Compact Subsets of the Glimm Space of a C^*-algebra

2012/03/15 by Aldo J. Lazar, Lazar, Aldo J.
Mathematics · #46L05 #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1203.3350

openalex publication_date 2012/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If A is a σ-unital C^*-algebra and a is a strictly positive element of A then for every compact subset K of the complete regularization Glimm(A) of Prim(A) there exists α> 0 such that K⊂ \G∈ Glimm(A) | ‖a + G‖≥ α\. This extends a 1974 result of J. Dauns to all σ-unital C^*-algebras. However, there is a C^*-algebra A and a compact subset of Glimm(A) that is not contained in any set of the form \G∈ Glimm(A) | ‖a + G‖≥ α\, a∈ A and α> 0.

Citations

• Locally Inner Actions on C₀(X)-Algebras

Related

• Glimm Spaces of Separable C*-algebras
• The Glimm space of the minimal tensor product of C^∗-algebras
• Restrictions of pure states to subspaces of C^*-algebras
• Linear maps preserving regularity in C∗-algebras
• Regular inclusions of simple unital C^*-algebras