Let A be a Banach algebra and let ϕ be a non-zero character on A. Suppose that A M is the closure of the faithful Banach algebra A in the multiplier norm. In this paper, topologically left invariant ϕ-means on A * M are defined and studied. Under some conditions on A, we will show that the set of topologically left invariant ϕ-means on A * and on A * M have the same cardinality. We also study the left uniformly continuous functionals associated with these algebras. The main applications are concerned with the Fourier algebra of an ultraspherical hypergroup H. In particular, we obtain some characterizations of discreteness of H.

Let $\mathbb{G}$ be a locally compact quantum group and let $I$ be a closed ideal of $L^{1}(\mathbb{G})$ with $y|_{I}\neq 0$ for some $y\in \text{sp}(L^{1}(\mathbb{G}))$ . In this paper, we give a characterization for compactness of $\mathbb{G}$ in terms of the existence of a weakly compact left or right multiplier $T$ on $I$ with $T(f)(y|_{I})\neq 0$ for some $f\in I$ . Using this, we prove that $I$ is an ideal in its second dual if and only if $\mathbb{G}$ is compact. We also study Arens regularity of $I$ whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of $\mathbb{G}$ in terms of the existence of some $I$ -module homomorphisms on $I^{\ast \ast }$ and on  $I^{\ast }$ .

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