Table Of ContentUniform Algebras Over Complete
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Jonathan W. Mason, MMath.
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. Thesis submitted to The University of Nottingham
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1FromRudyardKipling’spoemTheGipsyTrail.
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Abstract
UNIFORM algebras have been extensively investigated because of their importance in
thetheoryofuniformapproximationandasexamplesofcomplexBanachalgebras. An
interestingquestioniswhetheranalogousalgebrasexistwhenacompletevaluedfield
other than the complex numbers is used as the underlying field of the algebra. In the
Archimedean setting, this generalisation is given by the theory of real function alge-
brasintroducedbyS.H.KulkarniandB.V.Limayeinthe1980s. Thisthesisestablishes
abroadertheoryaccommodatinganycompletevaluedfieldastheunderlyingfieldby
involvingGaloisautomorphismsandusingnon-Archimedeananalysis. Theapproach
takenkeepsclosetotheoriginaldefinitionsfromtheArchimedeansetting.
Basicfunctionalgebrasaredefinedandgeneraliserealfunctionalgebrastoallcomplete
valuedfieldswhilstretainingtheobligatorypropertiesofuniformalgebras.
Several examples are provided. A basic function algebra is constructed in the non-
Archimedean setting on a p-adic ball such that the only globally analytic elements of
thealgebraareconstants.
Eachbasicfunctionalgebraisshowntohavealatticeofbasicextensionsrelatedtothe
field structure. In the non-Archimedean setting it is shown that certain basic function
algebrashaveresiduealgebrasthatarealsobasicfunctionalgebras.
A representation theorem is established. Commutative unital Banach F-algebras with
square preserving norm and finite basic dimension are shown to be isometrically F-
isomorphic to some subalgebra of a Basic function algebra. The condition of finite
basicdimensionisalwayssatisfiedintheArchimedeansettingbytheGel’fand-Mazur
Theorem. Thespectrumofanelementisconsidered.
The theory of non-commutative real function algebras was established by K. Jarosz in
2008. The possibility of their generalisation to the non-Archimedean setting is estab-
lishedinthisthesisandalsoappearedinapaperbyJ.W.Masonin2011.
In the context of complex uniform algebras, a new proof is given using transfinite
induction of the Feinstein-Heath Swiss cheese “Classicalisation” theorem. This new
proofalsoappearedinapaperbyJ.W.Masonin2010.
ii
Acknowledgements
I would particular like to thank my supervisor J. F. Feinstein for his guidance and en-
thusiasm over the last four years. Through his expert knowledge of Banach algebra
theoryhehashelpedmetoidentifyseveralproductivelinesofresearchwhilstalways
allowingmethefreedomrequiredtomaketheworkmyown.
In addition to my supervisor, I. B. Fesenko also positively influenced the direction of
myresearch. DuringmydoctoraltrainingIundertookapostgraduatetrainingmodule
on the theory of local fields given by I. B. Fesenko. With extra reading, this enabled
metoworkbothintheArchimedeanandnon-Archimedeansettingsasimplicitlysug-
gestedbymythesistitle.
ItwasapleasuretoknowmyfriendsinthealgebraandanalysisgroupatNottingham
andIthankthemfortheirinterestinmyworkandhospitality.
I am grateful to the School of Mathematical sciences at the University of Nottingham
forprovidingfundsinsupportofmyconferenceparticipationandvisits.
SimilarlyIappreciatethesupportgiventomebytheEPSRCthroughaDoctoralTrain-
ingGrant.
ThisPhDthesiswasexaminedbyA.G.O’FarrellandJ.ZachariaswhoIthankfortheir
timeandinterestinmywork.
iii
Contents
1 Introduction 1
1.1 BackgroundandOverview . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Completevaluedfields 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Seriesexpansionsofelementsofvaluedfields . . . . . . . . . . . 7
2.1.2 Examplesofcompletevaluedfields . . . . . . . . . . . . . . . . . 9
2.1.3 Topologicalpropertiesofcompletevaluedfields . . . . . . . . . . 12
2.2 Extendingcompletevaluedfields . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Galoistheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Functionsandalgebras 22
3.1 Functionalanalysisovercompletevaluedfields . . . . . . . . . . . . . . 22
3.1.1 Analyticfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 BanachF-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Spectrumofanelement . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Uniformalgebras 34
4.1 Complexuniformalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1.1 Swisscheesesetsinthecomplexplane . . . . . . . . . . . . . . . 36
4.1.2 Classicalisationtheorem . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Non-complexanalogsofuniformalgebras . . . . . . . . . . . . . . . . . . 48
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CONTENTS
4.2.1 Realfunctionalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 50
5 Commutativegeneralisationovercompletevaluedfields 53
5.1 Maindefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Generalisationtheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4 Non-Archimedeannewbasicfunctionalgebrasfromold . . . . . . . . . 65
5.4.1 Basicextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4.2 Residuealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6 Representationtheory 79
6.1 FurtherBanachringsandBanachF-algebras . . . . . . . . . . . . . . . . 79
6.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2.1 Establishedtheorems . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.2.3 Representationunderfinitebasicdimension . . . . . . . . . . . . 91
7 Non-commutativegeneralisationandopenquestions 98
7.1 Non-commutativegeneralisation . . . . . . . . . . . . . . . . . . . . . . . 98
7.1.1 Non-commutativerealfunctionalgebras . . . . . . . . . . . . . . 98
7.1.2 Non-commutativenon-Archimedeananalogs . . . . . . . . . . . 100
7.2 Openquestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
References 105
v
CHAPTER 1
Introduction
Thisshortchapterprovidesaninformaloverviewofthematerialinthisthesis. Justifi-
cationofthestatementsmadeinthischaptercanthereforebefoundinthemainbody
ofthethesiswhichstartsatChapter2.
