Larotonda spaces : homogeneous spaces and conditional expectations

Abstract

We define a Larotonda space as a quotient space $P=U_A /U_B$ of the unitary groups of C*-algebras $1\in B\subset A$ with a faithful unital conditional expectation $$ \Phi:A\to B. $$ In particular, $B$ is complemented in $A$, a fact which implies that $P$ has $C^\infty$ 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 $U_A$-invariant Finsler metric in $P$. given a point $\rho\in P$ and a tangent vector $X\in(T P)_\rho$, we consider the problem of wether the geodesic $\delta$ 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.