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Concept# Locally compact group

Summary

In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on G so that standard analysis notions such as the Fourier transform and L^p spaces can be generalized.
Many of the results of finite group representation theory are proved by averaging over the group. For compact groups, modifications of these proofs yields similar results by averaging with respect to the normalized Haar integral. In the general locally compact setting, such techniques need not hold. The resulting theory is a central part of harmonic analysis. The representation theory for locally compact abelian groups is described by Pontryagin duality.
Examples and counterexamples

Official source

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Let G a locally compact group, H a closed subgroup and 1 < p < ∞. It's well-known that the restriction of the functions from G to H is a surjective linear contraction from Ap(G) onto Ap(H). We prove, when H is amenable, that every element in Ap(H) with compact support can be extended to an element in Ap(G) of which we can check norm and support. This result is already known in the case of normal subgroups and also for compact subgroups. We obtain the existence of a quasi-coretract in the BAN category, as a substitute of a morphism ΓH such that ResH ◦ ΓH = idAp(H). Indeed, for an amenable subsgroup, the morphism ΓH, a priori, doesn't exist. So, we construct a net of morphismes in BAN from Ap(H) into Ap(G), that converge to idAp(H) for the strong operator's topology on Ap(H) (that's for us the notion of a quasi-coretract in BAN). Furthermore, if H is metrizable and σ-compact we obtain, more precisely, a sequence. Moreover, our approach allows us to extend to the non-abelian case some works of H. Reiter and C. Herz concerning the spectral synthesis of bounded uniformly continuous functions. My results are new even for the Fourier algebra.

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Let G be a locally compact group and H a closed amenable subgroup of G. We prove that every element in A(p)(H) with compact support can be extended to an element of A(p)(G) of which we control the norm and support. The result is new even for the Fourier algebra. Our approach gives us new results concerning the operator norm closure of the convolution operators of G with compact support.

2009We investigate how probability tools can be useful to study representations of non-amenable groups. A suitable notion of "probabilistic subgroup" is proposed for locally compact groups, and is valuable to induction of representations. Nonamenable groups admit nonabelian free subgroups in that measure-theoretical sense. Consequences for affine actions and for unitarizability are then drawn. In particular, we obtain a new characterization of amenability via some affine actions on Hilbert spaces. Along the way, various fixed-point properties for groups are studied. We also give a survey of several useful facts about group representations on Banach spaces, continuity of group actions, compactness of convex hulls in locally convex spaces, and measurability pathologies in Banach spaces.