Geometric characterization of hermitian algebras with continuous inversion

TL;DR

This paper characterizes hermitian algebras with continuous inversion through geometric properties of flag manifolds, linking algebraic involution with transitive unitary group actions.

Contribution

It establishes a geometric characterization of hermitian algebras with continuous inversion via transitivity of unitary group actions on flag manifolds.

Findings

Hermitian algebras with continuous inversion have smooth flag manifold structures.

The algebra is hermitian iff the unitary groups act transitively on flag manifolds.

Provides a geometric criterion for hermitian property in locally convex algebras.

Abstract

A hermitian algebra is a unital associative ${\mathbb C}$-algebra endowed with an involution such that the spectra of self-adjoint elements are contained in ${\mathbb R}$. In the case of an algebra ${\mathcal A}$ endowed with a Mackey-complete, locally convex topology such that the set of invertible elements is open and the inversion mapping is continuous, we construct the smooth structures on the appropriate versions of flag manifolds. Then we prove that if such a locally convex algebra ${\mathcal A}$ is endowed with a continuous involution, then it is a hermitian algebra if and only if the natural action of all unitary groups $U_n({\mathcal A})$ on each flag manifold is transitive.