Operator-valued measures, dilations, and the theory of frames

98 Pages·2014·0.63 MB·English

by  Deguang Han

MEMOIRS
of the
American Mathematical Society
Volume 229 • Number 1075 (second of 5 numbers) • May 2014
Operator-Valued Measures,
Dilations, and the Theory of Frames
Deguang Han
David R. Larson
Bei Liu
Rui Liu
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 229 • Number 1075 (second of 5 numbers) • May 2014
Operator-Valued Measures,
Dilations, and the Theory of Frames
Deguang Han
David R. Larson
Bei Liu
Rui Liu
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Han,Deguang,1959-author.
Operator-valued measures, dilations, and the theory of frames / Deguang Han, David R.
Larson,BeiLiu,RuiLiu.
pagescm. –(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;number1075)
“May2014,volume229,number1075(secondof5numbers).”
Includesbibliographicalreferences.
ISBN978-0-8218-9172-8(alk. paper)
1.Vector-valuedmeasures. 2.Operatorspaces. 3.VonNeumannalgebras. 4.Operator
theory. I.Title.
QA325.H36 2014
512(cid:2).556–dc23 2013051213
DOI:http://dx.doi.org/10.1090/memo/1075
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Contents
Introduction 1
Chapter 1. Preliminaries 7
1.1. Frames 7
1.2. Operator-valued Measures 12
Chapter 2. Dilation of Operator-valued Measures 17
2.1. Basic Definitions 17
2.2. Dilation Spaces and Dilations 21
2.3. Elementary Dilation Spaces 23
2.4. The Minimal Dilation Norm (cid:3)·(cid:3) 30
α
2.5. The Dual Space of the Minimal Dilation Space 38
2.6. The Maximal Dilation Norm (cid:3)·(cid:3) 39
ω
Chapter 3. Framings and Dilations 45
3.1. Hilbertian Dilations 45
3.2. Operator-Valued Measures Induced by Discrete Framings 48
3.3. Operator-Valued Measures Induced by Continuous Frames 52
Chapter 4. Dilations of Maps 59
4.1. Algebraic Dilations 59
4.2. The Commutative Case 60
4.3. The Noncommutative Case 66
Chapter 5. Examples 73
Bibliography 83
v

Abstract
Wedevelopelementsofageneraldilationtheoryforoperator-valuedmeasures.
Hilbert space operator-valued measures are closely related to bounded linear maps
on abelian von Neumann algebras, and some of our 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. We 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
linearmapsbetweenC∗-algebras. Inthisconnectionframetheoryitselfisidentified
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 σ-weakly, or w*)
continuous. Some of the results for maps extend to the case where the domain
algebra is non-commutative. It has been known for a long time that a necessary
and sufficient condition for a bounded linear map from a unital C*-algebra into
B(H) to have a Hilbert space dilation to a ∗-homomorphism is that the mapping
needstobecompletelybounded. Ourtheoryshowsthatevenifitisnotcompletely
bounded it still has a Banach space dilation to a homomorphism. For the special
case when the domain algebra is an abelian von Neumann algebra and the map
is normal, we show that the dilation can be taken to be normal with respect to
theusual Banachspace versionof ultraweak topologyonthe range space. We view
theseresultsasgeneralizationsoftheknownresultofCazzaza,HanandLarsonthat
arbitraryframingshaveBanachdilations,andalsotheknownresultthatcompletely
boundedmapshave Hilbertiandilations. Our methodsextendtosome caseswhere
the domain algebra need not be commutative, leading to new dilation results for
maps of general von Neumann algebras. This paper was motivated by some recent
resultsinframetheoryandtheobservationthatthereisacloseconnectionbetween
ReceivedbytheeditorJuly31,2011,and,inrevisedform,June26,2012.
ArticleelectronicallypublishedonOctober1,2013.
DOI:http://dx.doi.org/10.1090/memo/1075
2010MathematicsSubjectClassification. Primary46G10,46L07,46L10,46L51,47A20;Sec-
ondary42C15,46B15,46B25,47B48.
Keywordsandphrases. Operator-valuedmeasures,vonNeumannalgebras,dilations,normal
maps,completelyboundedmaps,frames.
Acknowledgements: TheauthorswereallparticipantsintheNSFfundedWorkshopinAnal-
ysisandProbabilityatTexasA&MUniversity. Thefirstauthoracknowledgespartialsupportby
agrantfromtheNSF.ThethirdandfourthauthorsreceivedpartialsupportfromtheNSFC.
(cid:3)c2013 American Mathematical Society
vii
viii ABSTRACT
the analysis of dual pairs of frames (both the discrete and the continuous theory)
and the theory of operator-valued measures.
Introduction
We investigate some natural associations between the theory of frames (in-
cluding continuous frames and framings), the theory of operator-valued measures
(OVM’s) on sigma algebras of sets, and the theory of normal (ultraweakly or w*
continuous) linear maps on von Neumann algebras. Our main focus is on the dila-
tion theory of these objects.
Generalized analysis-reconstruction schemes include dual pairs of frame se-
quences, framings, and continuous versions of these. We observe that all of these
induce operator-valued measures on an appropriate σ-algebra of Borel sets in a
natural way. The dilation theories for frames, dual pairs of frames, and framings,
have been studied in the literature and many of their properties are well known.
The continuous versions also have a dilation theory, but their properties are not as
wellunderstood. Weshowthatallthesecanbeperhapsbetterunderstoodinterms
of dilations of their operator-valued measures and their associated linear maps.
There is a well known dilation theory for those operator-valued measures that
arecompletely boundedinthe sense thattheirassociatedbounded linear maps be-
tweentheoperatoralgebraL∞ ofthesigmaalgebraandthealgebraofallbounded
linear operators on the underlying Hilbert space are completely bounded maps (cb
maps for short). In this setting the dilation theory for operator-valued measures is
obtained naturally from the dilation theory for cb maps, and cb maps dilate to *-
homomorphismswhileOVM’sdilatetoprojection-valuedmeasures(PVM’s),where
theprojectionsareorthogonalprojections. Wedevelopageneraldilationtheoryfor
operator valued measures acting on Banach spaces where operator-valued measure
(or maps) are are not necessarily completely bounded. Our first main result (The-
orem 2.31) shows that any operator-valued measure (not necessarily completely
bounded) always has a dilation to a projection-valued measure acting on a Banach
space. HerethedilationspaceoftenneedstobeaBanachspaceandtheprojections
are idempotents that are not necessarily self-adjoint (c.f. [14]).
Theorem A 1 Let E : Σ → B(X,Y) be an operator-valued measure. Then there
exist a Banach space Z, bounded linear operators S :Z →Y and T :X →Z, and
a projection-valued probability measure F :Σ→B(Z) such that
E(B)=SF(B)T
for all B ∈Σ.
1We are enumerating what we feel are perhaps the most important of our contributions by
labelingthemA,B,C,D,E,...,withtheordernotnecessarilybyorderofimportancebutsimply
by the order of appearance in this manuscript. We thank the referee for making a suggestion
alongtheselinesinordertohelpthereader.
1