
No comments yet.