Invariant function algebras on compact commutative homogeneous spaces

TL;DR

This paper characterizes invariant function algebras on compact commutative homogeneous spaces, linking antisymmetry to the multiplicativity of invariant measures and exploring implications for Stein manifolds with transitive group actions.

Contribution

It establishes a criterion for antisymmetry of invariant function algebras on commutative homogeneous spaces and connects this to the structure of Stein manifolds with transitive group actions.

Findings

A function algebra is antisymmetric iff the invariant measure is multiplicative on it.

If a compact orbit is a commutative homogeneous space, it is a real form of the Stein manifold.

Provides a characterization of invariant function algebras on these spaces.

Abstract

Let $M$ be a commutative homogeneous space of a compact Lie group $G$ and $A$ be a closed $G$-invariant subalgebra of the Banach algebra $C(M)$. A function algebra is called antisymmetric if it does not contain nonconstant real functions. By the main result of this paper, $A$ is antisymmetric if and only if the invariant probability measure on $M$ is multiplicative on $A$. This implies, for example, the following theorem: if $G^{\mathbb C}$ acts transitively on a Stein manifold $\cal M$, $v\in{\cal M}$, and the compact orbit $M=Gv$ is a commutative homogeneous space, then $M$ is a real form of $\cal M$.