Positivity (2005) 9:485–490 © Springer 2005 DOI 10.1007/s11117-005-8111-8 Equicontinuity in Measure Spaces and von Neumann Algebras J.K. BROOKS Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL 32611–8105 USA (E-mail: email@example.comﬂ.edu) Received 9 September 2004; accepted 25 October 2004 1. Introduction An announcement of the results presented here formed a portion of a talk given at the 2003 Positivity Conference dedicated to Y. A. Abramovich and A. C. Zaanen. In this note we shall present a few aspects of equicontinuity, uniform absolute continuity and weak compactness in measure space (the commu- tative setting) and von Neumann algebras (the non-commutative setting). Recall that a family K of ﬁnitely-additive Banach-valued set functions deﬁned on a ring of sets is equicontinuous if lim m(R ) = 0, uniformly for m ∈ K , whenever (R ) is a disjoint sequence of sets from the ring. The C -algebra notion of equicontinuity deals with K as a subset of the dual space of a C -algebra A, and (R ) is an orthogonal sequence of self-adjoint elements in the unit ball of A. The relationship between equicontinuity and uniform absolute continuity is the author’s main theorem in , which states that
Positivity – Springer Journals
Published: May 30, 2005
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