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

For C*-algebras A₁, A₂ the map (I₁, I₂) → ker(qI₁ qI₂) from Id(A₁) Id(A₂) into Id(A₁ ⊗_min A₂) 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(A) and q_I is the quotient map of A onto A/I. This result is used to show that any continuous function on Prim(A₁) × Prim(A₂) with values into a T₁ topological space can be extended to Prim(A₁ ⊗_min A₂). This enlarges the scope of [7, corollary 35] that dealt only with scalar valued functions. A new proof for a result of Archbold [3] about the space of minimal primal ideals of A₁ ⊗_min A ₂ is obtained also by using the homeomorphism mentioned above. New proofs of the equivalence of the property (F) of Tomiyama for A₁ ⊗_min A₂ with certain other properties are presented.