Larotonda spaces : homogeneous spaces and conditional expectations

• Autores: Andruchow, Esteban; Recht, Lázaro
• Año de publicación: 2016
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: Journal Of Mathematics. Feb. 2016; 27(2): 1-17; https://www.worldscientific.com/toc/ijm/27/02
• Materia: Finsler Metric; Geodesic; Homogeneous Space; Unitary Group of A C-Algebra
• Institución: Universidad Nacional de General Sarmiento

Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina.
Fil: Recht, Lázaro. Universidad de los Andes, Bogotá. Departamento de Matemáticas; Colombia.
Fil: Recht, Lázaro. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina.
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.