When absolute continuity on C*-algebras is automatically uniform

Let A be a C*-algebra and let K be a relatively weakly compact subset of the dual of A. Let psi be a positive linear functional on A such that, for each phi in K, phi is strongly absolutely continuous with respect to psi. Then, for each epsilon > 0, there exists delta > 0, such that for each x in the closed unit ball of A, psi (xx* + x*x) 1/2_less than or equal to delta implies \phi(x)\ less than or equal to epsilon for every phi is an element of K. This result is extended to the situation where K is a sigma-bounded set of weakly compact operators from A to a Banach space Y.