On the C*-algebra associated with the full adele ring of a number field
Chris Bruce, Takuya Takeishi
TL;DR
The paper analyzes C*-algebras arising from the action of the multiplicative group $K^*$ on the full adele ring $\mathbb{A}_K$ of a number field $K$, defining $\mathscr{A}_K=C_0(\mathbb{A}_K)\rtimes K^*$ and studying its primitive ideal space. It establishes an explicit description of $\mathrm{Prim}(\mathscr{A}_K)$ via the quasi-orbit space and introduces place-based subquotients of the auxiliary algebra $\mathscr{B}_K$, using K-theory boundary maps to distinguish real, complex, and finite places. From these K-theoretic invariants, it is shown that any *-isomorphism between $\mathscr{A}_K$ and $\mathscr{A}_L$ yields a field isomorphism between $K$ and $L$, constructed from the induced isomorphism; in particular, $\mathscr{A}_K$ and $\mathscr{A}_L$ are isomorphic if and only if $K\cong L$. This work provides a full rigidity result in C*-algebras of number-theoretic origin for actions on spaces that are not zero-dimensional, linking operator-algebraic invariants directly to arithmetic structure.
Abstract
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.