  The paper contains a description of the maximal ideal spaces (spectra)
$\cM_A$ of bi-invariant function algebras $A$ on a compact group $G$. There are
natural compatible structures in $\cM_A$: it is a compact topological semigroup
with involution, polar decomposition, and analytic structure. The paper
contains a description of $\cM_A$ and related results on function algebras; for
example, a bi-invariant function algebra on a connected compact Lie group is
antisymmetric if and only if the Haar measure of its maximal torus is
multiplicative on it. Some results are extended to the case of compact
commutative homogeneous spaces. As a consequence, we get an infinite
dimensional version of the Hilbert--Mumford criterion for commutative (as
homogeneous spaces) orbits of compact connected Lie groups, where one parameter
semigroup is replaced by a finite sequence of them.
  Keywords: Maximal ideal space, invariant function algebra, complex Lie
semigroup, commutative homogeneous space, Hilbert--Mumford criterion

Functional Analysis;Rings and Algebras