Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory

Bost, Jean-Benôıt; Connes, Alain

doi:10.1007/bf01589495

Cited by 251 publications

(534 citation statements)

References 11 publications

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“…The results of [2] show that the Galois theory of the cyclotomic field Q cycl appears naturally in the BC system when considering the action of the group of symmetries of the system on the extremal KMS states at zero temperature. In the case of 1-dimensional Q-lattices up to scaling, the algebra of coordinates C(Ẑ) can be regarded as the algebra of homogeneous functions of weight zero on the space of 1-dimensional Q-lattices.…”

Section: Tower Powermentioning

confidence: 95%

“…As proved in [9], the algebra A 1,Q is the same as the rational subalgebra considered in [2], generated over Q by the µ n and the exponential functions (3.17) e(r)(ρ) := exp(2πiρ(r)), for ρ ∈ Hom(Q/Z, Q/Z), and r ∈ Q/Z, with relations e(r + s) = e(r)e(s), e(0) = 1, e(r) * = e(−r), µ * n µ n = 1, µ k µ n = µ kn , and…”

Section: Tower Powermentioning

confidence: 99%

“…The construction of BC [2] was partially generalized to other global fields (i.e. number fields and function fields) in [18,14,4].…”

Section: Quantum Statistical Mechanics For Imaginary Quadratic Fieldsmentioning

confidence: 99%

See 2 more Smart Citations

KMS states and complex multiplication

Connes

Marcolli

Ramachandran

2005

Sel. math., New ser.

146

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Section: Tower Powermentioning

confidence: 95%

Section: Tower Powermentioning

confidence: 99%

See 1 more Smart Citation

KMS states and complex multiplication

Connes

Marcolli

Ramachandran

2005

Sel. math., New ser.

146

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“…We bypass this obstacle by breaking the complexity of the algebra of N O(A, α) into two steps, one of which is carried out at the automorphic dilation level. For the second part, we were intrigued by the ongoing program of Laca and Raeburn [24], as well as by the growing interest in the structure of KMS states on C*-algebras (deducting symmetry and/or phase transition breaking) from the seminal work of Bost and Connes [3]. See for example [1,8,13,15,21,22,23,24,25,26] to mention but a few inspiring works.…”

Section: Introductionmentioning

confidence: 99%

On Nica-Pimsner Algebras of C*-Dynamical Systems Over $\mathbb{Z}_+^n$

Kakariadis

2016

Int Math Res Notices

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Abstract. We examine Nica-Pimsner algebras associated with semigroup actions of Z n + on a C*-algebra A by * -endomorphisms. We give necessary and sufficient conditions on the dynamics for exactness and nuclearity of the Nica-Pimsner algebras. Furthermore we parameterize the KMS states at finite temperature by the tracial states on A. A parametrization is also shown for KMS states at zero temperature (resp. ground states) by the tracial states on A (resp. states on A). Finally we give a formula for obtaining tracial states on the Nica-Pimsner algebras. IntroductionThis project lies in the intersection of three ongoing programs on analysing C*-algebras associated with C*-dynamics. There has been a continuous interest for a generalised C*-crossed product construction of possibly noninvertible semigroup actions. (to mention only a few). The key idea is to consider the quotient of a Fock algebra by an ideal of redundancies. Our standing point of view follows Arveson's program on the C*-envelope [19], and suggests that this ideal is theŠilov ideal of a nonselfadjoint operator algebra in the sense of Arveson [2]. In contrast to the case of the usual C*-crossed products, these redundancies may be very complicated to allow an in-depth exploration. An alternate route is suggested in the joint work of the author with Davidson and Fuller [9, Introduction]: dilate a possibly non-invertible semigroup action α : P → End(A) to a group action β : G → Aut( B) such that the distinguished quotient related to α : Z n + → End(A) is a full corner of the usual C*-crossed product related to β : G → Aut( B). Consequently the examination of the distinguished quotient can be induced via the more exploit-able and more exploit-ed theory of usual C*-crossed products. This pathway was met with success for several semigroup actions including Ore semigroups and spanning cones in [9], and in earlier works of the author for P = Z + [16], and of the author with Katsoulis for P = F n + [18].2010 Mathematics Subject Classification. 46L05, 46L55, 46L30, 58B34.

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“…This result was not accessible without non-commutative geometric intuition. Bost-Connes and Connes have also shown [BC95], [Con99], that non-commutative geometry can be very useful for arithmetic questions. We refer the reader to Marcolli [Mar04] for a nice survey on non-commutative arithmetic geometry.…”

Section: Introductionmentioning

confidence: 99%

Three examples of non-commutative boundaries of Shimura varieties

Paugam¹

Noncommutative Geometry and Number Theory

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Abstract. Our modest aims in writing this paper were twofold: we first wanted to understand the linear algebra and algebraic group theoretic background of Manin's real multiplication program proposed in [Man]. Secondly, we wanted to find nice higher dimensional analogs of the non-commutative modular curve studied by Manin and Marcolli in [MM02]. These higher dimensional objects, that we call irrational or non-commutative boundaries of Shimura varieties, are double cosets spaces of the form Γ\G(R)/P (K), where G is a (connected) reductive Q-algebraic group,is a real parabolic subgroup corresponding to a rational parabolic subgroup P ⊂ G, and Γ ⊂ G(Q) is an arithmetic subgroup. Along the way, it also seemed clear that the spaces Γ\G(R)/M (K)A are of great interest, and sometimes more convenient to study. We study in this document three examples of these general spaces. These spaces describe degenerations of complex structures on tori in (multi)foliations.

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Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory

Cited by 251 publications

References 11 publications

KMS states and complex multiplication

KMS states and complex multiplication

On Nica-Pimsner Algebras of C*-Dynamical Systems Over $\mathbb{Z}_+^n$

Three examples of non-commutative boundaries of Shimura varieties

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