Multipliers and Ideals in Second Conjugate Algebras Related to Locally Compact Groups

Let G be a locally compact group. In this paper we study compact or weakly compact multipliers on the second conjugate algebras L1(G)** and M(G)**. We prove, among other things, that G is amenable if and only if there is a non-zero compact or weakly compact right multiplier on L1(G)** or M(G)**. We show that if G is σ-compact non-compact, then L1(G)** cannot have any non-zero weakly compact left multipliers T with T(n)(1) ≠ 0 for some n ∈ L1(G)**. We also study maximal regular ideals of L1(G, w) when G is a locally compact abelian group and w is a weight function on G, and Ap(G)**, 1 < p < ∞ on a locally compact group G.