Predual of the Multiplier Algebra of <i>A_p</i>(<i>G</i>) and Amenability

Abstract

Abstract For a locally compact group G and 1 &lt; p &lt; ∞, let A p ( G ) be the Herz-Figà-Talamanca algebra and let PM p ( G ) be its dual Banach space. For a Banach A p ( G )-module X of PM p ( G ), we prove that the multiplier space ℳ( A p ( G ); X *) is the dual Banach space of Q X , where Q X is the norm closure of the linear span A p ( G ) X of u f for u 2 A p ( G ) and f ∈ X in the dual of ℳ( A p ( G ); X *). If p = 2 and PF p ( G ) ⊆ X , then A p ( G ) X is closed in X if and only if G is amenable. In particular, we prove that the multiplier algebra MA p ( G ) of A p ( G ) is the dual of Q , where Q is the completion of L 1 ( G ) in the ‖ · ‖ M -norm. Q is characterized by the following: f ∈ Q if an only if there are u i ∈ A p ( G ) and f i ∈ PF p ( G ) ( i = 1; 2, … ) with such that on MA p ( G ). It is also proved that if A p ( G ) is dense in MA p ( G ) in the associated w *-topology, then the multiplier norm and ‖ · ‖ A p ( G ) -norm are equivalent on A p ( G ) if and only if G is amenable.

Locations

• Canadian Journal of Mathematics - View - PDF