Polynomial functions for locally compact group actions

TL;DR

This paper develops a theory of polynomial functions associated with local group actions on spaces, extending classical concepts to local actions and exploring their properties for applications like bicrossproducts.

Contribution

It introduces and analyzes polynomial functions for local group actions, generalizing existing notions from global actions and providing tools for studying bicrossproduct structures.

Findings

Defined polynomial functions in the context of local group actions

Established properties and structure of polynomial functions

Applied theory to actions of groups on themselves and related structures

Abstract

Consider a locally compact group $G$ and a locally compact space $X$. A local right action of $G$ on $X$ is a continuous map $(x,p)\mapsto x\cdot p$ from an open subset $\Gamma$ of the Cartesian product $X\times G$ to $X$ satisfying certain obvious properties. A global right action of $G$ on $X$ gives rise to a global left action of $G$ on the space $C_c(X)$ of continuous complex functions with compact support in $X$ by the formula $p\,\cdot f:x\mapsto f(x\cdot p)$. In the case of a local action, one still can define $p\,\cdot f$ in $C_c(X)$ by this formula for $f\in C_c(X)$ and $p$ in a neighborhood $V_f$ of the identity in $G$. This yields a local left action of $G$ on $C_c(X)$. Given a local right action of $G$ on $X$, a function $f\in C_c(X)$ is called polynomial if there is a neighborhood $V$ of the identity, contained in $V_f$, and a finite-dimensional subspace $F$ of $C_c(X)$ containing all the functions $v\cdot f$ for $v\in V$. In this paper we study such polynomial functions. If $G$ acts on itself by multiplication, we are also interested in the local actions obtained by restricting it to an open subset of $G$. This is the typical situation that is encountered in our paper on bicrossproducts of groups with a compact open subgroup. In fact, the need for a better understanding of polynomial functions for that case has led us to develop the theory in general here.