A classification of the commutative Banach perfect semi-fields of characteristic 1. Applications

We define and study the concept of commutative Banach perfect semi-field of characteristic 1 by using results from Connes-Consani. We prove a Gelfand-Naimark type theorem allowing to identify such a Banach semi-field F to the semi-field of all the continuous functions on a compact space which we define as the spectrum of F. We give many applications.