Cohomogeneity One Groupoid Analysis of the Dynamical System of Rings of Continuous Functions

Abstract

Using the group $G(1)$ of invertible elements and the maximal ideals $\mathfrak{m}_x$ of the commutative algebra $C(X)$ of real-valued functions on a compact regular space $X$, we define a Borel action of the algebra on the measure space $(X,\mu)$ with $\mu$ a Radon measure. The zero sets $Z(X)$ of the algebra $C(X)$ is used to study the ergodicity of the $G(1)$-action via its action on the maximal ideals $\mathfrak{m}_x$ which defines an action groupoid $\mathcal{G} = \mathfrak{m}_x \ltimes G(1)$ trivialized on $X$. The resulting measure groupoid $(\mathcal{G},\mathcal{C})$ is used to define a proper action on the generalized space $\mathcal{M}(X)$. The existence of slice at each point of $\mathcal{M}(X)$ present it as a cohomogeneity-one $\mathcal{G}$-space. The dynamical system of the algebra $C(X)$ is defined by the action of the measure groupoid $(\mathcal{G},\mathcal{C}) \times \mathcal{M}(X) \to \mathcal{M}(X)$.