[Paper Review] Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebra

Marcelo Laca, Nadia S. Larsen+1 more|ArXiv.org|Sep 11, 2006

Advanced Operator Algebra Research15 references4 citations

TL;DR

This paper investigates the finite part of the Connes-Marcolli C*-algebra via the reduced Hecke C*-algebra associated with a Hecke pair of semidirect product groups arising from 2×2 integer and rational matrices. It establishes a decomposition of the fixed point algebra under the action of finite integral ideles into a tensor product over primes, proving uniqueness of regular KMS states for β > 1 and conjecturing their completeness for β ≠ 0,1.

ABSTRACT

We study a C*-dynamical system arising from the ring inclusion of the 2 imes 2 integer matrices in the rational ones. The orientation preserving affine groups of these rings form a Hecke pair that is closely related to a recent construction of Connes and Marcolli; our dynamical system consists of the associated reduced Hecke C*-algebra endowed with a canonical dynamics defined in terms of the determinant function. We show that the Schlichting completion also consists of affine groups of matrices, over the finite adeles, and we obtain results about the structure and induced representations of the Hecke C*-algebra. In a somewhat unexpected parallel with the one dimensional case studied by Bost and Connes, there is a group of symmetries given by an action of the finite integral ideles, and the corresponding fixed point algebra decomposes as a tensor product over the primes. This decomposition allows us to obtain a complete description of a natural class of equilibrium states which conjecturally includes all KMS_β-states for β e 0,1.

Motivation & Objective

• To understand the structure of the reduced Hecke C*-algebra associated with the Hecke pair (P, P₀), where P is the semidirect product of Mat₂(ℤ) and SL₂(ℤ), and P₀ is the corresponding subgroup.
• To analyze the C*-dynamical system defined by the determinant function on this Hecke C*-algebra.
• To determine the equilibrium states—specifically KMS states—of this system, with a focus on regularity and uniqueness.
• To explore the role of the finite integral ideles in acting as symmetries of the system, leading to a tensor product decomposition of the fixed point algebra.
• To conjecture that the identified class of KMS states includes all such states for β ≠ 0,1, extending the Bost-Connes model to higher rank.

Proposed method

• The authors construct the reduced Hecke C*-algebra C*r(P, P₀) as a crossed product associated with the Hecke pair (P, P₀), where P acts on the upper half-plane and matrices over finite adeles.
• They compute the Schlichting completion of the Hecke pair, showing it consists of affine groups over the finite adeles, enabling a global analysis of the algebraic structure.
• They identify a canonical dynamics σ on C*r(P, P₀) defined via the determinant function, which allows the study of KMS states.
• They analyze the action of the finite integral ideles Ź* on the system, leading to a decomposition of the fixed point algebra as a tensor product over primes.
• They define equilibrium states φβ,w via averaging over cosets of SL₂(ℤ̂) in Mat₂(ℤ̂), weighted by det(s)−β, and prove regularity and extremality.
• They use the structure of the Toeplitz-Hecke algebra at each prime p and the representation πp to characterize states via trace-like conditions, such as φβ,p(v*p v_p) = p + 1.

Experimental results

Research questions

• RQ1 Does the reduced Hecke C*-algebra C*r(P, P₀) admit a canonical dynamics, and how does it relate to the determinant function?
• RQ2 How does the action of the finite integral ideles Ź* decompose the fixed point algebra of the C*-dynamical system?
• RQ3 Are the KMS states constructed via averaging over cosets extremal and regular at every prime?
• RQ4 Is the class of KMS states φβ,w for β > 1 unique and complete for β ≠ 0,1?
• RQ5 Can the uniqueness of KMS states on the finite part be established despite the presence of non-regular states in the Toeplitz-Hecke subalgebras?

Key findings

• The finite part of the Connes-Marcolli C*-algebra, identified as C*r(P, P₀), admits a canonical dynamics σ defined via the determinant function.
• The fixed point algebra under the action of Ź* decomposes as a tensor product over primes of the fixed algebras at each prime p.
• For β > 1, the state φβ,w defined by φβ,w(f) = ζ(β)⁻¹ζ(β−1)⁻¹∑s∈Γ\S det(s)⁻β f(sw) is a regular, extremal KMS state for the dynamics σ.
• The measure μβ,w induced by φβ,w is supported on a single SL₂(ℤ̂)-orbit, confirming extremality and regularity at every prime.
• For β < 0, no σ-KMS β-states exist on C*r(P, P₀), as shown by a contradiction involving the norm of isometries u_p.
• The conjecture is formulated that all KMS β-states for β ≠ 0,1 are regular and included in the class of states φβ,w, with uniqueness for β > 1.