[Paper Review] The ideal structures of crossed products of Cuntz algebras by quasi-free actions of abelian groups

Takeshi Katsura|ArXiv.org|Aug 7, 2001

Advanced Operator Algebra Research8 references4 citations

TL;DR

This paper completely determines the ideal structures of crossed products of Cuntz algebras by quasi-free actions of abelian groups, providing a necessary and sufficient condition for simplicity and primitivity, computing Connes spectra and K-groups, and establishing a correspondence between ideals and closed subsets of topological spaces derived from the dual group and action spectrum. The results generalize and reprove A. Kishimoto's simplicity criterion using a novel topological approach to ideal classification.

ABSTRACT

We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and K-groups of our algebras.

Motivation & Objective

• To fully characterize the ideal structure of crossed products of Cuntz algebras under quasi-free actions of abelian groups.
• To reprove and generalize A. Kishimoto’s result on the simplicity of such crossed products using a new topological framework.
• To determine a necessary and sufficient condition for these algebras to be primitive.
• To compute the Connes spectra and K-groups of the resulting C*-algebras.
• To establish a bijective correspondence between ideals and closed subsets of a topological space derived from the action’s spectrum.

Proposed method

• Use of the gauge action to classify ideals invariant under the group action, establishing a one-to-one correspondence with closed subsets of the dual group Γ satisfying specific conditions.
• Introduction of the spectrum Ω of the action as a finite subset of the dual group Γ, which governs the ideal structure.
• Application of the theory of Cuntz-Pimsner algebras to compute K-groups via K-theoretic techniques.
• Construction of a topological space Ω/Ω̄ and use of invariant sets under translation by Ω to classify ideals in non-regular cases.
• Use of the strong Connes spectrum as a key invariant, computed via the ideal structure and duality theory.
• Reduction of the general case to simpler cases via quotienting by the kernel of the action, leading to isomorphisms with simpler crossed products.

Experimental results

Research questions

• RQ1 What is the complete ideal structure of a crossed product of a Cuntz algebra by a quasi-free action of an abelian group?
• RQ2 Under what conditions is such a crossed product simple or primitive?
• RQ3 How can the Connes spectrum and K-theory of these algebras be computed?
• RQ4 What is the relationship between the spectrum of the action and the lattice of ideals?
• RQ5 How do the ideal structures differ when the action satisfies or fails to satisfy Condition 5.1 (analogous to Condition (K) in graph algebras)?

Key findings

• The set of ideals invariant under the gauge action is in one-to-one correspondence with closed subsets of the dual group Γ satisfying specific algebraic conditions, as formalized in Theorem 3.14.
• A necessary and sufficient condition for simplicity of the crossed product is that the action’s spectrum generates a dense subgroup of the dual group, which reproves and extends A. Kishimoto’s result (Theorem 4.8).
• The algebra is primitive if and only if the action’s spectrum satisfies a certain minimality condition on the dual group, as stated in Theorem 4.12.
• When the action satisfies Condition 5.1 (analogous to Condition (K)), all ideals are gauge-invariant, and the ideal lattice is completely described by closed subsets of the dual group Γ.
• In the general case (when Condition 5.1 fails), the ideals correspond bijectively to closed subsets of a topological space derived from the action’s spectrum, as shown in Theorem 5.49.
• For the case G = ℝ, the crossed product is simple if and only if ω is aperiodic and of type (+) or (−), with the primitive ideal space homeomorphic to 𝕋 in the non-simple case.