A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is then a comonoid object in this monoidal category. It's straightforward to write down what a Banach *-coalgebra is too. It's a little less obvious what a C*-coalgebra is, and I don't know if that term appears in the literature, but I've written down my definition.
Generally, the dual space of a coalgebra is an algebra (but not conversely), and that works here too: the dual of a C*-coalgebra is a C*-algebra. But not every C*-algebra arises in this way; obviously, since all of these C*-algebras have preduals (having been explicitly constructed so), they are W*-algebras. But I don't know what other conditions must be satisfied.
So my question is, and I'll be grateful for incomplete answers: Which W*-algebras arise (up to abstract isomorphism) as duals of C*-coalgebras?
Partial answers: The sequence space $l^\infty$ is the dual of $l^1$, and $l^1$ is a C*-coalgebra. But this doesn't work for $L^\infty(R)$; this is the dual of $L^1(R)$, but I can't make $L^1(R)$ into even a Banach coalgebra (in an appropriate way), essentially because the diagonal in $R^2$ has measure zero. (Unless I've miscalculated something, and I'm trying to do the wrong thing here.) Of course, these are quite limited examples: they're (co)commutative. I'd be grateful for more.