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KMS, Gibbs & zeta function

Time to wrap up this series on the Bost-Connes algebra. Here’s what we have learned so far : the convolution product on double cosets

[1Z01][1Q0Q>0]/[1Z01]

is a noncommutative algebra, the Bost-Connes Hecke algebra H, which is a bi-chrystalline graded algebra (somewhat weaker than ‘strongly graded’) with part of degree one the group-algebra Q[Q/Z]. Further, H has a natural one-parameter family of algebra automorphisms σt defined by σt(Xn)=nitXn and σt(Yλ)=Yλ.

For any algebra A together with a one-parameter family of automorphisms σt one is interested in KMS-states or Kubo-Martin-Schwinger states with parameter β, KMSβ (this parameter is often called the ‘invers temperature’ of the system) as these are suitable equilibria states. Recall that a state is a special linear functional ϕ on A (in particular it must have norm one) and it belongs to KMSβ if the following commutation relation holds for all elements a,bA

ϕ(aσiβ(b))=ϕ(ba)

Let us work out the special case when A is the matrix-algebra Mn(C). To begin, all algebra-automorphisms are inner in this case, so any one-parameter family of automorphisms is of the form

σt(a)=eitHaeitH

where eitH is the matrix-exponential of the nxn matrix H. For any parameter β we claim that the linear functional

ϕ(a)=1tr(eβH)tr(aeβH)

is a KMS-state.Indeed, we have for all matrices a,bMn(C) that

ϕ(aσiβ(b))=1tr(eβH)tr(aeβHbeβHeβH)

=1tr(eβH)tr(aeβHb)=1tr(eβH)tr(baeβH)=ϕ(ba)

(the next to last equality follows from cyclic-invariance of the trace map).
These states are usually called Gibbs states and the normalization factor 1tr(eβH) (needed because a state must have norm one) is called the partition function of the system. Gibbs states have arisen from the study of ideal gases and the place to read up on all of this are the first two chapters of the second volume of “Operator algebras and quantum statistical mechanics” by Ola Bratelli and Derek Robinson.

This gives us a method to construct KMS-states for an arbitrary algebra A with one-parameter automorphisms σt : take a simple n-dimensional representation π : AMn(C), find the matrix H determining the image of the automorphisms π(σt) and take the Gibbs states as defined before.

Let us return now to the Bost-Connes algebra H. We don’t know any finite dimensional simple representations of H but, sure enough, have plenty of graded simple representations. By the usual strongly-graded-yoga they should correspond to simple finite dimensional representations of the part of degree one Q[Q/Z] (all of them being one-dimensional and corresponding to characters of Q/Z).

Hence, for any uG=pZ^p (details) we have a graded simple H-representation Su=nN+Cen with action defined by

{πu(Xn)(em)=enmπu(Yλ)(em)=e2πinu.λem

Here, u.λ is computed using the ‘chinese-remainder-identification’ A/R=Q/Z (details).

Even when the representations Su are not finite dimensional, we can mimic the above strategy : we should find a linear operator H determining the images of the automorphisms πu(σt). We claim that the operator is defined by H(en)=log(n)en for all nN+. That is, we claim that for elements aH we have

πu(σt(a))=eitHπu(a)eitH

So let us compute the action of both sides on em when a=Xn. The left hand side gives πu(nitXn)(em)=nitemn whereas the right-hand side becomes

eitHπu(Xn)eitH(em)=eitHπu(Xn)mitem=

eitHmitemn=(mn)itmitemn=nitemn

proving the claim. For any parameter β this then gives us a KMS-state for the Bost-Connes algebra by

ϕu(a)=1Tr(eβH)Tr(πu(a)eβH)

Finally, let us calculate the normalization factor (or partition function) 1Tr(eβH). Because eβH(en)=nβen we have for that the trace

Tr(eβH)=nN+1nβ=ζ(β)

is equal to the Riemann zeta-value ζ(β) (at least when β>1).

Summarizing, we started with the definition of the Bost-Connes algebra H, found a canonical one-parameter subgroup of algebra automorphisms σt and computed that the natural equilibria states with respect to this ‘time evolution’ have as their partition function the Riemann zeta-function. Voila!

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