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The Quarterly Journal of Mathematics Advance Access published online on June 2, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap018
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© 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

NON-COMMUTATIVE LOCALLY CONVEX MEASURES

José Bonet{dagger}

Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universidad Politécnica de Valencia, E-46071 Valencia, Spain

J. D. Maitland Wright {ddagger}

Mathematical Sciences, Kings College, University of Aberdeen, Aberdeen AB24 3UE, UK

{dagger} Corresponding author. E-mail: jbonet{at}mat.upv.es

Received 9 February 2009; revised 29 April 2009
  Abstract

We study weakly compact operators from a C*-algebra with values in a complete locally convex space. They constitute a natural non-commutative generalization of finitely additive vector measures with values in a locally convex space. Several results of Brooks, Saîto and Wright are extended to this more general setting. Building on an approach due to Saîto and Wright, we obtain our theorems on non-commutative finitely additive measures with values in a locally convex space, from more general results on weakly compact operators defined on Banach spaces X whose strong dual X' is weakly sequentially complete. Weakly compact operators are also characterized by a continuity property for a certain ‘Right topology’ as in joint work by Peralta, Villanueva, Wright and Ylinen.

{ddagger} E-mail: j.d.m.wright{at}abdn.ac.uk


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