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Let *A* be a C^{*}‐algebra and let *K* be a relatively weakly compact subset of the dual of *A*. Let ψbe a positive linear functional on *A* such that, for each φin *K*, φis strongly absolutely continuous with respect to ψ. Then, for each ε> 0, there exists δ> 0, such that for each *x* in the closed unit ball of *A*, ψ(*xx*^{*} +*x*^{*}*x*)^{1/2} ≤δimplies |φ(*x*) |≤εfor every φ∈*K*. This result is extended to the situation where *K* is a σ‐bounded set of weakly compact operators from *A* to a Banach space *Y*.

*Journal Article.*
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*Subjects: *
Pure Mathematics

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