Larotonda spaces: Homogeneous spaces and conditional expectations

Abstract

We define a Larotonda space as a quotient space P = UA/UB of the unitary groups of C ∗ -algebras 1 ∈ B ⊂ A with a faithful unital conditional expectation Φ : A → B. In particular, B is complemented in A, a fact which implies that P has C∞ differentiable structure, with the topology induced by the norm of A. The conditional expectation also allows one to define a reductive structure (in particular, a linear connection) and a UAinvariant Finsler metric in P. given a point ρ ∈ P and a tangent vector X ∈ (TP)ρ, we consider the problem of wether the geodesic δ of the linear connection satisfying these inital data is (locally) minimal for the metric. We find a sufficient condition. Several examples are given, of locally minimal geodesics.