Tensorprodukte und Simplexe

Behrends, Ehrhard; Wittstock, Gerd

doi:10.1007/bf01404650

Cited by 4 publications

(6 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…2)-( 7) has been established [153]. A new proof of the implication (0)0(8) may be found in [92]. Property (5) admits a useful restatement proposed by Alfsen [35] on the basis of the concept of affine dependence, suggesting a certain analogy between the concepts of a simplex and a free group (or, in general, a free algebraic system).…”

Section: I~p (X)mentioning

confidence: 92%

See 1 more Smart Citation

Convex sets, extreme points, and simplexes

Shashkin¹

1975

J Math Sci

View full text Add to dashboard Cite

UDC 513/515+517.948 the central topic of the survey is "Choquet boundary theory," i.e., the sphere of problems pertaining to the representation of infinite-dimensional convex sets in terms of their extreme points.The classical foundation for this topic was set down in the following theorem established in 1940 by M. G. Krein and D. P. Mil'man: A compact convex set in a locally convex Hausdorff space coincides with the closed convex hull of its extreme points, i.e., every point of that set is the center of gravity of a probability measure concentrated on the closure of the extreme points.Interpreted as an integral representation theorem, the Kreirr-Mil'man theorem was later refined by Mil'man [20,21] and Choquet [129][130][131][132]. Choquet further introduced the concept of an infinite-dimensional simplex as a set in which every point has a unique representing "extremal" measure.The Choquet theory has a multiplicity of connections with other branches of mathematics. Within the scope of functional analysis it has bearing on ordered linear spaces, normed algebras (C*-algebras and functional algebras), and operator theory, not to mention ramifications in potential theory, probability theory, ergodic theory, the theory of functions (the problem of moments and completely monotonic functions), extremal problems (best-approximation theory, in particular), and the theory of representations of groups.We include in the present survey papers published in the period from 1967 through 1972. We are concerned exclusively with the "pure" Choquet theory, without delving, as a rule, into its applications to the aforementioned branches of mathematics. Books devoted specifically to Choquet theory have been written by Bauer [74], Phelps [411], Goullet de Rugy [253], Choquet [146], and Alfsen [44]; earlier surveys have been published by Choquet [143], Choquet and Meyer [153], Monna [379], D. A. Edwards [196], and Jacobs [290]. Books by Meyer [361], R. Edwards [29], and Semadeni [465] include sections devoted to the Krein--Mil'man theorem, integral representations, and simplexes. Finally, we mention the surveys of Bauer [76] on integral representations, Alfsen [40] on simplexes, Choquet [47] on conic measures, and D. P. Mil'man [22] on the facial structure of convex sets.We use standard terminology, noting, however, the use of the adjective "semibounded" with reference to a convex set that does not contain a straight line, as well as the interpretation of a "compactum" in the sense of a Hausdorff compact topological space.The following notation is used throughout the article: E is a real locally convex Hausdorff-linear topological space; X is a convex (usually closed) set in E; ~ is an arbitrary compactum; C(~) is the space of real continuous functions on ~; M(~) = C*(~) is the Radon measure space on g; M+(~) is the cone of positive measures; M+(~) is a set of probability measures; in general, L + denotes the positive cone of the linear space L; A(X) and P(X) are the subsets of C(X) formed by, respectively, affine and convex functions; ...

show abstract

Section: I~p (X)mentioning

confidence: 92%

“…(20) If! Y is any compact convex set, then the tensor products YAYand XDF coincide (Namioka and[ Phelps [385]; Behrends and Wittstock [92]).…”

Section: I~p (X)mentioning

confidence: 99%

Convex sets, extreme points, and simplexes

Shashkin¹

1975

J Math Sci

View full text Add to dashboard Cite

show abstract

“…Ein wesentliches Hilfsmittel fiir den Beweis sind die Tensorprodukte geordneter linearer R~iume und die Tensorprodukte kompakter konvexer Mengen [6,2,3]. Hier wird die Darstellung in [3] benutzt.…”

Section: Choquetsimplexe Und Nukleare R Iumeunclassified

“…~I(X, Y) ist konvex und in der punktweisen Topologie kompakt. Wir definieren (s. [3]) das geordnete projektive Tensorprodukt X | Y= d~' (X, Y). …”

Section: Vorbemerkungenunclassified

Choquetsimplexe und nukleare R�ume

Wittstock

1971

Invent Math

Self Cite

View full text Add to dashboard Cite

Es wird gezeigt, dab man nukleare Riiume mit Hilfe von Choquetsimplexen charakterisieren kann. Ein separierter lokalkonvexer Raum E ist dann und nur dann nuklear, wenn im dualen Raum E' jede gleichstetige Menge in einem gleichstetigen schwach* kompakten Choquetsimplex enthalten ist. Fiir einen F-oder vollst~indigen DF-Raum E folgt, dab E dann und nur dann nuklear ist, wenn jede beschr~inkte Menge in einem beschr~inkten Choquetsimplex enthalten ist.

show abstract

“…Some of these notions were studied in relation with the so-called Choquet Theory, as done by Z. Semadeni in [26], I. Namioka and R.R. Phelps in [20] or E. Behrends and G. Wittstock in [4]. More recently, tensor products of convex sets have been studied in relation with the so called quantum information theory, as can be found in [2] by G. Aubrun and S. Szarek.…”

Section: Introductionmentioning

confidence: 99%

A general theory of tensor products of convex sets in Euclidean spaces

Fernández-Unzueta

Higueras-Montaño

2020

Positivity

View full text Add to dashboard Cite

We introduce both the notions of tensor product of convex bodies that contain zero in the interior, and of tensor product of 0-symmetric convex bodies in Euclidean spaces. We prove that there is a bijection between tensor products of 0-symmetric convex bodies and tensor norms on finite dimensional spaces. This bijection preserves duality, injectivity and projectivity. We obtain a formulation of Grothendieck's Theorem for 0-symmetric convex bodies and use it to give a geometric representation (up to the KG-constant) of the Hilbertian tensor product. We see that the property of having enough symmetries is preserved by these tensor products, and exhibit relations with the Löwner and the John ellipsoids.

show abstract

Tensorprodukte und Simplexe

Cited by 4 publications

References 7 publications

Convex sets, extreme points, and simplexes

Convex sets, extreme points, and simplexes

Choquetsimplexe und nukleare R�ume

A general theory of tensor products of convex sets in Euclidean spaces

Contact Info

Product

Resources

About