Dimension of spaces of polynomials on abelian topological semigroups

TL;DR

This paper investigates the structure and dimensionality of polynomial spaces on abelian topological semigroups and their extensions to groups, highlighting conditions for finite dimensionality and differences between polynomial classes.

Contribution

It characterizes when polynomial spaces on semigroups are finite dimensional and compares Riss polynomials with general polynomial spaces, providing new insights into their structure.

Findings

Polynomial extension from semigroups to groups is unique under certain conditions.

Finite dimensionality of polynomial spaces depends on properties of the semigroup and the polynomial class.

Identifies classes of groups where polynomial spaces are finite dimensional.

Abstract

In this paper we study (continuous) polynomials $p: J\to X$, where $J$ is an abelian topological semigroup and $X$ is a topological vector space. If $J$ is a subsemigroup with non-empty interior of a locally compact abelian group $G$ and $G=J-J$, then every polynomial $p$ on $J$ extends uniquely to a polynomial on $ G$. It is of particular interest to know when the spaces $P^n (J,X)$ of polynomials of order at most $n$ are finite dimensional. For example we show that for some semigroups the subspace $P^n_{R} (J,\mathbf{C})$ of Riss polynomials (those generated by a finite number of homomorphisms $\alpha: J\to \mathbf{R}$) is properly contained in $P^n (G,\mathbf{C})$. However, if $P^1 (J,\mathbf{C})$ is finite dimensional then $P^n_{R} (J,\mathbf{C})= P^n (J,\mathbf{C})$. Finally we exhibit a large family of groups for which $P^n (G,\mathbf{C})$ is finite dimensional.