nLab
locally compact topological group

Context

Topology

Group Theory

Contents

Definition

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

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

One often says just “locally compact group”.

Properties

In harmonic analysis

We take here locally compact groups GG 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 ρ\rho and that L 1(ρ)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 HH of a locally compact topological group GG is itself locally compact (in induced topology) iff it is closed in GG.

References

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

Revised on February 2, 2012 11:50:53 by Urs Schreiber (82.169.65.155)