Topological properties of spaces of ideals of the minimal tensor product

TL;DR

This paper explores the topological structure of the space of ideals in minimal tensor products of C*-algebras, extending known results and providing new proofs about continuous functions and primal ideals.

Contribution

It generalizes Brown's 1977 result on extending continuous functions and offers a new proof regarding minimal primal ideals in tensor products of C*-algebras.

Findings

Continuous functions on Prim(A_1)×Prim(A_2) extend to Prim(A_1⊗A_2) in T_1 spaces.

New proof for the structure of minimal primal ideals in tensor products.

Topological analysis of prime ideals in minimal tensor products.

Abstract

One shows that for two C^*-algebras A_1 and A_2 any continuous function on Prim(A_1)\times Prim(A_2) can be continuously extended to Prim(A_1\otimes A_2) provided it takes its values in a T_1 topological space. This generalizes a 1977 result of L.G. Brown. A new proof is given for a result of R.J. Archbold about the space of minimal primal ideals of A_1\otimes A_2. To obtain these two results one makes use of the topological properties of the space of prime ideals of the minimal tensor product.