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.

A locally compact group G is called Lie projective if it is the projective limit underset ←lim_{α∈{A!!!G_α of Lie groups G_α:=G/K_α with a descending family (K_α)_(α∈𝔸) of compact normal subgroups K_α of G satisfying ∩_(α∈𝔸)K_α=e.}}

Let G be a locally compact group and H a closed subgroup of G, and suppose that π and τ are irreducible representations of G and H, respectively.

Locally compact group

Meaning

Noun