On the $C^{*}$-algebra associated with the full adele ring of a number field

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The multiplicative group of a number field acts by multiplication on the full adele ring of the field. Generalising a theorem of Laca and Raeburn, we explicitly describe the primitive ideal space of the crossed product C^{*} -algebra associated with this action. We then distinguish real, complex, and finite places of the number field using K-theoretic invariants. Combining these results with a recent rigidity theorem of the authors implies that any * -isomorphism between two such C^{*} -algebras gives rise to an isomorphism of the underlying number fields that is constructed from the * -isomorphism.