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Operator-Valued Measures, Dilations, and the Theory of Frames
Deguang Han, University of Central Florida, Orlando, Florida, David R. Larson, Texas A&M University, College Station, Texas, Bei Liu, Tianjin University of Technology, China, and Rui Liu, Nankai University, Tianjuin, China

Memoirs of the American Mathematical Society
2013; 84 pp; softcover
Volume: 229
ISBN-10: 0-8218-9172-3
ISBN-13: 978-0-8218-9172-8
List Price: US$65
Individual Members: US$39
Institutional Members: US$52
Order Code: MEMO/229/1075
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The authors develop elements of a general dilation theory for operator-valued measures. Hilbert space operator-valued measures are closely related to bounded linear maps on abelian von Neumann algebras, and some of their results include new dilation results for bounded linear maps that are not necessarily completely bounded, and from domain algebras that are not necessarily abelian. In the non-cb case the dilation space often needs to be a Banach space. They give applications to both the discrete and the continuous frame theory. There are natural associations between the theory of frames (including continuous frames and framings), the theory of operator-valued measures on sigma-algebras of sets, and the theory of continuous linear maps between \(C^*\)-algebras. In this connection frame theory itself is identified with the special case in which the domain algebra for the maps is an abelian von Neumann algebra and the map is normal (i.e. ultraweakly, or \(\sigma\) weakly, or w*) continuous.

Table of Contents

  • Introduction
  • Preliminaries
  • Dilation of operator-valued measures
  • Framings and dilations
  • Dilations of maps
  • Examples
  • Bibliography
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