locally compact topological group



Group Theory



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”.


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.


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


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

Revised on September 7, 2015 03:12:02 by Todd Trimble (