The space of ideals in the minimal tensor product of $C^{\ast }$-algebras

For $C^{\ast }$-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{m}\mathrm{i}\mathrm{n}$} A_2) is a homeomorphism onto its image which is dense in the range. Here, for a $C^{\ast }$-algebra $A$, the space of all proper closed two sided ideals endowed with an adequate topology is denoted $I{d}^{\mathrm{\prime }}(A)$ and $q_I$ is the quotient map of $A$ onto $A\mathrm{/}I$. New proofs of the equivalence of the property (F) of Tomiyama for $A_1\otimes_{$\mathrm{m}\mathrm{i}\mathrm{n}$} A_2$ with certain other properties are presented.