Esteban Andruchow

Projective spaces of aC*-algebra

Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C*-algebraA with a fixed projectionp. The resulting spaceP(p) admits a rich geometrical structure as a holomorphic manifold and a homogeneous reductive space of the invertible group ofA. Moreover, several metrics (chordal, spherical, pseudo-chordal, non-Euclidean-in Schwarz-Zaks terminology) are considered, allowing a comparison amongP(p), the Grassmann manifold ofA and the space of positive elements which are unitary with respect to the bilinear form induced by the reflection ε=2p−1. Among several metrical results, we prove that geodesics are unique and of minimal length when measured with the spherical and non-Euclidean metrics.

Geometry of the sphere of a Hilbert module

The sphere SX =fx2 X:hx;xi =1 g of a right Hilbert C-module X over a unital C-algebra B is studied using differential geometric techniques. An action of the unitary group of the algebra LB(X) of adjointable B-module operators of X makes SX a homogeneous space of this group. A reductive structure is introduced, as well as a Finsler metric. Metric properties of the geodesic curves are established. In the case B a von Neumann algebra and X self-dual, the fundamental group of SX is computed.

Finsler geometry and actions of the p-Schatten unitary groups

Let p be an even positive integer and U p (H) the Banach-Lie group of unitary operators u which verify that u — 1 belongs to the p-Schatten ideal B p (H). Let O be a smooth manifold on which U p (H) acts transitively and smoothly. Then one can endow O with a natural Finsler metric in terms of the p-Schatten norm and the action of U p (H). Our main result establishes that for any pair of given initial conditions x ∈ O and X ∈ (TO) x there exists a curve 6(t) = e t z · x in O, with z a skew-hermitian element in the p-Schatten class, such that δ(0) = x and δ(0) = X, which remains minimal as long as t∥z∥ p < π/4. Moreover, δ is unique with these properties. We also show that the metric space (O, d) (where d is the rectifiable distance) is complete. In the process we establish minimality results in the groups U p (H) and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit O = {uAu*: u ∈ U p (H)} of a self-adjoint operator A ∈ B(H).

GEOMETRY OF CONDITIONAL EXPECTATIONS AND FINITE INDEX

Let e be the Jones projection associated to a conditional expectation where are von Neumann algebras. We prove that the similarity orbit of e by invertibles of is an homogeneous space iff the index of E is finite. If also , then this orbit is a covering space for the orbit of E.

PROJECTIVE SPACE OF A C*-MODULE

Let X be a right Hilbert C*-module over A. We study the geometry and the topology of the projective space of X, consisting of the orthocomplemented submodules of X which are generated by a single element. We also study the geometry of the p-sphere Sp(X) and the natural fibration , where Sp(X) = {x ∈ X: = p}, for p ∈ A a projection. The projective space and the p-sphere are shown to be homogeneous differentiable spaces of the unitary group of the algebra ℒA(X) of adjointable operators of X. The homotopy theory of these spaces is examined.

The space of spectral measures is a homogeneous reductive space

We prove that the space of spectral measures on a W*-algebra is a smooth Banach manifold in a natural way and that the action of the group of invertible elements of the algebra by inner automorphisms makes it into a reductive homogeneous space. This gives a geometric structure for the set of normal operators with the same spectrum.

GEOMETRY OF UNITARIES IN A FINITE ALGEBRA: VARIATION FORMULAS AND CONVEXITY

Given a C*-algebra

Nonpositively Curved Metric in the Positive Cone of a Finite Von Neumann Algebra

\mathcal{A}

GRASSMANNIANS OF A FINITE ALGEBRA IN THE STRONG OPERATOR TOPOLOGY

with trace τ, we compute the first and second variation formulas for the p-energy functional Fp of the unitary group

Joint similarity orbits with local cross sections

\mathcal{U}_{\mathcal{A}}