On Conditions of Local Lineal Convexity Generalized to Commutative Algebras

Abstract

The notion of lineally convex domains in the finite-dimensional complex space $\mathbb {C}^n$ and some of their properties are generalized to the finite-dimensional space $\mathcal {A}^n$, $n\ge 2$, that is the Cartesian product of n commutative and associative algebras $\mathcal {A}$. Namely, a domain in $\mathcal {A}^n$ is said to be (locally)$\mathcal {A}$-lineally convex if, for every boundary point of the domain, there exists a hyperplane in $\mathcal {A}^n$ passing through the point but not intersecting the domain (in some neighborhood of the point). It is proved that $\mathcal {A}_3$-lineal convexity of bounded domains with a smooth boundary in the space $\mathcal {A}_3^n$ follows from their local $\mathcal {A}_3$-lineal convexity for a three-dimensional algebra $\mathcal {A}_3$.