Regular ideals under the ideal intersection property

TL;DR

This paper proves an isomorphism between regular ideals of a C*-algebra and its subalgebra under certain conditions, clarifying the structure of ideals in operator algebras.

Contribution

It establishes a new isomorphism between regular ideals of a C*-algebra and those of a subalgebra satisfying specific properties.

Findings

The map J ↦ J ∩ A is an isomorphism between ideal boolean algebras.

Regular ideals correspond precisely under the intersection map.

The result applies to subalgebras satisfying the ideal intersection property and axiom (INV).

Abstract

The goal of this short note is to prove that when $A$ is a closed *-subalgebra of a C*-algebra $B$ satisfying the ideal intersection property plus a mild axiom (INV), then the map $J\mapsto J\cap A$ establishes an isomorphism from the boolean algebra of all regular ideals of $B$ to the boolean algebra of all regular, invariant ideals of $A$.