The Glimm space of the minimal tensor product of C^∗-algebras

2012/12/17 by McConnell, David
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1212.3971

Abstract

We show that for C^∗-algebras A and B, there is a natural open bijection from Glimm(A) × Glimm (B) to Glimm(A ⊗_α B) (where A ⊗_α B denotes the minimal C^∗-tensor product), and identify a large class of C^∗-algebras A for which the map is continuous for arbitrary B. As a consequence we determine the structure space of the centre of the multiplier algebra ZM(A ⊗_α B) in terms of Glimm(A) and Glimm (B), and give necessary and sufficient conditions for the inclusion ZM(A) ⊗ ZM(B) ⊆ ZM(A ⊗_α B) to be surjective. Further we show that when the Glimm spaces are considered as sets of ideals, the map (G,H) ↦ G ⊗_α B + A ⊗_α H implements the above bijection, extending a result of Kaniuth from a 1996 paper by eliminating the assumption of property (F).

Related

• Glimm Spaces of Separable C*-algebras
• Exact C^∗-algebras and C₀(X)-structure
• Topological properties of spaces of ideals of the minimal tensor product
• Compact Subsets of the Glimm Space of a C^*-algebra
• Ideals in Operator Space Projective Tensor Product of C^*-algebras