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Boundary representations of operator spaces, and compact rectangular matrix convex sets

Fuller, Adam H. and Hartz, Michael and Lupini, Martino (2018) Boundary representations of operator spaces, and compact rectangular matrix convex sets. Journal of Operator Theory, 79 (1). pp. 139-172. ISSN 1841-7744. http://resolver.caltech.edu/CaltechAUTHORS:20180410-161452217

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Abstract

We initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein--Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed by its boundary representations. This yields a canonical construction of the triple envelope of an operator space.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.7900/jot.2017jan28.2165DOIArticle
http://www.mathjournals.org/jot/2018-079-001/2018-079-001-006.htmlPublisherArticle
https://arxiv.org/abs/1610.05828arXivDiscussion Paper
ORCID:
AuthorORCID
Lupini, Martino0000-0003-1588-7057
Additional Information:© 2018 Theta Foundation. M.H. was partially supported by an Ontario Trillium Scholarship and a Feodor Lynen Fellowship. M.L. was partially supported by the NSF Grant DMS-1600186. This work was initiated during a visit of M.H. at the California Institute of Technology in the Spring 2016, and continued during a visit of M.H. and M.L. at the Oberwolfach Mathematics Institute supported by an Oberwolfach Leibnitz Fellowship. The authors gratefully acknowledge the hospitality and the financial support of both institutions.
Funders:
Funding AgencyGrant Number
Ontario Trillium ScholarshipUNSPECIFIED
Alexander von Humboldt FoundationUNSPECIFIED
NSFDMS-1600186
CaltechUNSPECIFIED
Oberwolfach Mathematics InstituteUNSPECIFIED
Subject Keywords:operator space, operator system, boundary representation, compact matrix convex set, matrix-gauged space
Classification Code:2000 Mathematics Subject Classification. Primary 46L07, 47L25; Secondary 46E22, 47L07
Record Number:CaltechAUTHORS:20180410-161452217
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20180410-161452217
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85738
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:11 Apr 2018 14:59
Last Modified:03 Aug 2018 17:29

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