Regular Ideals, Ideal Intersections, and Quotients

Authors

Jonathan H. Brown, University of Dayton
Adam H. Fuller, Ohio University
David R. Pitts, University of Nebraska - LincolnFollow
Sarah A. Reznikoff, Virginia Tech

Document Type

Article

Date of this Version

12-15-2023

Citation

Integr. Equ. Oper. Theory (2024) 96:3 https://doi.org/10.1007/s00020-023-02753-4

Comments

Open access.

Abstract

Let B ⊆ A be an inclusion of C^∗ -algebras. We study the relationship between the regular ideals of B and regular ideals of A.We show that if B ⊆ A is a regular C^∗ -inclusion and there is a faithful invariant conditional expectation from A onto B, then there is an isomorphism between the lattice of regular ideals of A and invariant regular ideals of B. We study properties of inclusions preserved under quotients by regular ideals. This includes showing that if D ⊆ A is a Cartan inclusion and J is a regular ideal in A, then D/(J ∩D) is a Cartan subalgebra of A/J. We provide a description of regular ideals in the reduced crossed product of a C^∗ -algebra by a discrete group.