For a Banach algebra $A$ with a bounded approximate identity, we investigate the $A$-module homomorphisms of certain introverted subspaces of $A^*$, and show that all $A$-module homomorphisms of $A^*$ are normal if and only if $A$ is an ideal of $A^{**}$. We obtain some characterizations of compactness and discreteness for a locally compact quantum group $\G$. Furthermore, in the co-amenable case we prove that the multiplier algebra of $\LL$ can be identified with $\MG.$ As a consequence, we prove that $\G$ is compact if and only if $\LUC={\rm WAP}(\G)$ and $\MG\cong\mathcal{Z}({\rm LUC}(\G)^*)$; which partially answer a problem raised by Volker Runde.Comment: The detailed proof of Lemma 4.1 is added in addendum. 11 pages, To appear in J. Math. Anal. App

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. Let A and X be Banach algebras and let X be an algebraic Banach A−module. Then the ℓ 1 −direct sum A × X equipped with the multiplicationis a Banach algebra, denoted by A ⊲⊳ X, which will be called "a generalized module extension Banach algebra". Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. We characterize the structure of n−dual valued (n ∈ N)) derivations on A ⊲⊳ X from which we investigate the n−weak amenability for the algebra A ⊲⊳ X. We apply the results and the techniques of proofs for presenting some older results with simple direct proofs.

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Abstract. Let G be a von Neumann algebraic locally compact quantum group, in the sense of Kustermans and Vaes. In this paper, as a consequence of a notion of amenability for actions of Lau algebras, we show that G, the dual of G, is co-amenable if and only if there is a state m ∈ L ∞ ( G) * which is invariant under a left module action of L 1 (G) on L ∞ ( G) * . This is the quantum group version of a result by Stokke [17]. We also characterize amenable action of Lau algebras by several properties such as fixed point property. This yields in particular, a fixed point characterization of amenable groups and H-amenable representation of groups.