The Hecke algebra of Bost-Connes revisited
by
Magnus Landstad
NTNU, Trondheim, Norway

I shall look at some (but not all) developments after the introduction of the Hecke algebra of Bost and Connes. A general Hecke pair (G, H) where G is a group and H a subgroup can more easily described via its Schlichting completion (G*, H*) with H* a compact open subgroup of G* (due to Tzanev).

To a Hecke algebra there is also a Banach *-algebra and a C*-algebra and for the representation theory of these 3 algebras both Fell’s and Rieffel’s version of Morita equivalence is needed. Some old results about Banach *-algebras reappear.

The BC Hecke algebra ax+b-group has two nice properties we shall study separately:

1) The semigroup S={s| ad(s) maps H into H} gives an ordering of G.

2) G is a semi direct product QN, where H is a subgroup of the normal subgroup N.

Time permitting, we may also look at

3) Generalised Hecke algebras, here it turns out that there is a different Schlichting completion of (G, H).

4) Cuntz’ ax+b-semigroup and why does it contain the BC Hecke algebra?

Date received: May 12, 2008