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We define and study the concept of commutative Banach perfect semi-field $$({\mathcal {F}}, \oplus , +)$$(F,⊕,+) 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 $$E\in {\mathcal {F}}$$E∈F satisfying a structural assumption. We define the spectrum $$S_E({\mathcal {F}})$$SE(F) as the set of characters $$\phi : ({\mathcal {F}}, \oplus , +) \rightarrow ({\mathbb {R}}, \max , +)$$ϕ:(F,⊕,+)→(R,max,+) satisfying $$\phi (E)=1$$ϕ(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: $$\begin{aligned}&\Theta : ({\mathcal {F}}, \oplus , +) \rightarrow (C^0(S_E({\mathcal {F}}), {\mathbb {R}}), {\max }, +)\\&\quad X \mapsto \Theta _X: \phi \mapsto \phi (X)= \Theta _X (\phi )\,. \end{aligned}$$Θ:(F,⊕,+)→(C0(SE(F),R),max,+)X↦ΘX:ϕ↦ϕ(X)=ΘX(ϕ).In this way, $${\mathcal {F}}$$F is naturally identified with the set of real valued continuous functions on $$S_E({\mathcal {F}})$$SE(F). Our proof relies on a study of the congruences on $${\mathcal {F}}$$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 $$({\mathcal {F}}, E)$$(F,E) coincides with $$S_E({\mathcal {F}})$$SE(F). We give many other applications. Then we study the case of the commutative cancellative perfect semi-rings $$({\mathcal {R}}, \oplus , +)$$(R,⊕,+) 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 $$S_E({\mathcal {F}})$$SE(F) and the concept of valuation associated to a character $$\phi \in S_E({\mathcal {F}})$$ϕ∈SE(F). Then we come to the notions of $$v-$$v-local semi-ring and of scheme.

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