Structural transition between $L^{p}(G)$ and $L^{p}(G/H)$

Abstract

Let H be a compact subgroup of a locally compact group G. We consider the homogeneous space G/H equipped with a strongly quasi-invariant Radon measure µ. For 1 ≤ p ≤ +∞, we introduce a norm decreasing linear map from L p (G) onto L p (G/H, µ) and show that L p (G/H, µ) may be identified with a quotient space of L p (G). Also, we prove that L p (G/H, µ) is isometrically isomorphic to a closed subspace of L p (G). These help us study the structure of the classical Banach spaces constructed on a homogeneous space via those created on topological groups.

Abstract Structure of Measure Algebras on Coset Spaces of Compact Subgroups in Locally Compact Groups

“…Therefore, the linear map T H has a unique extension to a norm-decreasing linear map in L p -sense, denoted by [20]. By applying a canonical normalization of the linear operator T H given by (2.8), most results of [20] concerning analytic properties of the linear map T H extended to the case that the measure µ is not G invariant, see [35]. Authors of [35] did not clearly mention that they exteneded the results of Ghaani Farashahi in [18,20].…”

“…By applying a canonical normalization of the linear operator T H given by (2.8), most results of [20] concerning analytic properties of the linear map T H extended to the case that the measure µ is not G invariant, see [35]. Authors of [35] did not clearly mention that they exteneded the results of Ghaani Farashahi in [18,20]. Instead, the authors cited [20] in the introduction with an incomplete statement about the content therein.…”

“…By employing a canonical normalization of the linear operator T H given by (2.8), the convolution of functions introduced by (3.5) of [18] extended to the case that the measure µ is not relatively invariant, see [23,35]. Authors of [35] did not cite [18]. Also, authors of [23] did not cite [18].…”

Character amenability and contractibility of some Banach algebras on left coset spaces

“…ρ(xξ) dξ for almost all xH ∈ G/H. Then, it has been shown that L 1 (G/H, µ) becomes a Banach algebra by multiplication f * g = T 1 (f ρ * g ρ ), where f ρ , g ρ ∈ L 1 (G) are defined by f ρ (x) = ρ(x)f (xH) and g ρ (x) = ρ(x)g(xH) for almost all x ∈ G (see [12]). Also, one can easily show that for each f, g ∈ L 1 (G/H) we have…”

“…The term homogeneous space refers to a transitive G-space which is topologically isomorphic to G/H, the space of all left cosets of a closed subgroup H in a locally compact Hausdorff topological group G. In [3] and [6], the authors introduced and investigated the Fourier algebra A(G/H) and the Fourier-Stieltjes algebra B(G/H), where H is compact. In [12], a bounded surjective linear map T p : L p (G) → L p (G/H) was introduced using a compact subgroup H of G and equipping the homogeneous space G/H with a strongly quasi-invariant Radon measure. The authors also showed that the restriction of T p to a special closed subspace L p (G : H) of L p (G) is an isometric isomorphism for all 1 ≤ p ≤ ∞.…”