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Let ${\mathcal A}$ be a Banach algebra and let $\varphi $ be a non-zerocharacter on ${\mathcal A}$. Suppose that ${\mathcal A}_M$ is the closure ofthe faithful Banach algebra ${\mathcal A}$ in the multiplier norm. In thispaper, topologically left invariant $\varphi$-means on ${\mathcal A}_M^*$ aredefined and studied. Under some conditions on ${\mathcal A}$, we will show thatthe set of topologically left invariant $\varphi$-means on ${\mathcal A}^*$ andon ${\mathcal A}_M^*$ have the same cardinality. We also study the leftuniformly continuous functionals associated with these algebras. The mainapplications are concerned with the Fourier algebra of an ultrasphericalhypergroup $H$. In particular, we obtain some characterizations of discretenessof $H$.

Invariant $\varphi $-means on multiplier completion of Banach algebras with application to hypergroups