The space of ideals in the minimal tensor product of $C^*$-algebras

TL;DR

This paper studies the structure of ideals in the minimal tensor product of $C^*$-algebras, establishing a homeomorphism between ideal pairs and their tensor products, and providing new proofs of Tomiyama's property (F).

Contribution

It characterizes the ideal space of the minimal tensor product and offers new proofs of key properties related to tensor products of $C^*$-algebras.

Findings

Homeomorphism between ideal pairs and their tensor products

Density of the image of the ideal map in the ideal space

New proofs of the equivalence of property (F) with other properties

Abstract

For $C^*$-algebras $A_1, A_2$ the map $(I_1,I_2)\to ker(q_{I_1}\otimes q_{I_2})$ from $Id^{\prime}(A_1)\times Id^{\prime}(A_2)$ into $Id^{\prime}(A_1\otimes_{\mathrm{min}} A_2) is a homeomorphism onto its image which is dense in the range. Here, for a $C^*$-algebra $A$, the space of all proper closed two sided ideals endowed with an adequate topology is denoted $Id^{\prime}(A)$ and $q_I$ is the quotient map of $A$ onto $A/I$. New proofs of the equivalence of the property (F) of Tomiyama for $A_1\otimes_{\mathrm{min}} A_2$ with certain other properties are presented.