1.1 Background and Overview
Complexuniformalgebrashavebeenextensivelyinvestigatedbecauseoftheirimpor-
tanceinthetheoryofuniformapproximationandasexamplesofcomplexBanachalge-
bras. LetCC(X)denotethecomplexBanachalgebraofallcontinuouscomplex-valued
functionsdefinedonacompactHausdorffspace X. Acomplexuniformalgebra Aisa
subalgebra of CC(X) that is complete with respect to the sup norm, contains the con-
stantfunctionsmakingitaunitalcomplexBanachalgebraandseparatesthepointsofX
inthesensethatforall x ,x ∈ X with x (cid:54)= x thereis f ∈ Asatisfying f(x ) (cid:54)= f(x ).
1 2 1 2 1 2
Attempting to generalise this definition to other complete valued fields simply by re-
placingCwithsomeothercompletevaluedfield Lproducesverylimitedresults. This
isbecausethevariousversionsoftheStone-Weierstrasstheoremrestrictsourattention
toC (X)inthiscase.
L
However the theory of real function algebras introduced by S. H. Kulkarni and B. V.
Limaye in the 1980s does provide an interesting generalisation of complex uniform
algebras. Oneimportantdepartureinthedefinitionofthesealgebrasfromthatofcom-
plex uniform algebras is that they are real Banach algebras of continuous complex-
valued functions. Similarly the elements of the algebras introduced in this thesis are
also continuous functions that take values in some complete valued field or division
ringextendingthefieldofscalarsoverwhichthealgebraisavectorspace.
Aprominentaspectoftheemergingtheoryisthatithasalottodowithrepresentation.
As a very simple example the field of complex numbers itself is isometrically isomor-
phictoarealfunctionalgebra,allbeitonatwopointspace.
1
CHAPTER 1: INTRODUCTION
When considering the generalisation of complex uniform algebras over all complete
valuedfieldsInaturallywantedthecomplexuniformalgebrasandrealfunctionalge-
bras to appear directly as instances of the new theory. This resulted in the definition
ofbasicfunctionalgebrasinvolvingtheuseofaGaloisautomorphismandhomeomor-
phicendofunctionthatinteractinausefulway,seeDefinition5.1.2. Inretrospectthese
particular algebras should more appropriately be referred to as cyclic basic function
algebras since the functions involved take values in some cyclic extension of the un-
derlyingfieldofscalarsofthealgebra.
Necessarilythisthesisstartsbysurveyingcompletevaluedfieldsandtheirproperties.
ThetransitionfromtheArchimedeansettingtothenon-Archimedeansettingpreserves
in places several of the nice properties that complete Archimedean fields have. How-
ever all complete non-Archimedean fields are totally disconnected, some of them are
not locally compact and there is no non-Archimedean analog of the Gel’fand-Mazur
Theorem.
Ontheotherhandsomecompletenon-Archimedeanfieldshaveinterestingproperties
thatonlyappearinthenon-Archimedeansetting. Considerforexampletheclosedunit
discofthecomplexplane. Itisclosedundermultiplicationbutnotwithrespecttoad-
dition. In the non-Archimedean setting the closed unit ball O , of a complete valued
F
field F,isaringsinceinthiscasethevaluationinvolvedobservesthestrongversionof
thetriangleinequality,seeDefinition2.1.1. Theset M = {a ∈ F : |a| < 1} isamax-
F F
imal ideal of O from which the residue field F = O /M is obtained. The residue
F F F
fieldisofgreatimportanceinthestudyofsuchfields.
