Locally compact group

A locally compact group is a topological group in mathematics whose underlying topology is locally compact. This property allows us to generalize some known analytical concepts from Euclidean space in such a more general groups. These groups, particularly their representations are object of study of harmonic analysis.

Definition

A topological group is a group with join and neutral element provided with a topology such that both ( with the product topology ) and the inverses are continuous. A topological space is called locally compact if every point has a neighborhood base of compact sets. A locally compact group can, however, characterize less conditions also using: a group with a topology is exactly then a locally compact group, if

• The topology is präregulär,
• The neutral element has a compact neighborhood and
• Group forms a group semitopologische, i.e., the image in the two components is separately continuously, that is, for each picture, and the translation are continuous.

Due to the continuity of the left translation by the quantity is compact for each and due to the continuity of the left translation to is a around. Thus, each point has a compact environment, space is thus locally compact due to the Präregularität. Further considerations show that every locally compact group semitopologische actually has a simultaneous continuous operation ( ie a paratopologische group) and the inverses is continuous.

Some authors use the definition always precedes the Hausdorff property. It is usually sufficient (especially in representation theory ), be limited to such groups. Namely the Kolmogorov quotient is again a locally compact group that has essentially the same properties for every locally compact group. The formation of the Kolmogorov quotient is a functor linksadjungiert for embedding the Hausdorff locally compact groups in the category of locally compact groups ( with continuous homomorphisms as morphisms ).

Examples

• Each group is provided with the discrete topology, or the topology of a locally compact group lumps. ( The latter example, however, does not satisfy the Hausdorff axiom, which is provided with some authors in the definition of locally compact groups. )
• The Euclidean space forms with the addition, the multiplication and more generally, any Lie group is a locally compact group.
• For a lot of forms according to the rate of Tikhonov with the element-wise addition of a compact and thus locally compact group, for we speak of the Cantor group.
• The body of the p- adic numbers is the addition, the multiplication of a locally compact group. In general, this applies to all local bodies.
• A real or complex normed vector space with the addition of a topological group if and only lokalkomapkt if the space is finite-dimensional.
• More generally, a minimum of one-dimensional T ₀ topological vector space over a with respect to the induced by the addition of uniform structure complete, non- discrete topological division ring if and only with the addition of a locally compact group, if it is finite and the skew field locally compact.
• On the free product of at least two non-trivial groups, and in particular free groups, each Hausdorff locally compact group is discrete.
• The Kolmogorov quotient of each most countable locally compact group is discrete, this can be shown on the set of Baire or properties of the hair - measure.

Topological properties

Locally Compact Groups are like any locally compact space and every topological group completely regular. Moreover, they are even paracompact and thus normal. This can be deduced from the uniform local compactness, that is, from the fact that in the induced by the group structure on the left - or right-sided uniform structure exists a neighborhood such that for each is a compact neighborhood.

As for the left-side and right-side uniform structure locally compact topological groups are complete, ie every Cauchy filter converges. From the Metrisierbarkeitssatz of Urysohn follows as with any regular space that Metrisierbarkeit follows from the Zweitabzählbarkeit. Even exist for each zweitabzählbare locally compact group a metric that induces the topology invariant under left shifts and in which all the bounded, closed sets (as in the Euclidean space, by the theorem of Heine - Borel ) are compact, so that the metric in particular completely and thus zweitabzählbare any locally compact group is a Polish group.

Among groups and quotients

A subgroup of a locally compact group is locally compact if and only turn when it is completed. The execution is valid for arbitrary subsets of locally compact spaces are not (consider as a non-trivial open subset of Euclidean space ). It results from the fact that each complete subspace of a uniform space is complete. If completed, the space of left cosets with the quotient topology is a locally compact homogeneous space, operates on the by left multiplication. Is a closed subgroup even a normal subgroup, then the quotient group is again a locally compact group.

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

For each group, and a topological locally compact subset of the space of the left side of the classes with respect to the quotient of the right side of uniform structure by, i.e., with respect to the canonical Finaluniformität surjection of after entirely. For each discrete subgroup, a topological group is locally compact if and only if the space is locally compact.

Structure

Every Hausdorff locally compact group can be approximated by Lie groups in a certain sense: Each such group has an open subgroup such that for every neighborhood of the neutral element exists a subset, the compact normal subgroup of is so that a Lie group. Thus, Every connected, locally compact Hausdorff group has a compact normal subgroup, so that a Lie group and subgroup is a product of Lie groups.

Even before this statement was shown was proved that every connected locally compact group which meets these approximation property (ie, every Hausdorff, as we know today ) is homeomorphic to a natural number and a compact group ( with neutral element ). A homeomorphism can be chosen so that all constraints and are isomorphisms of topological groups.

For coherent maximum of almost -periodic groups, ie groups whose finite-dimensional unitary representations are separating points, this includes all abelian groups can even choose entirely as a topological isomorphism of groups.

Products, Limites and colimits

The forgetful, of a locally compact group functions for the underlying group has a left and a Rechtsadjunktion, the linksadjungierte functor equips the group with the discrete topology, the rechtsadjungierte functor with the lump topology. Thus, the forgetful Limites and colimits receives, that is, each Limes (about a product ) or colimit (about a coproduct ) is if he exists, is the corresponding limit or colimit in the category of groups provided with a suitable topology.

The category of locally compact groups actually has finite products and their topology is the product topology. If one limits on ( the forgetful then gets into the category of groups continue Limites ) on the category of Hausdorff locally compact groups even exist any fiber products ( for morphisms as the core of ) and the corresponding category is finally complete. The product topology on the product of an infinite number of locally compact groups, however, is generally not locally compact - it is exactly then locally compact if all are compact but finitely many factors. In some cases, however, with a finer topology on the Cartesian product gives a product in the category of Hausdorff locally compact groups. This is exactly the case when all but finitely many factors have a compact open normal subgroup, so that the corresponding quotient is torsion. The topology of the categorical product of such factors with compact open normal subgroups can be characterized by the requirement that the product may form an open subspace with the product topology. On the product, the topology is then given as the sum of the topology of the cosets of the normal divider, which is independent of the choice of. For example, the categorical product of any family of discrete, torsion-free groups, in turn, is (like ) in this category discreet.

Hair measure

On every Hausdorff locally compact group there exists a unique up to scaling regular Borel measure which is positive on non-empty open sets and invariant under left shifts, the so-called left Hair measure. Similarly, the right Hair measure which is invariant under right shifts exist. An important special case of locally compact groups with special properties form groups that share left and right Hair measure and thus are left and rechtsinvariant called unimodular groups. The Hair measure allows the integration on locally compact groups and plays a crucial role in the representation theory of locally compact groups.

Automatic continuity of homomorphisms

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

Representations

For a locally compact group and a Hilbert space is a unitary representation of a continuous homomorphism, where the unitary group equipped with the strong (or the matching weak) operator topology call. Some of the key sets of harmonic analysis allow by considering such representation far-reaching generalizations of the Fourier transform to functions on certain locally compact groups.