WEAKLY COMPACT MULTIPLIERS AND φ-AMENABLE BANACH ALGEBRAS

Let A be a Banach algebra and let J be a closed ideal of A such that phi vertical bar J not equal 0 for some nonzero character phi on A. We obtain some relations between the existence of compact and weakly compact multipliers on J and on A in some sense. Then we apply these results to hypergroup algebra L-1(K) when K is a locally compact hypergroup. In particular, for a closed ideal J in L-1 (K) we prove that K is compact if and only if there is f is an element of J such that phi(1) (f) not equal 0 and the multiplication operator lambda(f) : g bar right arrow g * f weakly compact on J. Using this, we study Arens regularity of J whenever it has a bounded left approximate identity. Finally, we apply these results on some abstract Segal algebras with respect to the L-1(k).