Local compact group

In mathematics, a locally compact group is a topological group whose underlying topology is locally compact . This property allows some analytical concepts known from Euclidean space to be generalized to such more general groups. These groups, especially their representations , are objects of study in harmonic analysis .

definition

A topological group is a group with a link and a neutral element provided with a topology so that both (with the product topology on ) and the inverse formation are continuous . A topological space is called locally compact if every point has a surrounding basis of compact sets . A locally compact group can, however, also be characterized using a few prerequisites: A group with a topology is a locally compact group if and only if ${\displaystyle G}$ ${\displaystyle \cdot }$ ${\displaystyle e}$ ${\displaystyle \cdot \colon G\times G\to G}$ ${\displaystyle G\times G}$ ${\displaystyle x\mapsto x^{-1}}$ ${\displaystyle G}$

• the topology is pre-regular ,
• the neutral element has a compact environment and ${\displaystyle e}$ ${\displaystyle K}$
• the group forms a semitopological group , d. H. the mapping is continuous in both components separately, i.e. the translation maps and are continuous for each . ${\displaystyle \cdot }$ ${\displaystyle x\in G}$ ${\displaystyle y\mapsto x\cdot y}$ ${\displaystyle y\mapsto y\cdot x}$

Because of the continuity of the left translation um , the set is compact for each, and because of the continuity of the left translation um , a neighborhood of . Each point has a compact environment, the space is locally compact due to the pre-regularity. Further considerations show that every locally compact semitopological group actually has a simultaneous continuous link ( i.e. it is a paratopological group ) and that the formation of the inverse is also continuous. ${\displaystyle x}$ ${\displaystyle x\in G}$ ${\displaystyle xK}$ ${\displaystyle x^{-1}}$ ${\displaystyle xK}$ ${\displaystyle x}$ ${\displaystyle \cdot \colon G\times G\to G}$

Some authors always assume the Hausdorff property in the definition . It is usually sufficient (especially in representation theory) to restrict oneself to such groups. For each locally compact group, the Kolmogorow quotient is in turn a locally compact group that has essentially the same properties. The formation of the Kolmogorow quotient is left adjoint as a functor to embed the Hausdorff locally compact groups in the category of the locally compact groups (with continuous homomorphisms as morphisms).

Examples

• Each group is provided with the discrete topology or the cluster topology , a locally compact group. (The latter example does not, however, satisfy the Hausdorff axiom, which some authors assume when defining locally compact groups.)
• The Euclidean space forms a locally compact group with addition, with multiplication and more generally with each Lie group. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} \setminus \left\{0\right\}}$
• According to Tichonow's theorem, every set forms a compact and therefore locally compact group with the addition of elements, which is called the Cantor group . ${\displaystyle X}$ ${\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{X}}$ ${\displaystyle X=\mathbb {N} }$
• The field of the p-adic numbers forms a locally compact group with addition and multiplication. In general, this applies to all local bodies . ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle \mathbb {Q} _{p}\setminus \left\{0\right\}}$
• A real or complex normalized vector space is with the addition a topological group which is locally compact if and only if the space is finite dimensional.
• More generally: A at least one-dimensional T ₀ topological vector space over a related induced by the addition. Uniform structure complete, non- discrete topological skew field is exactly then with the addition when he and the swash body is locally compact a locally compact group finite.
• On the free product of at least two nontrivial groups, especially on free groups, each Hausdorff locally compact group is discrete.
• The Kolmogorow quotient of each at most countable locally compact group is discrete, this can be shown using Baire's theorem or properties of the hair measure .

Topological properties

Locally compact groups, like any locally compact space and topological group, are completely regular . In addition, they are even paracompact and therefore normal . This can be inferred from the uniform local compactness , i.e. H. from the fact that in the left- or right-sided uniform structure induced by the group structure there is a neighborhood , so that for each there is a compact neighborhood of . ${\displaystyle U}$ ${\displaystyle U[x]}$ ${\displaystyle x\in G}$ ${\displaystyle x}$

With regard to the left-hand and right-hand uniform structure, locally compact topological groups are complete , i.e. H. each Cauchy filter converges. From Urysohn's metrizability theorem it follows, as with every regular space , that metrizability follows from second countability . For every two-countable locally compact group there is even a metric that induces the topology, is invariant under left shifts and in which all bounded , closed sets (as in Euclidean space according to the Heine-Borel theorem ) are compact, making the metric especially complete and thus every second countable locally compact group is a Polish group .

