dual group of G is homeomorphic to the character space of L1(G)


Let G be a locally compact abelianMathworldPlanetmath (http://planetmath.org/AbelianGroup2) group (http://planetmath.org/TopologicalGroup) and L1(G) its group algebraPlanetmathPlanetmath.

Let G^ denote the Pontryagin dual of G and Δ the character space of L1(G), i.e. the set of multiplicative linear functionals of L1(G) endowed with the weak-* topologyMathworldPlanetmath.

Theorem - The spaces G^ and Δ are homeomorphic. The homeomorphism is given by

ωϕω,ωG^

where ϕωΔ is defined by

ϕω(f):=Gf(s)ω(s)𝑑μ(s),fL1(G)
Title dual group of G is homeomorphic to the character space of L1(G)
Canonical name DualGroupOfGIsHomeomorphicToTheCharacterSpaceOfL1G
Date of creation 2013-03-22 17:42:49
Last modified on 2013-03-22 17:42:49
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 5
Author asteroid (17536)
Entry type Theorem
Classification msc 46K99
Classification msc 43A40
Classification msc 43A20
Classification msc 22D20
Classification msc 22D15
Classification msc 22D35
Classification msc 22B10
Classification msc 22B05
Related topic L1GIsABanachAlgebra