Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On a lemma of Milutin concerning averaging operators in continuous function spaces


Author: Seymour Z. Ditor
Journal: Trans. Amer. Math. Soc. 149 (1970), 443-452
MSC: Primary 47B37
MathSciNet review: 0435921
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that any infinite compact Hausdorff space S is the continuous image of a totally disconnected compact Hausdorff space $ S'$, having the same topological weight as S, by a map $ \varphi $ which admits a regular linear operator of averaging, i.e., a projection of norm one of $ C(S')$ onto $ {\varphi ^ \circ }C(S)$, where $ {\varphi ^\circ }:C(S) \to C(S')$ is the isometric embedding which takes $ f \in C(S)$ into $ f \circ \varphi $. A corollary of this theorem is that if S is an absolute extensor for totally disconnected spaces, the space $ S'$ can be taken to be the Cantor space $ {\{ 0,1\} ^\mathfrak{m}}$, where $ \mathfrak{m}$ is the topological weight of S. This generalizes a result due to Milutin and Pełczyński. In addition, we show that for compact metric spaces S and T and any continuous surjection $ \varphi :S \to T$, the operator $ u:C(S) \to C(T)$ is a regular averaging operator for $ \varphi $ if and only if u has a representation $ uf(t) = \smallint _0^1f(\theta (t,x))$ for a suitable function $ \theta :T \times [0,1] \to S$.


References [Enhancements On Off] (What's this?)

  • [1] Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, New Jersey, 1952. MR 0050886
  • [2] John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
  • [3] John L. Kelley, General topology, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. MR 0070144
  • [4] E. Michael, Selected Selection Theorems, Amer. Math. Monthly 63 (1956), no. 4, 233–238. MR 1529282, 10.2307/2310346
  • [5] A. A. Milutin, On spaces of continuous functions, Dissertation, Moscow State University, Moscow, 1952. (Russian)
  • [6] A. A. Miljutin, Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 2 (1966), 150–156. (1 foldout) (Russian). MR 0206695
  • [7] A. Pełczyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Dissertationes Math. Rozprawy Mat. 58 (1968), 92. MR 0227751
  • [8] Z. Semadeni, Inverse limits of compact spaces and direct limits of spaces of continuous functions, Studia Math. 31 (1968), 373–382. MR 0240602

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B37

Retrieve articles in all journals with MSC: 47B37


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0435921-2
Keywords: Averaging operator, extension operator, linear exave, projection operator, regular operator, continuous function space, Milutin space, totally disconnected space, inverse and direct limits, integral representation of regular averaging and extension operators
Article copyright: © Copyright 1970 American Mathematical Society