Subgroups and quotients

A subgroup of a locally compact group is again locally compact if and only if it is closed. The execution does not apply to any subsets of locally compact spaces (consider a nontrivial open subset of Euclidean space, for example). It results from the fact that every complete sub-space of a uniform room is closed off. If it is closed, the space of the left secondary classes with the quotient topology is a locally compact, homogeneous space that operates on through left multiplication. If a closed subgroup is even a normal divisor , then the quotient group is again a locally compact group. ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle G/H}$ ${\displaystyle G}$

Every locally compact group has a subgroup that is open (equivalent to the environment of the neutral element), closed (which follows from the openness) and σ-compact . It is thus a disjoint union of σ-compact subspaces (namely the left secondary classes or right secondary classes of this group) with the sum topology .

For each topological group and a locally compact subgroup , the space of the left subclasses with respect to the quotient of the right-hand uniform structure of by , i.e. H. the final uniformity with regard to the canonical surjection from to , complete. For every discrete subgroup a topological group is locally compact if and only if the space is locally compact. ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle G/H}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle G/H}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle G/H}$

structure

Every Hausdorff locally compact group can be approximated in a certain sense by Lie groups : Every such group has an open subgroup , so that for every neighborhood of the neutral element there is a subset that is a compact normal divisor of , so that is a Lie group. Every connected, Hausdorff locally compact group thus has a compact normal divisor , so that is a Lie group, and is a subgroup of a product of Lie groups. ${\displaystyle G}$ ${\displaystyle G^{\prime }}$ ${\displaystyle K}$ ${\displaystyle G^{\prime }}$ ${\displaystyle G^{\prime }/N}$ ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle G/K}$

Even before this statement was shown, it had been proven that every connected locally compact group that fulfills this approximation property (i.e. every Hausdorffian, as we know today) is homeomorphic to a natural number and a compact group (with a neutral element ). A homeomorphism can be chosen so that all constraints and isomorphisms are topological groups. ${\displaystyle G}$ ${\displaystyle \mathbb {R} ^{n}\times K}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle 0}$ ${\displaystyle \phi \colon \mathbb {R} ^{n}\times K\to G}$ ${\displaystyle \phi {\upharpoonright }\{0\}^{i}\times \mathbb {R} \times \{0\}^{n-i}}$ ${\displaystyle \phi {\upharpoonright }\{0\}^{n}\times K}$

For connected maximally almost periodic groups , i. H. Groups whose finite-dimensional unitary representations are point-separating , including all Abelian groups, can even be chosen entirely as an isomorphism of topological groups. ${\displaystyle \phi }$

Products, Limites and Kolimites

The forgetting functor , which assigns the underlying group to a locally compact group, has a left and a right adjunction, the left adjoint functor provides the group with the discrete topology, and the right adjoint functor with the lump topology. Thus, the forgetful functor gets Limites and Kolimites, i.e. H. every Limes (e.g. a product ) or Kolimes ( e.g. a coproduct ) is, if it exists, the corresponding Limes or Kolimes in the category of groups provided with a suitable topology.

The category of locally compact groups actually has finite products and its topology is the product topology . If one restricts oneself to the category of Hausdorff locally compact groups (the forget function in the category of groups then continues to receive limits), there are even any fiber products (for morphisms as the core of ) and the corresponding category is finitely complete . The product topology for a product of an infinite number of locally compact groups, on the other hand, is generally no longer locally compact - it is locally compact if and only if all but a finite number of factors are compact. In some cases, however, with a finer topology on the Cartesian product, a product in the category of Hausdorff locally compact groups is obtained. This is exactly the case when all but a finite number of factors have a compact, open normal divider, so that the associated quotient is torsion-free . The topology of the categorical product of such factors with compact, open normal parts can be characterized by the requirement that the product and the product topology form an open subspace. On the product , the topology is then given as the sum topology of the secondary classes of the normal divider , which is independent of the choice of . For example, the categorical product of any family of discrete, torsion-free groups (such as ) in that category is again discrete. ${\displaystyle f\colon F\to S,g\colon G\to S}$ ${\displaystyle F\times G\to S,(x,y)\mapsto f(x)g(x)^{-1}}$ ${\displaystyle G_{i}}$ ${\displaystyle K_{i}}$ ${\displaystyle \textstyle \prod _{i}K_{i}}$ ${\displaystyle \textstyle \prod _{i}G_{i}}$ ${\displaystyle \textstyle \prod _{i}K_{i}}$ ${\displaystyle K_{i}}$ ${\displaystyle \mathbb {Z} }$

Hair measure

→ Main article : Hair measure

On every Hausdorff locally compact group there is a regular Borel measure that is unique except for scaling , which is positive on non-empty open sets and is invariant under left shifts, the so-called left hair measure . Analogous to this, there is the right hair measure , which is invariant under right shifts . An important special case of locally compact groups with special properties are groups in which the left and right hair measurements coincide and are thus left and right invariant, so-called unimodular groups . The hair measure allows the integration on locally compact groups and plays a decisive role in the representation theory of locally compact groups.

