Connes-amenability of multiplier Banach algebras

Abstract

Let $B$ be a Banach algebra with bounded approximate identity, and let $M\left(B\right)$ be its multiplier algebra. If there exists a continuous linear injection $B^{*}\to M\left(B\right)$ such that, for every $b\in B$ and every $u,v\in B^{*}$, $\langle u,vb{\rangle }_B=\langle v,bu{\rangle }_B$, then $M\left(B\right)$ is a dual Banach algebra and the following are equivalent:

(i) $B$ is amenable;

(ii) $M\left(B\right)$ is Connes amenable;

(iii) $M\left(B\right)$ has a normal, virtual diagonal.