I will introduce the Bost-Connes C*-dynamical
system from Number Theory exhibiting a phase
transition with spontaneous symmetry breaking.
The key ingredient in their construction
is a Hecke algebra, obtained from the ring
inclusion of the integers in the rationals.
I will show how this Hecke algebra
can be realized as a semigroup crossed product,
and how this alternative description
can be used to elucidate its structure
and to simplify the study of equilibrium.

Created: 2/21/01 Updated: 2/21/01