On maximal ideal spaces of invariant function algebras on homogeneous spaces of compact Lie groups

Let $H$ be a homogeneous space of a compact connected Lie group $G$ and $A$ be a closed $G$-invariant subalgebra of $C(H)$ which contains all constant functions. Every invariant algebra has the greatest invariant ideal $J$. We consider two extreme cases, $J=0$ and ${\mathop{\rm{codim}} J}=1$.
Let $J=0$. Then under some additional conditions it is possible to prove that $A$ and $H$ admit a finite dimensional realization: $G$ is a linear group, $H=Gv$ is the orbit of a vector $v$, and the orbit $H_{\mathbb C}=G^{\mathbb C}v$ is closed. Then $A$ is the uniform closure of the algebra of holomorphic polynomials on $H$ and its maximal ideal space ${\cal M}_{A}$ is the polynomial hull of $H$ which is contained in $H_{\mathbb C}$.
If $H$ is a commutative homogeneous space (i.e., the algebra of invariant differential operators on it is abelian) and $A$ is antisymmetric (i.e., it does not contain nonconstant real functions), then ${\mathop{\rm{codim}} J}=1$. In particular this is true for $H=G\times G/G$, and then ${\cal M}_{A}$ can be described in detail.