Similarly in the non-Archimedean setting we will see that certain basic function al-
gebras have residue algebras that are also basic function algebras. In the process of
proving this result an interesting fact is shown concerning a large class of complete
non-Archimedeanfields. ForsuchafieldFandeveryfiniteextensionLofF,extending
F asa valuedfield, itis shownthat foreachGalois automorphism g ∈ Gal(L/ ) there
F
exists a set R ⊆ O of residue class representatives such that the restriction of g to
L,g L
R isanendofunction,i.e. aselfmap,onR . Thisfactisprobablyknowntocertain
L,g L,g
numbertheorists.
Thisthesisalsoincludesseveralexamplesofbasicfunctionalgebrasandthesearecon-
sideredatdepth. Anewproofofanexistingtheoreminthesettingofcomplexuniform
algebrasisgivenandtheoryinthenon-commutativesettingisalsoconsidered.
With respect to commutative Banach algebra theory, Chapter 6 presents an interest-
ingnewGel’fandrepresentationresultextendingthoseoftheArchimedeansetting. In
particular we have the following theorem where the condition of finite basic dimen-
sionisautomaticallysatisfiedintheArchimedeansettingandcompensatesforthelack
of a Gel’fand-Mazur Theorem in the non-Archimedean setting. See Chapter 6 for full
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CHAPTER 1: INTRODUCTION
details.
Theorem1.1.1. LetFbealocallycompactcompletevaluedfieldwithnontrivialvaluation. Let
AbeacommutativeunitalBanachF-algebrawith(cid:107)a2(cid:107) = (cid:107)a(cid:107)2 foralla ∈ Aandfinitebasic
A A
dimension. Then:
(i) if F is the field of complex numbers then A is isometrically F-isomorphic to a complex
uniformalgebraonsomecompactHausdorffspace X;
(ii) if F is the field of real numbers then A is isometrically F-isomorphic to a real function
algebraonsomecompactHausdorffspace X;
(iii) if F is non-Archimedean then A is isometrically F-isomorphic to a non-Archimedean
analog of the real function algebras on some Stone space X where by a Stone space we
meanatotallydisconnectedcompactHausdorffspace.
Inparticular A isisometrically F-isomorphictosomesubalgebra Aˆ ofabasicfunctionalgebra
and Aˆ separatesthepointsof X.
Notethat(i)and(ii)ofTheorem1.1.1arethewellknownresultsfromtheArchimedean
setting. Thisbringsustothefollowingsummary.
1.2 Summary
Chapter2: Therelevantbackgroundconcerningcompletevaluedfieldsisprovided. Several
examples are given and the topological properties of complete valued fields are
comparedanddiscussed. Aparticularlyusefulandwellknownwayofexpress-
ing the extension of a valuation is considered and the relevant Galois theory is
introduced.
Chapter3: Some background concerning functional analysis over complete valued fields is
given. Analytic functions are discussed. Banach F-algebras are introduced and
thespectrumofanelementisconsidered.
Chapter4: Complex uniform algebras are introduced. In the context of complex uniform
algebras, a new proof is given using transfinite induction of the Feinstein-Heath
Swisscheese“Classicalisation”theorem. Thisnewproofalsoappearedinapaper
byJ.W.Masonin2010. Thisisfollowedbyapreliminarydiscussionconcerning
non-complexanalogsofuniformalgebras. Realfunctionalgebrasareintroduced.
Chapter5: Basic function algebras are defined providing the required generalisation of real
functionalgebrastoallcompletevaluedfields. Ageneralisationtheoremproves
3
CHAPTER 1: INTRODUCTION
thatBasicfunctionalgebrashavetheobligatorypropertiesofuniformalgebras.
Several examples are provided. Complex uniform algebras and real function al-
gebras now appear as instances of the new theory. A basic function algebra is
constructed in the non-Archimedean setting on a p-adic ball such that the only
globallyanalyticelementsofthealgebraareconstants.
Each basic function algebra is shown to have a lattice of basic extensions related
to the field structure. Further, in the non-Archimedean setting it is shown that
certain basic function algebras have residue algebras that are also basic function
algebras. ToprovethiseachGaloisautomorphism,forcertainfieldextensions,is
showntorestricttoanendofunctiononsomesetofresidueclassrepresentatives.
Chapter6: A representation theorem is established in the context of locally compact com-
plete fields with nontrivial valuation. For such a field F, commutative unital
Banach F-algebras with square preserving norm and finite basic dimension are
shown to be isometrically F-isomorphic to some subalgebra of a Basic function
algebra. The condition of finite basic dimension is automatically satisfied in the
ArchimedeansettingbytheGel’fand-MazurTheorem.
Chapter7: The theory of non-commutative real function algebras was established by K.
Jarosz in 2008. The possibility of their generalisation to the non-Archimedean
setting is established in this thesis having been originally pointed out in a pa-
per by J. W. Mason in 2011. The thesis concludes with a list of open questions
highlightingthepotentialforfurtherinterestingdevelopmentsofthistheory.
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