The author develops two kinds of cohomology theories for Hopf \(C^*\)-algebras and Hopf von Neumann algebras. The first cohomology theory, termed a natural cohomology, is based on an approach that uses the ‘dual analogue’ of the Banach algebra cohomology. The second theory, termed a dual cohomology, is a generalisation of the group cohomology. The author begins by studying multipliers of operator bimodules and then gives the definition and examples of comodules of Hopf \(C^*\)-algebras. Next, cohomology theories of Hopf \(C^*\)-algebras are developed.
Among the main results, it is shown that the dual cohomology of a saturated unital Hopf \(C^*\)-algebra \(S\) is concentrated in degree zero if and only if \(S\) has a counit and \(S\otimes S\) has a codiagonal. This is a Hopf \(C^*\)-algebra version of the fact that the Cartier cohomology of a coalgebra \(C\) is concentrated in degree zero if and only if \(C\) is a coseparable coalgebra (cf. Theorem 3 in [Y. Doi, J. Math. Soc. Japan 33, 31–50 (1981; Zbl 0459.16007)]). A codiagonal in a Hopf \(C^*\)-algebra corresponds to a cointegral in a coseparable coalgebra. It is also proven that, for a locally compact group \(G\), the vanishing of the first dual cohomology group of the reduced group \(C^*\)-algebra \(C^*_r(G)\) is equivalent to the amenability of \(G\).
Next, the author defines comodules of Hopf von Neumann algebras and constructs cohomology theories in this case. In difference to the Hopf \(C^*\)-algebra case, it is shown that there is a natural Hopf von Neumann algebra comodule structure on the dual space of a comodule, and that the dual cohomology with coefficients in a given comodule coincides with the natural cohomology with coefficients in the corresponding dual comodule.