We define the concept of module Connes amenability for dual Banach algebras which are also Banach modules with a compatible action. We distinguish a closed subhypergroup K0 of a locally compact measured hypergroup K, and show that, under different actions, amenability of K, M.K0/-module Connes amenability of M.K/, and existence of a normal M.K0/-module virtual diagonal are related.
We show that an unbounded measure on a strong commutative hypergroup is transformable if and only if its convolution with any positive definite function of compact support is positive definite.
For a completely contractive Banach algebra , we find conditions under which the completely bounded multiplier algebra is a dual Banach algebra and the operator amenability of is equivalent to the operator Connes-amenability of . We also show that, in this case, these are equivalent to the existence of a normal virtual operator diagonal.
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