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.
For a locally compact group G and a compact subgroup H, we show that the Banach space M (G/H) may be considered as a quotient space of M (G). Also, we define a convolution on M (G/H) which makes it into a Banach algebra. It may be identified with a closed subalgebra of the involutive Banach algebra M (G), and there is no involution on M (G/H) compatible with this identification unless H is a normal subgroup of G. In other words,As well, it is a unital Banach algebra just when H is a normal subgroup. Furthermore, when G/H is attached to a strongly quasi-invariant measure, L 1 (G/H) is a Banach subspace of M (G/H). Using the restriction of the convolution on M (G/H), we obtain a Banach algebra L 1 (G/H), which may be considered as a Banach subalgebra of L 1 (G), with a right approximate identity. It has no involution and no left approximate identity except for a normal subgroup H. Consequently, the Banach algebra L 1 (G/H) is amenable if and only if H is a normal subgroup and G is amenable.
Associated with a locally compact group G and a G-space X there is a Banach subspace LU C(X , G) of C b (X ), which has been introduced and studied by Lau and Chu in [4]. In this paper, we study some properties of the first dual space of LU C(X , G). In particular, we introduce a left action of LU C(G) * on LU C(X , G) * to make it a Banach left module and then we investigate the Banach subalgebra Z(X , G) of LU C(G) * , as the topological centre related to this module action, which contains M (G) as a closed subalgebra. Also, we show that the faithfulness of this module action is related to the properties of the action of G on X and we extend the main results of Lau [14] from locally compact groups to G-spaces. Sufficient and/or necessary conditions for the equality Z(X , G) = M (G) or LU C(G) * are given. Finally, we apply our results to some special cases of G and X for obtaining various examples whose topological centres Z(X , G) are M (G), LU C(G) * or neither of them.
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