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Recent results of Zsidó, based on his previous work with Niculescu and Ströh, on actions of topological semigroups on von Neumann algebras, give a Jacobs–de Leeuw–Glicksberg splitting theorem at the von Neumann algebra (rather than Hilbert space) level. We generalize this to the framework of actions of quantum semigroups, namely Hopf–von Neumann algebras. To this end, we introduce and study a notion of almost periodic vectors and operators that is suitable for our setting.

Ergodic theory for quantum semigroups