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KMS STATES ON THE C∗ALGEBRAS OF NONPRINCIPAL GROUPOIDS
"... Abstract. We describe KMSstates on the C∗algebras of etale groupoids in terms of measurable fields of traces on the C∗algebras of the isotropy groups. We use this description to analyze tracial states on the transformation groupoid C∗algebras and to give a short proof of recent results of Cuntz, ..."
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Abstract. We describe KMSstates on the C∗algebras of etale groupoids in terms of measurable fields of traces on the C∗algebras of the isotropy groups. We use this description to analyze tracial states on the transformation groupoid C∗algebras and to give a short proof of recent results of Cuntz, Deninger and Laca on the Toeplitz algebras of the ax+ b semigroups of the rings of integers in number fields.
Type III1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers
 J. Funct. Anal
"... Abstract. We complete the analysis of KMSstates of the Toeplitz algebra T (N o N×) of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature β in the critical interval 1 ≤ β ≤ 2, the unique KMSβstate is of type ..."
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Abstract. We complete the analysis of KMSstates of the Toeplitz algebra T (N o N×) of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature β in the critical interval 1 ≤ β ≤ 2, the unique KMSβstate is of type III1. We prove this by reducing the type classification from T (N o N×) to that of the symmetric part of the BostConnes system, with a shift in inverse temperature. To carry out this reduction we first obtain a parametrization of the Nica spectrum of N o N × in terms of an adelic space. Combining a characterization of traces on crossed products due to the second author with an analysis of the action of N o N × on the Nica spectrum, we can also recover all the KMSstates of T (N o N×) originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMSβstates for β> 2 that does not ostensibly come from an action of T on the C∗algebra.
BostConnes systems, Hecke algebras, and induction
"... Abstract. We consider a Hecke algebra naturally associated with the affine group with totally positive multiplicative part over an algebraic number field K and we show that the C∗algebra of the BostConnes system for K can be obtained from our Hecke algebra by induction, from the group of totally p ..."
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Abstract. We consider a Hecke algebra naturally associated with the affine group with totally positive multiplicative part over an algebraic number field K and we show that the C∗algebra of the BostConnes system for K can be obtained from our Hecke algebra by induction, from the group of totally positive principal ideals to the whole group of ideals. Our Hecke algebra is therefore a full corner, corresponding to the narrow Hilbert class field, in the BostConnes C∗algebra of K; in particular, the two algebras coincide if and only if K has narrow class number one. Passing the known results for the BostConnes system for K to this corner, we obtain a phase transition theorem for our Hecke algebra. In another application of induction we consider an extension L/K of number fields and we show that the BostConnes system for L embeds into the system obtained from the BostConnes system for K by induction from the group of ideals in K to the group of ideals in L. This gives a C∗algebraic correspondence from the BostConnes system for K to that for L. Therefore the construction of BostConnes systems can be extended to a functor from number fields to C∗dynamical systems with equivariant correspondences as morphisms. We use this correspondence to induce KMSstates and we show that for β> 1 certain extremal KMSβstates for L can be obtained, via induction and rescaling, from KMS[L:K]βstates for K. On the other hand, for 0 < β ≤ 1 every KMS[L:K]βstate for K induces to an infinite weight.
BOSTCONNES SYSTEMS ASSOCIATED WITH FUNCTION FIELDS
"... Abstract. With a global function field K with constant field Fq, a finite set S of primes in K and an abelian extension L of K, finite or infinite, we associate a C∗dynamical system. The systems, or at least their underlying groupoids, defined earlier by Jacob using the ideal action on Drinfeld mod ..."
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Abstract. With a global function field K with constant field Fq, a finite set S of primes in K and an abelian extension L of K, finite or infinite, we associate a C∗dynamical system. The systems, or at least their underlying groupoids, defined earlier by Jacob using the ideal action on Drinfeld modules and by ConsaniMarcolli using commensurability of Klattices are isomorphic to particular cases of our construction. We prove a phase transition theorem for our systems and show that the unique KMSβstate for every 0 < β ≤ 1 gives rise to an ITPFIfactor of type IIIq−βn, where n is the degree of the algebraic closure of Fq in L. Therefore for n = + ∞ we get a factor of type III0. Its flow of weights is a scaled suspension flow of the translation by the Frobenius element on Gal(F̄q/Fq).
ERGODICITY OF THE ACTION OF K ∗ ON AK
, 2013
"... Connes gave a spectral interpretation of the critical zeros of zeta and Lfunctions for a global field K using a space of square integrable functions on the space AK/K ∗ of adele classes. It is known that for K = Q the spaceAK/K ∗ cannot be understood classically, or in other words, the action ofQ ..."
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Connes gave a spectral interpretation of the critical zeros of zeta and Lfunctions for a global field K using a space of square integrable functions on the space AK/K ∗ of adele classes. It is known that for K = Q the spaceAK/K ∗ cannot be understood classically, or in other words, the action ofQ ∗ onAQ is ergodic. We prove that the same is true for any global field K, in both the number field and function field cases.