Table Of ContentTrends in Mathematics
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Positivity
Karim Boulabiar
Gerard Buskes
Abdelmajid Triki
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Karim Boulabiar Gerard Buskes
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Université de Carthage Hume Hall 305
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Contents
Preface ................................................................... vii
B. Banerjee and M. Henriksen
Ways in which C(X) mod a Prime Ideal Can be
a Valuation Domain; Something Old and Something New ............ 1
D.P. Blecher
Positivity in Operator Algebras and Operator Spaces ................ 27
K. Boulabiar, G. Buskes, and A. Triki
Results in f-algebras ................................................ 73
Q. Bu, G. Buskes, and A.G. Kusraev
Bilinear Maps on Products of Vector Lattices: A Survey ............. 97
G.P. Curbera and W.J. Ricker
Vector Measures, Integration and Applications ....................... 127
J. Mart´ınez
The Role of Frames in the Development of Lattice-ordered
Groups: A PersonalAccount ......................................... 161
B. de Pagter
Non-commutative Banach Function Spaces ........................... 197
A.R. Schep
Positive Operators on Lp-spaces ..................................... 229
A.W. Wickstead
Regular Operators between Banach Lattices ......................... 255
Preface
This collection of surveys is an outflow from the 2006 conference Carthapos06
in Tunis (Tunisia). Apart from regular conference talks, five survey talks formed
the core of a workshop in Positivity, supported by the National Science Founda-
tion.Theconferenceorganizers(KarimBoulabiar,GerardBuskes,andAbdelmajid
Triki) decided to expand on the idea of core surveys and the nine surveys in this
book are the harvest from that idea.
Positivity derives from an order relation. Order relations are the mathemat-
ical tool for comparison. It is no surprise that seen in such very general light, the
historyofPositivityisancient.Archimedes,certainly,hadtheveryessenceofposi-
tivityinmindwhenhediscoveredthelawofthelever.Hismethodofexhaustionto
calculate areas uses a principle that nowadays carries his name, the Archimedean
property. The surveys in this book are slanted into the direction that Archimedes
took.Functionalanalysisisheavilyrepresented.Butthereismore.Latticeordered
groupsappearinthearticlebyMartinezinthemodernjacketofframes.Henriksen
and Banerjee write their survey on rings of continuous functions. Blecher and de
Pagterineachoftheirpaperssurveypartsofnon-commutativefunctionalanalysis.
Positiveoperatorsarethemaintopicinthe papersbyCurberaandRicker,Schep,
and Wickstead. And positive bilinear maps are the protagonists in the survey by
Bu, Buskes, and Kusraev. The conference organizers (and editors of this volume)
write about f-algebras.
Carthapos06 was more than just a conference and workshop in Africa. It
brought together researchers in Positivity from many directions of Positivity and
form many corners of the world. This book can be seen as a culmination of their
paths meeting in Tunisia, Africa.
June 5, 2007 G. Buskes
Oxford, U.S.A.
Positivity
TrendsinMathematics,1–25
(cid:1)c 2007Birkh¨auserVerlagBasel/Switzerland
C(X)
Ways in which mod a Prime Ideal
Can be a Valuation Domain;
Something Old and Something New
Bikram Banerjee (Bandyopadhyay) and Melvin Henriksen
Abstract. C(X) denotes the ring of continuous real-valued functions on a
Tychonoff space X and P a prime ideal of C(X). We summarize a lot of
what is known about the reside class domains C(X)/P and add many new
resultsaboutthissubjectwithanemphasisondeterminingwhentheordered
C(X)/P is a valuation domain (i.e., when given two nonzero elements, one
of them must divide the other). The interaction between the space X and
the prime ideal P is of great importance in this study. We summarize first
what is known when P is a maximal ideal, and then what happens when
C(X)/P is a valuation domain for every prime ideal P (in which case X
is called an SV-space and C(X) an SV-ring). Two new generalizations are
introduced and studied. The first is that of an almost SV-spaces in which
each maximalideal containsaminimalprimeideal P suchthatC(X)/P isa
valuationdomain.Inthesecond,weassumethateachrealmaximalidealthat
fails tobe minimal contains a nonmaximal prime ideal P such that C(X)/P
is a valuation domain. Some of ourresults dependon whetheror not βω \ω
contains a P-point.Some concluding remarks include unsolved problems.
1. Introduction
Throughout, C(X) will denote the ring of real-valued continuous functions on a
TychonoffspaceX withtheusualpointwiseringandlatticeoperationsandC∗(X)
willdenote its subringofboundedfunctions,andalltopologicalspacesconsidered
areassumedtobeTychonoffspacesunlessthecontraryisstatedexplicitly.(Recall
that X is called a Tychonoff space if it is a subspace of a compact (Hausdorff)
space.EquivalentlyifX isaT spaceandwheneverK isaclosedsubspaceofX not
1
containing a point x, there is an f ∈ C(X) such that f(x) = 0 and f[K]= {1}.)
An element of C(X) is nonnegative in the usual pointwise sense if and only if it
a square. So algebraic operations automatically preserve order. This makes the
2 B. Banerjee and M. Henriksen
notionofpositivityessentialforstudyingC(X).This simpleobservationwasused
with great ingenuity by M.H. Stone in 1937 to make the first thorough study of
C(X)asaring.ItwasrestrictedtothecasewhenX iscompact.Amongthemany
interesting results in this seminal paper is that C(X) determines X. That is, if X
andY arecompactspacesand C(X)andC(Y)arealgebraicallyisomorphic,then
X and Y are homeomorphic.
