Monotone Complete C*-algebras and Generic Dynamics

265 Pages·2015·2.77 MB·English

by  Kazuyuki Saitô, J. D. Maitland Wright

Springer Monographs in Mathematics
Kazuyuki Saitô
J. D. Maitland Wright
Monotone
Complete
C*-algebras
and Generic
Dynamics
Springer Monographs in Mathematics
Moreinformationaboutthisseriesathttp://www.springer.com/series/3733
Kazuyuki Saitô (cid:129) J.D. Maitland Wright
Monotone Complete
C*-algebras and Generic
Dynamics
123
KazuyukiSaitô J.D.MaitlandWright
Sendai,Japan UniversityofAberdeen
Aberdeen,Scotland
ChristChurch
UniversityofOxford
Oxford,UK
ISSN1439-7382 ISSN2196-9922 (electronic)
SpringerMonographsinMathematics
ISBN978-1-4471-6773-0 ISBN978-1-4471-6775-4 (eBook)
DOI10.1007/978-1-4471-6775-4
LibraryofCongressControlNumber:2015958093
MathematicsSubjectClassification:46L05,46L35,37A20,37A55,46L36,06A12,54G05,06F05
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Abstract
This bookis aboutmonotonecomplete C(cid:2)-algebras,their properties,and the new
classification theory (usingspectroid invariantsand a classification semigroup).A
basicaccountofgenericdynamicsisincludedbecauseofitsimportantconnections
to these algebras.Eachbounded,upward-directednetof realnumbershasa limit.
Monotone complete algebras of operators have a similar property. In particular,
every von Neumannalgebra is monotonecomplete, but the converse is false. The
small von Neumann factors can be labelled by the set of real numbers. But there
aremanymore(2R)smallmonotonecompleteC(cid:2)-algebraswhicharefactors.The
aim of this book is to give an account of monotone complete C(cid:2)-algebras which
includesrecentadvancesbutalsoindicatesthemanymysteriesandopenproblems
whichremain.
v

Preface
First, we wish to thank ProfessorTakesaki who encouragedus to write this book.
Withouthim, we would never have started;withoutthe unfailingsupportof Yuko
andHarvinder,wewouldneverhavefinished.
Monotone complete C(cid:2)-algebras are now entering a new and exciting stage of
their development. The new classification semigroup and spectroid invariants are
useful, but we are a very long way from a full understanding; many mysteries
remain. The second author particularly wishes to thank Dennis Sullivan for stim-
ulatingdiscussionsongenericdynamicsandrelatedtopics.
We wouldboth like to thank the manydistinguishedmathematicianswho have
sharedtheirinsightswithus. Therearetoo manyto list themall, butamongthem
are J.Bonet, J.K. Brooks, L.J. Bunce, Cho-Ho Chu, C.M. Edwards, P. de Lucia,
MasanaoOzawa,A.Peralta,P.Ptak,B.Weiss,andK.Ylinen.
Sendai,Japan KazuyukiSaitô
Aberdeen,Scotland J.D.MaitlandWright
vii

Contents
1 Introduction .................................................................. 1
1.1 MonotoneCompleteAlgebrasofOperators.......................... 1
1.2 GenericDynamics..................................................... 4
2 OrderFundamentals ........................................................ 7
2.1 OrderStructuresandOrderConvergence ............................ 8
2.1.1 OperatorMonotone............................................ 19
2.1.2 MonotoneClosedSubspaces ................................. 19
2.1.3 RegularSubalgebrasandSubspaces.......................... 21
2.1.4 On`2-SummableSequences.................................. 23
2.2 Monotone(cid:2)-CompleteC(cid:2)-Algebras ................................. 24
2.2.1 OpenProblem ................................................. 30
2.2.2 PolarDecomposition.......................................... 30
2.2.3 SequentiallyClosedSubspaces............................... 31
2.3 BasicsforCommutativeAlgebras .................................... 36
2.3.1 ExtensionTheorems........................................... 41
2.4 MatrixAlgebrasoveraMonotoneCompleteC(cid:2)-Algebra........... 43
3 ClassificationandInvariants ............................................... 49
3.1 C(cid:2)-AlgebrasofSmallSize............................................ 49
3.2 ClassificationSemigroup.............................................. 51
3.3 SpectroidandRepresentingFunctions................................ 58
4 CommutativeAlgebras:ConstructionsandClassifications ............ 63
4.1 BooleanAlgebraPreliminaries ....................................... 63
4.1.1 Basics .......................................................... 63
4.1.2 RegularOpenSets............................................. 66
4.1.3 StructureSpace................................................ 68
4.2 CommutativeAlgebras:GeneralConstructions...................... 69
4.2.1 MeasurableFunctions......................................... 69
4.2.2 BaireSpacesandCategory.................................... 71
4.2.3 NormalStatesandWildAlgebras............................ 76
ix