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Additional info for Operator Algebras: The Abel Symposium 2004 (Abel Symposia)
When n is prime one has Φn (u(1/n)) = n πn . d. of the polynomials σm (xd ), where d divides n and m = n/d. For x = u(1/n), one has σm (xd ) = m πm ∈ J0 , for any divisor d|n. Thus, we obtain Φn (u(1/n)) ∈ J0 as required. Recall that πm = µm µ∗m and µ∗m µm = 1. Thus, the KMS∞ states of the BC system vanish identically on the πm . The KMSβ condition for 1 < β < ∞ shows that KMSβ states take value m−β on πm . It follows that the restriction of KMS∞ states to the abelian part vanish identically on J.
This unique KMS state takes values ϕβ (e(m/n)) = f−β+1 (n)/f1 (n), where µ(d)(n/d)k , fk (n) = d|n • with µ the Möbius function, and f1 is the Euler totient function. For 1 < β ≤ ∞, elements of Eβ are indexed by the classes of invertible ˆ hence by the classical points (16) of the ˆ ∗ = GL1 (Z), Q-lattices ρ ∈ Z noncommutative Shimura variety (26), Eβ ∼ = GL1 (Q)\GL1 (A)/R∗+ ∼ = CQ /DQ ∼ = IQ /Q∗+ , (34) with IQ as in (1). In this range of temperatures, the values of states ϕβ,ρ ∈ Eβ on the elements e(r) ∈ A1,Q is given, for 1 < β < ∞ by polylogarithms evaluated at roots of unity, normalized by the Riemann zeta function, ∞ ϕβ,ρ (e(r)) = 1 n−β ρ(ζrk ).
It is also compatible with the rational subalgebras A1,Q and A2,Q . This compatibility can be seen using the cyclotomic condition deﬁned in (62) and in Deﬁnition 11. The compatibility between the BC system and the CM system for imaginary quadratic ﬁelds is discussed in Proposition 22. 44 Alain Connes, Matilde Marcolli, and Niranjan Ramachandran 4 Quantum Statistical Mechanics for Imaginary Quadratic Fields In the Kronecker–Weber case, the maximal abelian extension of Q is generated by the values of the exponential function at the torsion points Q/Z of the group C/Z = C∗ .
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