Isotropy fibers of ideals in groupoid C${}^{\ast }$-algebras

Abstract

Given a locally compact étale groupoid and an ideal $I$ in its groupoid C${}^{\ast }$-algebra, we show that $I$ defines a family of ideals in group C${}^{\ast }$-algebras of the isotropy groups and then study to which extent $I$ is determined by this family. As an application we obtain the following results: (a) prove that every proper ideal is contained in an induced primitive ideal; (b) describe the maximal ideals; (c) classify the primitive ideals for a class of graded groupoids with essentially central isotropy.

arXiv comment22 pages; v3: minor corrections; to appear in Adv. Math. v2: Minor corrections and improvements, and a new section has been added with a new result on fibers of the singular ideal

References in corpus (6)

• C*-simplicity and the unique trace property for discrete groups
• KMS states on the C*-algebras of non-principal groupoids
• Cartan subalgebras in C*-algebras of Hausdorff etale groupoids
• A non-amenable groupoid whose maximal and reduced $C^{\ast }$-algebras are the same
• Non-amenable principal groupoids with weak containment
• A new uniqueness theorem for the tight C*-algebra of an inverse semigroup

Cited by in corpus (2)

• Maximal ideals of reduced group C*-algebras and Thompson's groups
• Relative topological principality and the ideal intersection property for groupoid C*-algebras