On the structure group of an infinite dimensional JB-algebra

TL;DR

This paper extends the understanding of the structure group, automorphism group, and cone-preserving group of infinite dimensional JB-algebras, describing their Banach-Lie group structures and components, especially for self-adjoint operators on Hilbert spaces.

Contribution

It provides a comprehensive description of the structure, automorphism, and cone-preserving groups as embedded Banach-Lie groups in the infinite dimensional setting, including for operator algebras.

Findings

The groups are embedded Banach-Lie groups of GL(V).

Descriptions of the components via cones, isotopes, and projections.

Automorphism group forms a smooth principal bundle over the unitary group.

Abstract

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 $G(\Omega)$ and the automorphism group $Aut(V)$ of the algebra $V$ are embedded Banach-Lie groups of $GL(V)$, and that each of the inclusions $Aut(V)\subset G(\Omega)\subset Str(V)$ 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 $Str(V), G(\Omega), Aut(V)$, 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.