Operator Connes-amenability of completely bounded multiplier Banach algebras

Abstract

For a completely contractive Banach algebra B, we find conditions under which the completely bounded multiplier algebra Mcb(B) is a dual Banach algebra and the operator amenability of B is equivalent to the operator Connes-amenability of Mcb(B). We also show that, in this case, these are equivalent to the existence of a normal virtual operator diagonal.