Table Of ContentLecture Notes in Mathematics 2021
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
•
Andreas Defant
Classical Summation
in Commutative
and Noncommutative
L -Spaces
p
123
AndreasDefant
CarlvonOssietzkyUniversityOldenburg
DepartmentofMathematics
CarlvonOssietzkyStrasse7-9
26111Oldenburg
Germany
defant@mathematik.uni-oldenburg.de
ISBN978-3-642-20437-1 e-ISBN978-3-642-20438-8
DOI10.1007/978-3-642-20438-8
SpringerHeidelbergDordrechtLondonNewYork
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2011931784
MathematicsSubjectClassification(2011):46-XX;47-XX
(cid:2)c Springer-VerlagBerlinHeidelberg2011
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations
areliabletoprosecutionundertheGermanCopyrightLaw.
Theuseofgeneral descriptive names,registered names, trademarks, etc. inthis publication does not
imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective
lawsandregulationsandthereforefreeforgeneraluse.
Coverdesign:deblik,Berlin
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface
In the theory of orthogonal series the most important coefficient test for almost
everywhere convergence of orthonormal series is the fundamental theorem of
MenchoffandRademacher.Itstatesthatwheneverasequence(α)ofcoefficients
k
satisfies the “test” ∑ |α logk|2 < ∞, then for every orthonormal series ∑ αx
k k k k k
in L (μ ) we have that ∑ α x converges μ -almost everywhere. The aim of this
2 k k k
researchistodevelopasystematicschemewhichallowsustotransformimportant
parts of the now classical theory of almost everywhere summation of general
orthonormal series in L (μ ) into a similar theory for series in noncommutative
2
L -spaces L (M,ϕ) or even symmetric spaces E(M,ϕ) constructed over a
p p
noncommutativemeasure space (M,ϕ), a von Neumannalgebra M of operators
actingonaHilbertspaceH togetherwithafaithfulnormalstateϕonthisalgebra.
In Chap.2 we present a new and modern understandingof the classical theory
on pointwise convergenceof orthonormalseries in the Hilbert spaces L (μ ), and
2
show that large parts of the classical theory transfer to a theory on pointwise
convergenceofunconditionallyconvergentseriesinspacesL (μ ,X)ofμ -integrable
p
functionswithvaluesinBanachspacesX,ormoregenerallyBanachfunctionspaces
E(μ ,X) of X-valued μ -integrablefunctions. Here our tools are strongly based on
Grothendieck’smetric theoryof tensor productsandin particularon his the´ore`me
fondamental. In Chap.3 this force turns out to be even strong enough to extend
ourschemetothesettingofsymmetricspacesE(M,ϕ)ofoperatorsandHaagerup
L -spacesL (M,ϕ).Incomparisonwiththeoldclassicalcommutativesettingthe
p p
newnoncommutationsettinghighlightsnewphenomena,andourtheoryasawhole
unifies,completesandextendsvariousresults,bothin the commutativeandin the
noncommutativeworld.
Oldenburg,Germany AndreasDefant
v
Contents
1 Introduction .................................................................. 1
1.1 Step1.................................................................... 2
1.2 Step2.................................................................... 4
1.3 Step3.................................................................... 6
1.3.1 CoefficientTestsintheAlgebraItself ......................... 7
1.3.2 Tests in the Hilbert Spaces L (M,ϕ)
2
andMoreGenerallyHaagerupL ’s............................ 9
p
1.3.3 TestsinSymmetricSpacesE(M,τ),τaTrace............... 11
1.4 Preliminaries ............................................................ 13
2 CommutativeTheory ....................................................... 15
2.1 MaximizingMatrices................................................... 15
2.1.1 SummationofScalarSeries.................................... 16
2.1.2 MaximalInequalitiesinBanachFunctionSpaces ............ 19
2.1.3 (p,q)-MaximizingMatrices.................................... 21
2.1.4 MaximizingMatricesandOrthonormalSeries................ 24
2.1.5 MaximizingMatricesandSummation:TheCaseq<p...... 25
2.1.6 BanachOperatorIdeals:ARepetitorium...................... 28
2.1.7 MaximizingMatricesandSummation:TheCaseq≥p...... 33
2.1.8 AlmostEverywhereSummation............................... 37
2.2 BasicExamplesofMaximizingMatrices.............................. 40
2.2.1 TheSumMatrix................................................. 40
2.2.2 RieszMatrices .................................................. 46
2.2.3 Cesa`roMatrices................................................. 50
