The set of ideals of a C*-algebra can be given a natural topology, which restricts to the hull-kernel topology on the primitive ideals. Our primary interest is to study continuous maps on the space of all ideals, rather than on the subset of primitive ideals. We show how the properties of a map between primitive ideal spaces carry over to properties of the extended mapb etween their ideal spaces. As an application of these results we determine a number of properties of maps between ideal spaces of tensor products, both minimal and maximal. For example, for C.-algebras A and B, themap (ker π, ker η) ↦ ker (π ⊗ η) : Id(A)×Id (B) → Id (A ⊗ B) is a homeomorphism onto its range. Finally, we apply these results to tensor products of continuous C*- bundles.
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