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Multiplication of convex sets in $C(K)$ spaces

Volume 232 / 2016

José Pedro Moreno, Rolf Schneider Studia Mathematica 232 (2016), 173-187 MSC: Primary 52A05; Secondary 46J10, 46E15. DOI: 10.4064/sm8509-4-2016 Published online: 21 April 2016

Abstract

Let $C(K)$ denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space $K$. For two subsets of $C(K)$, one can define their product by pointwise multiplication, just as the Minkowski sum of the sets is defined by pointwise addition. Our main interest is in correlations between properties of the product of closed order intervals in $C(K)$ and properties of the underlying space $K$. When $K$ is finite, the product of two intervals in $C(K)$ is always an interval. Surprisingly, the converse of this result is true for a wide class of compacta. We show that a first-countable space $K$ is finite whenever it has the property that the product of two nonnegative intervals is closed, or the property that the product of an interval with itself is convex. That some assumption on $K$ is needed can be seen from the fact that if $K$ is the Stone–Čech compactification of $\mathbb N$, then the product of two intervals in $C(K)$ with continuous boundary functions is always an interval. For any $K$, it is proved that the product of two positive intervals is always an interval, and that the product of two nonnegative intervals is always convex. Finally, square roots of intervals are investigated, with results of similar type.

Authors

  • José Pedro MorenoDepartamento de Matemáticas
    Facultad de Ciencias
    Universidad Autónoma de Madrid
    E-28049 Madrid, Spain
    e-mail
  • Rolf SchneiderMathematisches Institut
    Albert-Ludwigs-Universität
    D-79104 Freiburg i. Br., Germany
    e-mail

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