Let $A$ be an algebra over $\mathbb{R}$. Say a norm on $A$ is a map $$ | \cdot | : A \rightarrow \mathbb{R}_{\geq 0}$$ such that $|ab| = |a| \cdot |b|$, $|a + b | \leq |a| + |b|$, and $|1| = 1$.
Say a ball is a subset of $A$ of the form $\{ a \in A : |a| \leq 1 \}$ for some norm $| \cdot |$.
Can we characterize subsets of $A$ which are intersections of balls?
$1$ and $0$ are in any such intersecction, and any intersection is closed under convex combinations. Does this characterize intersections of balls in $A$?