Tensor Products, Infinite Products, and Projective Limits of Choquet Simplexes.

Davies, E. B.; Vincent-Smith, G. F.

doi:10.7146/math.scand.a-10879

Cited by 19 publications

(15 citation statements)

References 3 publications

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“…Thus our theorem also implies the result of F. Jellett, E. B. Davies, and G. F. Vincent-Smith [4,1] that the inverse limit of a system of Choquet simplexes is a Choquet simplex.…”

supporting

confidence: 64%

Inverse Systems of Abstract Lebesgue Spaces

Christianson¹

1990

Proceedings of the American Mathematical Society

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Abstract.We show that inverse limits exist in the category of L spaces and positive linear contractions between them. This result generalizes the wellknown classical results for inverse systems of Choquet Simplexes and of L-balls, but our proof is simple and more purely geometrical. The result finds physical application in the study of random fields.Let Z be a Banach lattice with positive cone C and norm || • ||. We say that Z is an AL-space (short for Abstract Lebesgue space) iff || • || is affine on C. See [7, §8, p. 112] for further information about ^4F-spaces.Let / be a linear map between two ,4F-spaces Z2 and Zx . We say that f is a contraction iff \\fx\\x < ||jt||2 for all x e C2. A positive linear contraction f:Z2^Zx is called an AL-morphism .Note that for an v4Z.-morphism / we have ||/jc||, < ||x||2 for all x e Z2 since ll/xii^iiiAi.ii^n/ixy^iiWjii^iwi.; however, an ^IL-morphism need not be a lattice homomorphism, an isometry, or even one-to-one.Let F be a directed set, and suppose that we have an AL-space Zi associated with each / e D and an v47_-morphism fi : Z( -» Z associated with each pair i, j e D where t> j .If this system is consistent in the sense that fji = fjk fu for a11 i>k> j in D, then we call (Z(., f¡¡)D an inverse ,4L-system. Now suppose further that Z is an ,4L-space, and f. : Z -► Z( for i e D are /4F-morphisms satisfying fj -fji fi for a11 ' > J in D ■ Then we call (Z , f.)D a prefix of (Z(., fn)D .

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“…Thus our theorem also implies the result of F. Jellett, E. B. Davies, and G. F. Vincent-Smith [4,1] that the inverse limit of a system of Choquet simplexes is a Choquet simplex.…”

supporting

confidence: 64%

Inverse Systems of Abstract Lebesgue Spaces

Christianson¹

1990

Proceedings of the American Mathematical Society

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“…Dans le cas où V est un espace simplicial (resp. avec unité) la démons-tration peut être aménagée pour obtenir le résultat suivant, qui se trouve aussi dans [1] et [8]. …”

Section: Propriétés Des Espacesunclassified

Approximation par des opérateurs compacts ou faiblement compacts à valeurs dans $C(X)$

Fakhoury¹

1977

Annales De L’institut Fourier

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“…Die Ordnungstopologie in U wird vonder Norm Ilull =inf{2>OI-2ev<-<_u<2ev}, ftir alle uEU, (1) erzeugt. Projektives Tensorprodukt kompakter konvexer M engen.…”

Section: Tensorprodukte Kompakter Konvexer Mengenunclassified

“…Phelps [7], S. 71). Da nach (1) (~r (q'))(rr v)= <v, ~r(~o)>K ist, folgt ~=rcx(~o ). Also lassen sich m K und r x eindeutig linear fortsetzen: (r~ x f)(k)= <mr(k), f>r =f(rr ~ mr(k))=f(k).…”

Section: Die Einbettungen (2) Erzeugen Eine Lineare Einbettungunclassified

Tensorprodukte kompakter konvexer Mengen

Behrends

Wittstock

1970

Invent Math

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EHRHARD BEHRENDS (Berlin) und GERD WrrTsTocK (Saarbriicken) Wir untersuchen das projektive Tensorprodukt K (~L zweier kompakter konvexer Mengen K, L. Dieses Tensorprodukt wurde yon Semadeni [9] und, ftir den Fall yon Simplexen, yon Davies und Vincent-Smith [1] und Lazar [4] eingeftihrt. Die Menge der stetigen affinen Funktionen ~K auf K ist ein archimedisch geordneter linearer Raum mit Einheit und vollst~indig in der Ordnungstopologie. Popa [8], Hulanicki und Phelps [2] und Peressini und Sherbert [6] haben Tensorprodukte geordneter linearer Riiume untersucht. Man kennt hier ein projektives Tensorprodukt U (~v V und ein injektives (biprojektives) Tensorprodukt U ~ V. Es sei K kompaktes Simplex und L kompakt konvex. Dann gilt f'tir die Menge ~ (K, L) der stetigen biaffinen reellen Funktionen auf K x L, dab ~(K, L)=dK ~r ~v~tL ist.Sind K, L Simplexe, so ist K ~L ein Simplex, wie bereits yon Davies

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Tensor Products, Infinite Products, and Projective Limits of Choquet Simplexes.

Cited by 19 publications

References 3 publications

Inverse Systems of Abstract Lebesgue Spaces

Inverse Systems of Abstract Lebesgue Spaces

Approximation par des opérateurs compacts ou faiblement compacts à valeurs dans $C(X)$

Tensorprodukte kompakter konvexer Mengen

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