### Banach manifolds of algebraic elements in the algebra $\mathcal{L}$ (H) of bounded linear operatorsof bounded linear operators

Given a complex Hilbert space H, we study the manifold $$\mathcal{A}$$ of algebraic elements in $$Z=\mathcal{L}\left(H\right)$$ . We represent $$\mathcal{A}$$ as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine...