Metric Geometry in Homogeneous Spaces of the Unitary Group of a C*-Algebra. Part II. Geodesics Joining Fixed Endpoints

Abstract.

This article focuses on the study of the metric geometry of homogeneous spaces $\mathcal{P} = U(\mathcal{A})/U(\mathcal{B})$ (the unitary group of a C*-algebra $\mathcal{A}$ modulo the unitary group of a C*-subalgebra $\mathcal{B}$) where the invariant Finsler metric in $\mathcal{P}$ is induced by the quotient norm of $\mathcal{A}/\mathcal{B}.$ Under the assumption that $\mathcal{B}$ is of compact type, i.e. when the unitary group is relatively compact in the strong operator topology, this work presents local and global versions of Hopf-Rinow-like theorems: given points $\rho_0,\rho_1 \in \mathcal{P},$ there exists a minimal uniparametric group curve joining ρ₀ and ρ₁.