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For a compact Hausdorff space $X$, the space $SC(X\times X)$ of separatelycontinuous complex valued functions on $X$ can be viewed as a $C^*$-subalgebraof $C(X)^{**}\overline\otimes C(X)^{**}$, namely those elements which sliceinto $C(X)$. The analogous definition for a non-commutative $C^*$-algebra doesnot necessarily give an algebra, but we show that there is always a greatest$C^*$-subalgebra. This thus gives a non-commutative notion of separatecontinuity. The tools involved are multiplier algebras and row/column spaces,familiar from the theory of Operator Spaces. We make some study of morphismsand inclusions. There is a tight connection between separate continuity and thetheory of weakly almost periodic functions on (semi)groups. We use ournon-commutative tools to show that the collection of weakly almost periodicelements of a Hopf von Neumann algebra, while itself perhaps not a$C^*$-algebra, does always contain a greatest $C^*$-subalgebra. This allows usto give a notion of non-commutative, or quantum, semitopological semigroup, andto briefly develop a compactification theory in this context.

Non-commutative separate continuity and weakly almost periodicity for Hopf von Neumann algebras