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