arXiv subjects

José Luna

Publications and source records attributed to José Luna.

2 recordsLinked to original sources

arXiv2024

Connections and Finsler geometry of the structure group of a JB-algebra

We endow the Banach-Lie structure group $Str(V)$ of an infinite dimensional JB-algebra $V$ with a left-invariant connection and Finsler metric, and we compute all the quantities of its connection. We show how this connection reduces to $G(Ω)$, the group of transformations that preserve the positive cone $Ω$ of the algebra $V$, and to $Aut(V)$, the group of Jordan automorphisms of the algebra. We present the cone $Ω$ as an homogeneous space for the action of $G(Ω)$, therefore inducing a quotient Finsler metric and distance. With the techniques introduced, we prove the minimality of the one-parameter groups in $Ω$ for any symmetric gauge norm in $V$. We establish that the two presentations of the Finsler metric in $Ω$ give the same distance there, which helps us prove the minimality of certain paths in $G(Ω)$ for its left-invariant Finsler metric.

math.DG↗

arXiv2022

On the structure group of an infinite dimensional JB-algebra

We extend several results for the structure group of a real Jordan algebra $V$, to the setting of infinite dimensional JB-algebras. We prove that the structure group $Str(V)$, the cone preserving group $G(Ω)$ and the automorphism group $Aut(V)$ of the algebra $V$ are embedded Banach-Lie groups of $GL(V)$, and that each of the inclusions $Aut(V)\subset G(Ω)\subset Str(V)$ are of embedded Banach-Lie subgroups. We give a full description of the components of $Str(V)$ via cones, isotopes and central projections. We apply these results to $V=B(H)_{sa}$ the special JB-algebra of self-adjoint operators on an infinite dimensional complex Hilbert space, describing the groups $Str(V), G(Ω), Aut(V)$, their Banach-Lie algebras and their connected components. We show that the action of the unitary group of $H$ on $Aut(V)$ has smooth local cross sections, thus $Aut(V)$ is a smooth principal bundle over the unitary group, with circle structure group.

math.FA↗