Automatic continuity of homomorphisms

Every measurable homomorphism between locally compact groups is continuous. The condition can be further weakened that only the archetypes of open sets may be measurable and that homomorphism need not be guaranteed on certain zero sets .

Representations

For a locally compact group and a Hilbert space , a unitary representation of is a continuous homomorphism , where denotes the unitary group equipped with the strong (or the corresponding weak ) operator topology. Some central theorems of harmonic analysis allow far-reaching generalizations of the Fourier transformation to functions on certain locally compact groups by considering such a representation . ${\displaystyle G}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle G}$ ${\displaystyle \pi \colon G\to U({\mathcal {H}})}$ ${\displaystyle U({\mathcal {H}})}$

See also

• Group C * algebra
• Theorem of Open Mapping (Local Compact Groups)

literature

• Markus Stroppel: Locally Compact Groups . European Mathematical Society, 2006, ISBN 3-03719-016-7 .
• Yves de Cornulier, Pierre de la Harpe : Metric geometry of locally compact groups. normalesup.org (PDF).

Individual evidence

1. ↑ Ahmed Bouziad: Every Čech-Analytic Baire Semitopological Group is a Topological Group . In: Proceedings of the American Mathematical Society . tape 124 , no. 3 . American Mathematical Society, 1996, ISSN  0002-9939 , doi : 10.1090 / S0002-9939-96-03384-9 .
2. ^ Nicolas Bourbaki : V. Topological Vector Spaces (=  Elements of Mathematics ). Springer, Berlin 2003, ISBN 3-540-42338-9 , I, p. 15 (Original title: Éspaces vectoriels topologiques . Paris 1981. Translated by HG Eggleston and S. Madan).
3. ^ Sidney Allen Morris, Peter Nickolas: Locally compact group topologies on an algebraic free product of groups . In: Journal of Algebra . tape 38 , no. 2 . Academic Press, 1976, ISSN  0021-8693 , pp. 393-397 , doi : 10.1016 / 0021-8693 (76) 90229-5 .
4. ^ Nicolas Bourbaki : Topologie Générale (=  Éléments de mathématique ). Springer, Berlin 2007, ISBN 3-540-33936-1 , chap. 3 , p. 35 .
5. ↑ John L. Kelley : General Topology . Springer, New York 1955, ISBN 0-387-90125-6 , pp. 214-215 .
6. ↑ ^{a b} Bourbaki: Topologie Générale , Chapter 3, p. 22.
7. ^ Raimond A. Struble: Metrics in locally compact groups . In: Compositio Mathematica . tape 28 , no. 3 , 1974, ISSN  0010-437X , pp. 217–222 ( numdam.org [accessed February 13, 2013]).
8. ↑ Uffe Haagerup, Agata Przybyszewska: Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces . June 30, 2006, arxiv : math / 0606794 .
9. ↑ Gerald Budge Folland: A Course in Abstract Harmonic Analysis . CRC Press, 1995, ISBN 0-8493-8490-7 , pp. 33 .
10. ^ Walter Roelcke, Susanne Dierolf: Uniform structures on topological groups and their quotients . McGraw-Hill, ISBN 0-07-053412-8 , pp. 199 ( books.google.de ).
11. ↑ Roelcke, Dierolf, p. 95.
12. ↑ Hidehiko Yamabe : A Generalization of A Theorem of Gleason . In: Annals of Mathematics . tape 58 , no. 2 , 1953, ISSN  0003-486X , p. 351-365 , JSTOR : 1969792 .
13. ^ Sidney Allen Morris: Pontryagin Duality and the Structure of Locally Compact Abelian Groups . Cambridge University Press, Cambridge 1977, ISBN 978-0-521-21543-5 , pp. 125 .
14. ^ Iwasawa Kenkichi : On Some Types of Topological Groups . In: Annals of Mathematics . tape 50 , no. 3 , 1949, ISSN  0003-486X , pp. 507-558 , JSTOR : 1969548 .
15. ^ Morris: Pontryagin Duality and the Structure of Locally Compact Abelian Groups. P. 117.
16. ↑ Stephen Willard: General Topology . Addison-Wesley, 1970, pp. 131 .
17. ^ Karl Heinrich Hofmann , Sidney Allen Morris: Locally compact products and coproducts in categories of topological groups . In: Bulletin of the Australian Mathematical Society . tape 17 , no. 3 . Australian Mathematical Society, 1977, ISSN  0004-9727 , pp. 401-417 , doi : 10.1017 / S0004972700010674 .
18. ^ Adam Kleppner: Measurable Homomorphisms of Locally Compact Groups . In: Proceedings of the American Mathematical Society . tape 106 , no. 2 . American Mathematical Society, 1989, ISSN  0002-9939 , pp. 391-395 , JSTOR : 2048818 .