2.2.4 KroneckerMatrices............................................. 55
2.2.5 AbelMatrices................................................... 61
2.2.6 SchurMultipliers ............................................... 62
vii
viii Contents
2.3 LimitTheoremsinBanachFunctionSpaces .......................... 65
2.3.1 CoefficientTestsinBanachFunctionSpaces ................. 65
2.3.2 LawsofLargeNumbersinBanachFunctionSpaces......... 74
3 NoncommutativeTheory ................................................... 79
3.1 TheTracialCase ........................................................ 79
3.1.1 SymmetricSpacesofOperators................................ 80
3.1.2 MaximalInequalitiesinSymmetricSpacesofOperators..... 86
3.1.3 TracialExtensionsofMaximizingMatrices................... 95
3.1.4 TheRow+ColumnMaximalTheorem........................ 100
3.1.5 AlmostUniformConvergence ................................. 109
3.1.6 CoefficientTestsinSymmetricOperatorSpaces.............. 111
3.1.7 LawsofLargeNumbersinSymmetricOperatorSpaces ..... 116
3.1.8 ACounterexample.............................................. 118
3.2 TheNontracialCase .................................................... 123
3.2.1 HaagerupL -Spaces............................................ 123
p
3.2.2 MaximalInequalitiesinHaagerupL -Spaces................. 126
p
3.2.3 NontracialExtensionsofMaximizingMatrices............... 128
3.2.4 AlmostSureConvergence...................................... 129
3.2.5 CoefficientTestsinHaagerupL -Spaces...................... 134
p
3.2.6 LawsofLargeNumbersinHaagerupL -Spaces ............. 139
p
3.2.7 CoefficientTestsintheAlgebraItself ......................... 141
3.2.8 LawsofLargeNumbersintheAlgebraItself................. 148
3.2.9 MaximalInequalitiesintheAlgebraItself .................... 150
References ........................................................................ 159
Symbols ........................................................................... 165
AuthorIndex ..................................................................... 167
SubjectIndex .................................................................... 169
Chapter 1
Introduction
In the theory of orthogonal series the most important coefficient test for almost
everywhereconvergenceoforthonormalseriesisthefundamentaltheoremofMen-
choffandRademacherwhichwasobtainedindependentlyin[60]and[81].Itstates
thatwheneverasequence(α)ofcoefficientssatisfiesthe“test”∑ |α logk|2<∞,
k k k
thenforeveryorthonormalseries∑ αx inL (μ )wehavethat
k k k 2
∑α x convergesμ -almosteverywhere. (1.1)
k k
k
The aim of this research is to develop a systematic scheme which allows us
to transform important parts of the now classical theory of almost everywhere
summation of orthonormalseries in L (μ ) into a similar theory for series in non-
2
commutativeL -spacesL (M,ϕ)orevensymmetricspacesE(M,ϕ)buildupover
p p
anoncommutativemeasurespace(M,ϕ),avonNeumannalgebraM ofoperators
actingonaHilbertspaceH togetherwithafaithfulnormalstateϕonthisalgebra.
The theory of noncommutative L -spaces – the analogs of ordinary Legesgue
p
spaces L (μ ) with a noncommutative von Neumann algebra playing the role of
p
L∞(μ ) – for semifinite von Neumann algebras was laid out in the early 1950s
by Dixmier [10], Kunze [49], Segal [84], and Nelson [66]. Later this theory was
extended in various directions. In a series of papers P.G. Doods, T.K. Doods and
dePagter[11,12,13,14],Ovchinnikov[69,70],ChilinandSukochev[88],Kalton
andSukochev[45],Sukochev[85,86,87,88]andXu[96]developedamethodfor
constructing symmetric Banach spaces of measurable operators, and Haagerup in
[24]presentedatheoryofL -spacesassociatedwithnotnecessarilysemifinitevon
p
Neumannnalgebras.See[80]forabeautifulsurveyonthewholesubject.
Inthe1980stheschoolofŁo´dz´ startedasystematicstudy,mainlyduetoHensz
and Jajte, of Menchoff-Rademacher type theorems within the Gelfand-Naimark
constructionL (M,ϕ),thenaturalHilbertspaceconstructedoveravonNeumann
2
algebra M and a faithful normal state ϕ on M, and most of their results were
latercollectedinthetwoSpringerLectureNotes[37,38].Basedonnewtechniques
mainlyduetoJunge,Pisier,andXu(seee.g.[39,42,79])someoftheseresultswere
A.Defant,ClassicalSummationinCommutativeandNoncommutativeLp-Spaces, 1
LectureNotesinMathematics2021,DOI10.1007/978-3-642-20438-8 1,
©Springer-VerlagBerlinHeidelberg2011
2 1 Introduction
generalizedin[7],andweplantousesimilartoolstocomplementandextendthis
workinvariousdirections.
