Maximal ideal spaces of invariant function algebras on compact groups

Let $G$ be a compact group and $A$ be a closed subalgebra of $C(G)$ which is invariant under the left and right shifts in $G$. We consider maximal ideal spaces (spectra) $\mathcal{M}_A$ of these algebras. They can be defined as closed sub-bialgebras of $C(G)$. There is a natural semigroup structure in $\mathcal{M}_A$ that admits an involutive anti-automorphism and a polar decomposition. If $\mathcal{M}_A\ne G$ then $\mathcal{M}_A$ has a nontrivial analytic structure. If $G$ is a Lie group then every idempotent in $\mathcal{M}_A$ is the identity element of a complex Lie semigroup embedded to $\mathcal{M}_A$. The semigroup $\mathcal{M}_A$ admits an analogue of Cartan's decomposition $KAK$, namely, $\mathcal{M}_A=G\widehat{T}G$, where $\widehat{T}$ is an abelian semigroup that is a hull of the maximal torus $T$.