locally compact group

locally compact group (plural locally compact groups)

1. (topology) A topological group whose underlying topology is both locally compact and Hausdorff.
   • 1988, J. M. G. Fell, R. S. Doran, Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Volume 1, Academic Press, page 63:

     Indeed, if there is one property of locally compact groups more responsible than any other for the rich positive content of their representation theory, it is their possession of left-invariant and right-invariant (Haar) measures. The connection between the representation theory of a locally compact group and that of general Banach algebras proceeds directly from Lebesgue integration with respect to Haar measure on the group.

   • 2012, H. Heyer, Probability Measures on Locally Compact Groups, Springer, page 12:

     A locally compact group ${\displaystyle G}$ is called Lie projective if it is the projective limit ${\displaystyle \textstyle {\underset {\leftarrow }{\lim }}_{\alpha \in \mathbb {A} }\!\!\!G_{\alpha }}$ of Lie groups ${\displaystyle \textstyle G_{\alpha }:=G/K_{\alpha }}$ with a descending family ${\displaystyle \textstyle (K_{\alpha })_{\alpha \in \mathbb {A} }}$ of compact normal subgroups ${\displaystyle K_{\alpha }}$ of ${\displaystyle G}$ satisfying ${\displaystyle \textstyle \cap _{\alpha \in \mathbb {A} }K_{\alpha }=\{e\}}$.

   • 2013, Eberhard Kaniuth, Keith F. Taylor, Induced Representations of Locally Compact Groups, Cambridge University Press, page 269:

     Let ${\displaystyle G}$ be a locally compact group and ${\displaystyle H}$ a closed subgroup of ${\displaystyle G}$, and suppose that ${\displaystyle \pi }$ and ${\displaystyle \tau }$ are irreducible representations of ${\displaystyle G}$ and ${\displaystyle H}$, respectively.

• Lie group