Hecke algebras and the finite part of the Connes-Marcolli C*-algebra
Seminar Room 1, Newton Institute
In a joint work with M. Laca and S. Neshveyev, we study the C*-algebra of a Hecke pair arising from the ring inclusion of the 2x2 integer matrices in the rational ones. This algebra is included in the multiplier algebra of the C*-algebra considered by Connes and Marcolli in their work on quantum statistical mechanics of Q-lattices. We uncover structural properties of our Hecke algebra which somewhat surprinsingly mirror the one dimensional case of the C*-algebra of Bost and Connes associated to the inclusion of the integers in the rationals; there is a symmetry group given by an action of the finite integral ideles, and the fixed-point algebra decomposes as a tensor product over the primes. In classifying KMS-states, we are again guided by the one-dimensional case, but have to deal with difficulties intrinsic to our two-dimensional set-up.