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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&lt; r&lt;∞) are real-analytic direct submanifolds of Z. Using the C*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine connection ∇ are defined on M, and the geodesics are computed. If M is the orbit of a finite rank projection, then a G-invariant Riemann structure is defined with respect to which ∇ is the Levi-Civita connection.

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