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

José Pedro Moreno¹ ; Rolf Schneider²

¹ Departamento de Matemáticas Facultad de Ciencias Universidad Autónoma de Madrid E-28049 Madrid, Spain
² Mathematisches Institut Albert-Ludwigs-Universität D-79104 Freiburg i. Br., Germany

Studia Mathematica, Tome 232 (2016) no. 2, pp. 173-187

Résumé

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.

Détail

DOI : 10.4064/sm8509-4-2016

Mots-clés : denote banach algebra continuous real functions supremum norm compact hausdorff space subsets define their product pointwise multiplication just minkowski sum sets defined pointwise addition main interest correlations between properties product closed order intervals properties underlying space finite product intervals always interval surprisingly converse result wide class compacta first countable space finite whenever has property product nonnegative intervals closed property product interval itself convex assumption needed seen the stone ech compactification mathbb product intervals continuous boundary functions always interval nbsp proved product positive intervals always interval product nonnegative intervals always convex finally square roots intervals investigated results similar type