Connes-amenability of multiplier Banach algebras

Abstract

Let B be a Banach algebra with bounded approximate identity, and let M(B) be its multiplier algebra. If there exists a continuous linear injection B∗ → M(B) such that, for every b ∈ B and every u, v ∈ B∗, 〈u, vb〉B = 〈v, bu〉B , then M(B) is a dual Banach algebra and the following are equivalent: (i) B is amenable; (ii) M(B) is Connes amenable; (iii) M(B) has a normal, virtual diagonal.