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Concept# Compact group

Summary

In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.
In the following we will assume all groups are Hausdorff spaces.
Compact Lie groups
Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include

- the circle group T and the torus groups Tn,
- the orthogonal group O(n), the special orthogonal group SO(n) and its covering spin group Spin(n),
- the unitary group U(n) and the special unitary group SU(n),
- the compact forms of the exceptional Lie groups: G2, F4, E6, E7, and

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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.

2009Let 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 1 < p < infinity, let G and H be locally compact groups and let c) be a continuous homomorphism of G into H. We prove, if G is amenable, the existence of a linear contraction of the Banach algebra CVp (G) of the p-convolution operators on G into CVp (H) which extends the usual definition of the image of a bounded measure by omega. We also discuss the uniqueness of this linear contraction onto important subalgebras of CVp(G). Even if G and H are abelian, we obtain new results. Let G(d) denote the group G provided with a discrete topology. As a corollary, we obtain, for every discrete measure, vertical bar parallel to mu vertical bar parallel to CVp(G)