In a first step we give a new and modern understandingof the classical theory
on pointwise convergence of general orthonormal series in the Hilbert spaces
L (μ ). Then in a second step we show that large parts of the classical theory
2
transfertoatheoryonpointwiseconvergenceofunconditionallyconvergentseries
in spaces L (μ ,X) of μ -integrable functions with values in Banach spaces X,
p
or more generally Banach function spaces E(μ ,X) of X-valued μ -integrable
functions. Here our tools are strongly based on Grothendieck’s metric theory of
tensor products and in particular on his the´ore`me fondamental – this force in a
third step is evenstrong enoughto extendourscheme to the setting of symmetric
spacesE(M,ϕ)ofoperatorsandHaagerupL -spacesL (M,ϕ). Itturnsoutthat
p p
incomparisonwiththeoldclassicalcommutativesettingthenewnoncommutative
settinghighlightsnewphenomena,andourtheoryasawholeunifies,completesand
extends the original theory, both in the commutative and in the noncommutative
setting:
Step1 Step2 Step3
classical summationof summationof
summationof unconditionally unconditionally
=⇒ =⇒
orthonormal convergentseriesin convergentseriesin
seriesinL2(μ ) Banachfunctionspaces operatoralgebras
1.1 Step 1
Let us start by describing more carefully some of the prototypical results of the
classicaltheoryofalmosteverywheresummationofgeneralorthogonalserieswhich
form the historical background of these notes; as standard references we use the
monographs[1]and[47].SeeagaintheMenchoff-Rademachertheoremfrom(1.1)
which is the most fundamental one. As quite often happens in Fourier analysis,
this fact comes with an at least formally stronger result on maximal functions.
Kantorovitchin [46] provedthat the maximalfunction (of the sequence of partial
sums)ofeachorthonormalseries∑ αx inL (μ )satisfying∑ |α logk|2<∞,is
k k k 2 k k
square-integrable,
(cid:2) (cid:2)
(cid:2) j (cid:2)
sup(cid:2)∑αx (cid:2)∈L (μ ), (1.2)
k k 2
j k=0
or in other terms: For each choice of finitely many orthonormal functions
x ,...,x ∈L (μ )andeachchoiceoffinitelymanyscalarsα,...α wehave
0 n 2 0 n
1.1 Step1 3
(cid:3)(cid:3)(cid:3)sup(cid:2)(cid:2)∑j αx (cid:2)(cid:2)(cid:3)(cid:3)(cid:3) ≤C(cid:4)∑n |α logk|2(cid:5)1/2, (1.3)
k k k
j k=0 2 k=0
whereCisanabsoluteconstant.ThisinequalityisalsoknownastheKantorovitch-
Menchoff-Rademacherinequality.
Results ofthistypecanandshouldbeseeninthecontextoflimittheoremsfor
random variables. To illustrate this by an example, we mention that by a simple
lemma usually attributed to Kronecker, the Menchoff-Rademacher theorem (1.1)
impliesthefollowingstronglawoflargenumbers:Givenasequenceofuncorrelated
randomvariablesX onaprobabilityspace(Ω ,P)wehavethat
k
(cid:4) (cid:5)
1 ∑j X −E(X ) convergesP-almosteverywhere, (1.4)
j+1k=0 k k j
provided
∑log2kVar(X )<∞.
k2 k
k
There is a long list of analogs of the Menchoff-Rademacher theorem for various
classicalsummationmethods,ase.g.Cesa`ro,Riesz,orAbelsummation.Recallthat
asummationmethodformallyisascalarmatrixS=(s )withpositiveentriessuch
jk
thatforeachconvergentseriess=∑ x ofscalarswehave
n n
k
s=lim ∑sjk∑x(cid:2). (1.5)
j k (cid:2)=0
Letusexplainwhatwemeanbyacoefficienttestforalmosteverywhereconvergence
of orthonormal series with respect to a summation method S = (s ) and an
kj
increasing and unboundedsequence (ω) of positive scalars. We mean a theorem
k
whichassuresthat
k
∑αkxk=lim ∑sjk∑α(cid:2)x(cid:2) μ -a.e. (1.6)
k j k (cid:2)=0
whenever ∑ αx is an orthonormal series in some L (μ ) with coefficients (α)
k k k 2 k
satisfyingthetest∑ |αω|2<∞;intheliteraturesuchasequenceωisthenoften
k k k
calledaWeylsequence.
Forexample,thecoefficienttest∑ |α loglogk|2<∞assuresthatallorthonor-
k k
malseries∑ αx arealmosteverywhereCesa`ro-summable,i.e.
k k k
∑αkxk= j+11 ∑j ∑k α(cid:2)x(cid:2) μ -a.e., (1.7)
k k=0(cid:2)=0
