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