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A Selection Principle for Mappings of Bounded Variation

✍ Scribed by S.A. Belov; V.V. Chistyakov

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
141 KB
Volume
249
Category
Article
ISSN
0022-247X

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✦ Synopsis

E. Helly's selection principle states that an infinite bounded family of real functions on the closed inter¨al, which is bounded in ¨ariation, contains a pointwise con¨ergent sequence whose limit is a function of bounded ¨ariation. We extend this theorem to metric space valued mappings of bounded variation. Then we apply the extended Helly selection principle to obtain the existence of regular selections of Ž . non-convex set-valued mappings: any set-¨alued mapping from an inter¨al of the real line into nonempty compact subsets of a metric space, which is of bounded ¨ariation with respect to the Hausdorff metric, admits a selection of bounded ¨ariation. Also, we show that a compact-valued set-valued mapping which is Lipschitzian, absolutely continuous, or of bounded Riesz ⌽-variation admits a selection which is Lipschitzian, absolutely continuous, or of bounded Riesz ⌽-variation, respectively.

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Given an operator space X and a von Neumann algebra A, we consider a contractive mapping q: A eh X eh A Ä NCB(X\*, A) formally defined by q( a j x jk b k )= x jk a j b k , from the extended Haagerup tensor product A eh X eh A into the space of w\*-continuous completely bounded maps from X\* into A.