Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory

Abstract

In this paper, we construct a natural C*-dynamical system whose partition function is the Riemann ζ function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated a x+b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings ℤ⊂ℚ provides the desired C*-dynamical system, which admits the ζ function as partition function and the Galois group Gal(ℚcycl/ℚ) of the cyclotomic extension ℚcycl of ℚ as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature β=1 (cf. [Bos-C]). The original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe]). © 1995 Birkhäuser Verlag.