A classification of the commutative Banach perfect semi-fields of characteristic 1: applications

Abstract

We define and study the concept of commutative Banach perfect semi-field $$({\mathcal {F}}, \oplus , +)$$ of characteristic 1. The metric allowing to define the Banach structure comes from Connes (Contemp Math 546:83–113, 2011) and is constructed from a distinguished element $$E\in {\mathcal {F}}$$ satisfying a structural assumption. We define the spectrum $$S_E({\mathcal {F}})$$ as the set of characters $$\phi : ({\mathcal {F}}, \oplus , +) \rightarrow ({\mathbb {R}}, \max , +)$$ satisfying $$\phi (E)=1$$ . This set is shown to be naturally a compact space. Then we construct an isometric isomorphism of Banach semi-fields of Gelfand–Naimark type: $$\begin{aligned}&\Theta : ({\mathcal {F}}, \oplus , +) \rightarrow (C^0(S_E({\mathcal {F}}), {\mathbb {R}}), {\max }, +)\\&\quad X \mapsto \Theta _X: \phi \mapsto \phi (X)= \Theta _X (\phi )\,. \end{aligned}$$ In this way, $${\mathcal {F}}$$ is naturally identified with the set of real valued continuous functions on $$S_E({\mathcal {F}})$$ . Our proof relies on a study of the congruences on $${\mathcal {F}}$$ and on a new Gelfand–Mazur type Theorem. As a first application, we prove that the spectrum of the Connes–Consani Banach algebra of the Witt vectors of $$({\mathcal {F}}, E)$$ coincides with $$S_E({\mathcal {F}})$$ . We give many other applications. Then we study the case of the commutative cancellative perfect semi-rings $$({\mathcal {R}}, \oplus , +)$$ and also give structure theorems in the Banach case. Lastly, we use these results to propose the foundations of a new scheme theory in the characteristic 1 setting. We introduce a topology of Zariski type on $$S_E({\mathcal {F}})$$ and the concept of valuation associated to a character $$\phi \in S_E({\mathcal {F}})$$ . Then we come to the notions of $$v-$$ local semi-ring and of scheme.