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

Abstract

For a compact Hausdorff space $X$, the space $SC(X\times X)$ of separately continuous complex valued functions on $X$ can be viewed as a $C^{\ast }$-subalgebra of $C(X{)}^{\ast \ast }\overline{\otimes }C(X{)}^{\ast \ast }$, namely those elements which slice into $C(X)$. The analogous definition for a non-commutative $C^{\ast }$-algebra does not necessarily give an algebra, but we show that there is always a greatest $C^{\ast }$-subalgebra. This thus gives a non-commutative 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 $C^{\ast }$-algebra, does always contain a greatest $C^{\ast }$-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.

17 pages; minor corrections; to appear in the Journal of Functional Analysis

References in corpus (1)

• Remarks on the Quantum Bohr Compactification

Cited by in corpus (3)

• Invariant means on subspaces of quantum weakly almost periodic functionals
• Quantum semigroups: A survey
• Quantum semigroups generated by locally compact semigroups