[Paper Review] C*-algebras associated with integral domains and crossed products by actions on adele spaces

Joachim Cuntz, Xin Li|ArXiv.org|Jun 26, 2009

Advanced Operator Algebra Research11 references4 citations

TL;DR

This paper establishes a duality theorem for C*-algebras associated with global fields, proving that crossed products over finite and infinite adele spaces are isomorphic. Using this duality, it computes the K-theory of ring C*-algebras for rings of integers in number fields, showing it is isomorphic to an exterior algebra over the torsion-free part of the multiplicative group, with a Z/2Z twist when the number of real embeddings is even and at least two.

ABSTRACT

We compute the K-theory for C*-algebras naturally associated with rings of integers in number fields. The main ingredient is a duality theorem for arbitrary global fields. It allows us to identify the crossed product arising from affine transformations on the finite adeles with the analogous crossed product algebra over the infinite adele space.

Motivation & Objective

• To provide complete proofs for earlier conjectured results on the K-theory of C*-algebras associated with the integers and global fields.
• To establish a general duality theorem between crossed products over finite and infinite adele spaces for arbitrary global fields.
• To compute the K-theory of ring C*-algebras associated with rings of integers in number fields under specific conditions on roots of unity.
• To determine the K-theory of a subalgebra of the ring C*-algebra excluding torsion units, clarifying its structure in terms of the multiplicative group.
• To show that the full crossed product over the full adele space is Morita equivalent to the group C*-algebra of the ax+b-group, enabling K-theory computation via group algebra methods.

Proposed method

• Prove a duality isomorphism between $ C_0(\mathbb{A}_f) \rtimes P_K $ and $ C_0(\mathbb{A}_\infty) \rtimes P_K $, where $ \mathbb{A}_f $ and $ \mathbb{A}_\infty $ are the finite and infinite adele spaces, and $ P_K $ is the ax+b-group over a global field $ K $.
• Use the duality to transfer K-theoretic computations from the totally disconnected finite adele space to the connected infinite adele space, where homotopy arguments are applicable.
• Apply the Pimsner-Voiculescu exact sequence iteratively to compute $ K_* $ of crossed products involving the action of $ \mu \times \Gamma_m' $, where $ \mu $ is the group of roots of unity.
• Use the elementary divisor theorem and the fact that the induced map on K-theory satisfies $ \beta_*^2 = \text{id} $ to analyze torsion in $ K_* $, particularly $ \mathbb{Z}/2\mathbb{Z} $-torsion.
• Establish that $ C_0(\mathbb{A}) \rtimes P_K \sim_M C^*(P_K) $, the group C*-algebra of the ax+b-group, via an equivariant Morita equivalence.
• Leverage the Morita equivalence to compute $ K_* $ of the full crossed product as $ K_0(C^*(\{\pm1\})) \otimes_\mathbb{Z} \Lambda^*(\mathbb{Q}_{>0}) $ in the case $ K = \mathbb{Q} $.

Experimental results

Research questions

• RQ1 What is the K-theory of the C*-algebra $ \mathfrak{A}[\mathcal{O}] $ associated with the ring of integers $ \mathcal{O} $ in a number field $ K $, particularly when $ K $ contains only $ \pm1 $ as roots of unity?
• RQ2 How does the duality between crossed products over the finite and infinite adele spaces facilitate K-theoretic computations?
• RQ3 Can the K-theory of the subalgebra of $ \mathfrak{A}[\mathcal{O}] $ generated by addition and multiplication operators (excluding roots of unity) be described in terms of the multiplicative group of $ K $?
• RQ4 What is the relationship between the full adele space crossed product $ C_0(\mathbb{A}) \rtimes P_K $ and the group C*-algebra $ C^*(P_K) $?
• RQ5 Does the K-theory of $ \mathfrak{A}[\mathcal{O}] $ encode arithmetic invariants like the class number of $ K $? If not, what finer structure is needed to recover such invariants?

Key findings

• The K-theory of $ \mathfrak{A}[\mathcal{O}] $ for a number field $ K $ with only $ \pm1 $ as roots of unity is isomorphic to $ \Lambda^*(\Gamma) \oplus (\mathbb{Z}/2\mathbb{Z} \otimes_\mathbb{Z} \Lambda^*(\Gamma)) $, where $ \Gamma $ is the torsion-free part of $ K^\times $, with the $ \mathbb{Z}/2\mathbb{Z} $-twist appearing if $ K $ has at least two real embeddings.
• The K-theory of the subalgebra of $ \mathfrak{A}[\mathcal{O}] $ excluding roots of unity is isomorphic to $ \Lambda^*(\Gamma) \oplus (\mathbb{Z}/2\mathbb{Z} \otimes_\mathbb{Z} \Lambda^*(\Gamma)) $, where $ \Gamma $ is the torsion-free part of $ K^\times $, and the $ \mathbb{Z}/2\mathbb{Z} $-factor appears precisely when the number of real embeddings of $ K $ is even and at least two.
• The crossed product $ C_0(\mathbb{A}_\infty) \rtimes P_K $ is isomorphic to $ \mathfrak{A}[\mathcal{O}] $ up to Morita equivalence, enabling the use of homotopy methods in K-theory computation.
• The crossed product $ C_0(\mathbb{A}) \rtimes P_K $ is Morita equivalent to the group C*-algebra $ C^*(P_K) $, and this equivalence is equivariant with respect to the multiplicative action.
• The K-theory of $ C^*(P_K) $ for $ K = \mathbb{Q} $ is $ K_0(C^*(\{\pm1\})) \otimes_\mathbb{Z} \Lambda^*(\mathbb{Q}_{>0}) $, showing a connection to exterior algebras over the positive rationals.
• The K-theory of $ \mathfrak{A}[\mathcal{O}] $ does not contain information about the class number of $ K $, suggesting that finer algebraic structures within $ \mathfrak{A}[\mathcal{O}] $ are required to recover such invariants.