Absolute continuity on C*-algebras

In an earlier work the notion of absolute continuity was extended from finitely additive measures to non-commutative C*-algebras. But to obtain a generalisation of the Vitali–Hahn–Saks theorem valid for all C*-algebras it was necessary to introduce ‘weak’ and ‘strong’ absolute continuity. For commutative algebras, these two notions of absolute continuity coincide but, given recent work by Chetcuti and Hamhalter, it is reasonable to ask if there are wider classes of C*-algebras for which weak and strong absolute continuity coincide.We show here that this is not true. If weak and strong absolute continuity coincide for a given algebra then the algebra must be commutative.