  A hermitian algebra is a unital associative ${\mathbb C}$-algebra endowed
with an involution such that the spectra of self-adjoint elements are contained
in ${\mathbb R}$. In the case of an algebra ${\mathcal A}$ endowed with a
Mackey-complete, locally convex topology such that the set of invertible
elements is open and the inversion mapping is continuous, we construct the
smooth structures on the appropriate versions of flag manifolds. Then we prove
that if such a locally convex algebra ${\mathcal A}$ is endowed with a
continuous involution, then it is a hermitian algebra if and only if the
natural action of all unitary groups $U_n({\mathcal A})$ on each flag manifold
is transitive.
