Multiplication of convex sets in C(K) spaces

José Pedro Moreno, and Rolf Schneider. "Multiplication of convex sets in C(K) spaces." Studia Mathematica 232.2 (2016): 173-187. <http://eudml.org/doc/286442>.

@article{JoséPedroMoreno2016,
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 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.},
author = {José Pedro Moreno, Rolf Schneider},
journal = {Studia Mathematica},
keywords = {C(K) space; order interval; product of sets},
language = {eng},
number = {2},
pages = {173-187},
title = {Multiplication of convex sets in C(K) spaces},
url = {http://eudml.org/doc/286442},
volume = {232},
year = {2016},
}

TY - JOUR
AU - José Pedro Moreno
AU - Rolf Schneider
TI - Multiplication of convex sets in C(K) spaces
JO - Studia Mathematica
PY - 2016
VL - 232
IS - 2
SP - 173
EP - 187
AB - 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 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.
LA - eng
KW - C(K) space; order interval; product of sets
UR - http://eudml.org/doc/286442
ER -