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BC stands for Bi-Crystalline graded

Towards the end of the Bost-Connes for ringtheorists post I freaked-out because I realized that the commutation morphisms with the Xn were given by non-unital algebra maps. I failed to notice the obvious, that algebras such as Q[Q/Z] have plenty of idempotents and that this mysterious ‘non-unital’ morphism was nothing else but multiplication with an idempotent…

Here a sketch of a ringtheoretic framework in which the Bost-Connes Hecke algebra H is a motivating example (the details should be worked out by an eager 20-something). Start with a suitable semi-group S, by which I mean that one must be able to invert the elements of S and obtain a group G of which all elements have a canonical form g=s1s21. Probably semi-groupies have a name for these things, so if you know please drop a comment.

The next ingredient is a suitable ring R. Here, suitable means that we have a semi-group morphism
ϕ : SEnd(R) where End(R) is the semi-group of all ring-endomorphisms of R satisfying the following two (usually strong) conditions :

  1. Every ϕ(s) has a right-inverse, meaning that there is an ring-endomorphism ψ(s) such that ϕ(s)ψ(s)=idR (this implies that all ϕ(s) are in fact epi-morphisms (surjective)), and

  2. The composition ψ(s)ϕ(s) usually is NOT the identity morphism idR (because it is zero on the kernel of the epimorphism ϕ(s)) but we require that there is an idempotent EsR (that is, Es2=Es) such that ψ(s)ϕ(s)=idREs

The point of the first condition is that the S-semi-group graded ring A=sSXsR is crystalline graded (crystalline group graded rings were introduced by Fred Van Oystaeyen and Erna Nauwelaarts) meaning that for every sS we have in the ring A the equality XsR=RXs where this is a free right R-module of rank one. One verifies that this is equivalent to the existence of an epimorphism ϕ(s) such that for all rR we have rXs=Xsϕ(s)(r).

The point of the second condition is that this semi-graded ring A can be naturally embedded in a G-graded ring B=g=s1s21GXs1RXs2 which is bi-crystalline graded meaning that for all rR we have that rXs=Xsψ(s)(r)Es.

It is clear from the construction that under the given conditions (and probably some minor extra ones making everything stand) the group graded ring B is determined fully by the semi-group graded ring A.

what does this general ringtheoretic mumbo-jumbo have to do with the BC- (or Bost-Connes) algebra H?

In this particular case, the semi-group S is the multiplicative semi-group of positive integers N×+ and the corresponding group G is the multiplicative group Q×+ of all positive rational numbers.

The ring R is the rational group-ring Q[Q/Z] of the torsion-group Q/Z. Recall that the elements of Q/Z are the rational numbers 0λ<1 and the group-law is ordinary addition and forgetting the integral part (so merely focussing on the ‘after the comma’ part). The group-ring is then

Q[Q/Z]=0λ<1QYλ with multiplication linearly induced by the multiplication on the base-elements Yλ.Yμ=Yλ+μ.

The epimorphism determined by the semi-group map ϕ : N×+End(Q[Q/Z]) are given by the algebra maps defined by linearly extending the map on the base elements ϕ(n)(Yλ)=Ynλ (observe that this is indeed an epimorphism as every base element Yλ=ϕ(n)(Yλn).

The right-inverses ψ(n) are the ring morphisms defined by linearly extending the map on the base elements ψ(n)(Yλ)=1n(Yλn+Yλ+1n++Yλ+n1n) (check that these are indeed ring maps, that is that ψ(n)(Yλ).ψ(n)(Yμ)=ψ(n)(Yλ+μ).

These are indeed right-inverses satisfying the idempotent condition for clearly ϕ(n)ψ(n)(Yλ)=1n(Yλ++Yλ)=Yλ and

ψ(n)ϕ(n)(Yλ)=ψ(n)(Ynλ)=1n(Yλ+Yλ+1n++Yλ+n1n)=Yλ.(1n(Y0+Y1n++Yn1n))=YλEn

and one verifies that En=1n(Y0+Y1n++Yn1n) is indeed an idempotent in Q[Q/Z]. In the previous posts in this series we have already seen that with these definitions we have indeed that the BC-algebra is the bi-crystalline graded ring

B=H=mnQ×+XmQ[Q/Z]Xn

and hence is naturally constructed from the skew semi-group graded algebra A=mN×+XmQ[Q/Z].

This (probably) explains why the BC-algebra H is itself usually called and denoted in C-algebra papers the skew semigroup-algebra Q[Q/Z]N×+ as this subalgebra (our crystalline semi-group graded algebra A) determines the Hecke algebra completely.

Finally, the bi-crystalline idempotents-condition works well in the settings of von Neumann regular algebras (such as all limits of finite dimensional semi-simples, for example Q[Q/Z]) because such algebras excel at idempotents galore

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