Non-commutative separate continuity and weakly almost periodicity for Hopf von Neumann algebras
نویسنده
چکیده
For a compact Hausdorff space X , the space SC(X ×X) of separately continuous complex valued functions on X can be viewed as a C∗-subalgebra of C(X)∗∗⊗C(X)∗∗, namely those elements which slice into C(X). The analogous definition for a non-commutative C∗-algebra does not necessarily give an algebra, but we show that there is always a greatest C∗-subalgebra. This thus gives a noncommutative notion of separate continuity. The tools involved are multiplier algebras and row/column spaces, familiar from the theory of Operator Spaces. We make some study of morphisms and inclusions. There is a tight connection between separate continuity and the theory of weakly almost periodic functions on (semi)groups. We use our non-commutative tools to show that the collection of weakly almost periodic elements of a Hopf von Neumann algebra, while itself perhaps not a C∗-algebra, does always contain a greatest C∗-subalgebra. This allows us to give a notion of non-commutative, or quantum, semitopological semigroup, and to briefly develop a compactification theory in this context.
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