  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.

Rings and Algebras;Algebraic Geometry;Number Theory;Operator Algebras