This study was broadened to include unbounded functions in [Hew48] by
Stone’sstudentE.Hewitt.Whilethispapercontainsanumberofseriouserrors,it
setthetoneforalotofthe researchthatledtothe book[GJ76].(Itwaspublished
originally in 1960 by Van Nostrand). For more background and history of this
subject,see [Wa74],[We75],[Hen97],and[Hen02].Ourgeneralsourcesforgeneral
topology are [E89] and [PW88].
Sections2and3surveysomeofwhathasbeendoneinthepastaboutintegral
domainsthatarehomomorphicimagesofa C(X)andtheprimeidealsP thatare
kernels of such homomorphisms. We concentrate especially on the cases when
C(X)/P is a valuation domain. In Section 2, we review some of what is known
when P is maximal; i.e., when C(X)/P is a field. Section 3 recalls what is known
aboutspacesX suchthatatC(X)/P isavaluationdomainwheneverP isaprime
ideal of C(X). They are called SV-spaces.The remainder of the paper focuses on
new research beginning with the study in Section 4 of almost SV-spaces; that is,
spacesX andringsC(X)inwhicheverymaximalidealofC(X)containsaminimal
prime ideal P such that C(X)/P is a valuation domain. Section 5 is devoted to
the study of products of almost SV-spaces and logical considerations concerning
thevalidityofsomeresults.Theone-pointcompactificationofacountablediscrete
space is not an SV-space, but the consequences of the assumption that it is an
almost SV-space are studied in Section 6. Spaces X and rings C(X) in which
every real maximal ideal of C(X) contains a prime ideal such that C(X)/P is a
valuation domain are examined in Section 7. In the final Section 8, two related
papers and the contents of a book are discussed briefly, some sufficient conditions
are given to say more about valuation domains that are homomorphic images of
a ring C(X), and some unsolved problems are posed.
2. What happens when the valuation domains are fields?
A commutative ring A such that whenever a and b are nonzero elements of A, it
followsthatoneofthemdividestheother,iscalledavaluation ring.Below,weare
interested only in the case when A is also an integral domain, in which case such
a ring A is called a valuation domain. We begin with the case when the valuation
domain is a field, and recall that the kernel of a homomorphism onto a field is
a maximal ideal. Let M(A) denote the set of maximal ideals of A. This set is
nonempty as long as A has an identity element
RecallthatafieldF issaidtobereal-closed ifitssmallestalgebraicextension
is algebraicallyclosed. Equivalently, F is real-closedif it is totally ordered, its set
Ways in which C(X) mod a Prime Ideal Can be a Valuation Domain 3
F+ of nonnegative elements is exactly the set of all squares of elements of F, and
each polynomial of odd degree with coefficients in F vanishes at some point of
F. As is shown in Chapter 13 of [GJ76], if M ∈ M(C(X)), then C(X)/M is a
real-closed field. We continue to quote facts from [GJ76].
If f ∈ C(X), then Z(f) denotes {x ∈ X : f(x) = 0}, and we let coz(f) =
X\Z(f). If S ⊂ C(X), we let Z[S] = {Z(f) : f ∈ S}. Thus Z[C(X)] (which we
abbreviate by Z[X]) is the family of all zerosets of functions in C(X). A subfam-
ily F of Z[X] that is closed under finite intersection, contains Z(g) whenever if
contains some Z(f) ∈ F, and does not contain the empty set is called a z-filter.
Note that an element f is in some proper ideal if and only if Z(f)(cid:3)=∅. It follows
that if I is a proper ideal of C(X), Then Z[I] is a z-filter.
AnidealI isfixed orfree accordingas∩{Z(f):f ∈I}isnonemptyorempty.
A maximal ideal M is fixed if and only if Z[M] = {x} for some x ∈ X, in which
case M is denoted by M . Clearly, C(X)/M always contains a copy of R. The
x
maximalidealM iscalledhyper-real ifC(X)/M containsR properlyandiscalled
real otherwise.Every fixed maximal ideal is real, but the converse fails to hold. If
every real maximal ideal of C(X) is fixed, then X is called a realcompact space.
Subsequent to the appearance of [GJ76], hyper-real fields are also called H-fields.
Recall that the continuum hypothesis CH is the assumption that the least
uncountable cardinal ω is equal to the cardinality 2ω of the continuum.
1
2.1 Definition. Suppose that an ordered set L satisfies: If A and B are countable
subsets of L such that a < b whenever a ∈ A and b ∈ B, then there is an x ∈ L
such that a < x < b whenever a ∈ A and b ∈ B (Symbolically we write this
conclusion as A<x<B.) Then L is called an η -set.
1
Much of what is known about H-fields of cardinality no larger than 2ω is
summarized next.
2.2 Theorem
(a) Every H-field is both real-closedand an η -set.
1
(b) All real-closed fields that are η -sets of cardinality ω are (algebraically)
1 1
isomorphic.
(c) Every η -set has cardinality at least 2ω.
1
(d) All H-fields of cardinality 2ω are isomorphic if and only if CH holds.
Allbutpartof(d)areshowninChapter13of[GJ76].Thatthereisonlyone
H-field (in the sense of isomorphism) of cardinality 2ω implies CH is due to A.
Dow in [D84]. Some more detail about what may happen if CH fails: see [R82].
There are a large number of results concerning H-fields of large cardinal-
ity in [ACCH81] that depend on various set-theoretic hypotheses and use proof
techniques involvingcombinatorialsettheory.Mostofits contentsarebeyondthe
scope of this article.(Some errorsin [ACCH81] are pointed out by A. Blass in his
review in Math. Sci. Net. None of them affect what is written above.)
