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 $G$.

Completeness

Again taking locally compact groups $G$ to be Hausdorff, such are complete both with respect to their left uniformity and their right uniformity. For if $\{x_\alpha\}$ is a Cauchy net in $G$ and $U$ is a compact neighborhood of the identity $e$, then there is $\alpha$ so large that $x_\beta x_\alpha^{-1} \in U$ for all $\beta \geq \alpha$. Those elements converge to a point $x \in U$ since $U$ is compact, and the original net converges to $x \cdot x_\alpha$. A similar argument is used for the right uniformity.

Pontrjagin duality

See at Pontrjagin duality.

Kazhdan’s property (T)

Kazhdan's property (T)

Related concepts

• compact topological group, compact Lie group

• maximal compact subgroup, closed subgroup

• locally compact groupoid

• coset space coprojections admitting local sections

References

• Linus Kramer, Locally Compact Groups, 2017 (pdf, pdf)

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

See also:

• Wikipedia, Locally compact group