nLab
locally compact topological group

Definition

A priori a locally compact topological group is a topological group $G$ whose underlying topological space is locally compact.

Typically it is also assumed that $G$ is Hausdorff. (Notice that if not, then $G/\overline{\{1\}}$ is Hausdorff.).

One often says just “locally compact group”.

Properties

In harmonic analysis

We take here locally compact groups $G$ to be also Hausdorff.

Locally compact topological groups are the standard object of study in classical abstract harmonic analysis. The crucial properties of locally compact groups is that they posses a left (right) Haar measure $\rho$ and that $L^1(\rho )$ has a structure of a Banach $*$-algebra.

A left (right) Haar measure on a locally compact topological group is a nonzero Radon measure which is invariant under the left (right) multiplications by elements in the group. A topological subgroup $H$ of a locally compact topological group $G$ is itself locally compact (in induced topology) iff it is closed in $G$.

Related concepts

• compact topological group, compact Lie group

• maximal compact subgroup

References

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995