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

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.