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