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The Quarterly Journal of Mathematics 2004 55(1):31-40; doi:10.1093/qmath/hag037
© 2004 by Oxford University Press
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When absolute continuity on C-algebras is automatically uniform

J. K. Brooks1, Kazuyuki SAITÔ2 and J. D. Maitland Wright3

1 Department of Mathematics, University of Florida, 358 Little Hall, Gainsville FL32611-8105, USA 2 Mathematical Institute, Tôhoku University, Sendai 980-8578, Japan 3 Mathematics Department, University of Reading, Reading RG6 6AX

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 {varepsilon}> 0, there exists {delta}> 0, such that for each x in the closed unit ball of A, {psi}(xx* +x*x)1/2 ≤{delta}implies |{phi}(x) |≤{varepsilon}for every {phi}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.

Received 16 December 2002. Revised 10 June 2003.


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