Invariant function algebras on compact commutative homogeneous spaces

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 $\mathcal{M}$, $v\in \mathcal{M}$, and the compact orbit $M=Gv$ is a commutative homogeneous space, then $M$ is a real form of $\mathcal{M}$.