[Paper Review] The Cuntz Algebra Q_N and C*-Algebras of Product Systems

Jeong Hee Hong, Nadia S. Larsen+1 more|arXiv (Cornell University)|Aug 3, 2011

Advanced Operator Algebra Research10 references3 citations

TL;DR

This paper establishes that the Cuntz-Pimsner algebra of a specific product system over the multiplicative semigroup ℕ× is isomorphic to Cuntz's algebra 𝒬_ℕ. Using universal properties and Nica covariant representations, the authors provide a presentation of the Nica-Toeplitz algebra associated to the product system, proving that its Cuntz-Nica-Pimsner quotient is precisely 𝒬_ℕ, thereby linking operator algebras of product systems to a key object in noncommutative geometry and K-theory.

ABSTRACT

We consider a product system over the multiplicative semigroup N^x of Hilbert bimodules which is implicit in work of S. Yamashita and of the second named author. We prove directly, using universal properties, that the associated Nica-Toeplitz algebra is an extension of the C^*-algebra Q_N introduced recently by Cuntz.

Motivation & Objective

• To clarify the structure of the C*-algebra 𝒬_ℕ by realizing it as a Cuntz-Nica-Pimsner algebra of a product system over ℕ×.
• To provide a direct, explicit presentation of the Nica-Toeplitz algebra associated with a product system of Hilbert bimodules over ℕ×.
• To establish the isomorphism between the Cuntz-Nica-Pimsner algebra of the product system and Cuntz's algebra 𝒬_ℕ using universal properties and Nica covariance.
• To fill in technical details of Yamashita’s construction of a ∞-graph whose C*-algebra is 𝒬_ℕ, by identifying the underlying product system.
• To compare and contrast the present approach with alternative constructions, such as Laca-Raeburn’s Toeplitz algebra and Brownlowe et al.’s boundary representations.

Proposed method

• Construct a product system X of Hilbert bimodules over the multiplicative semigroup ℕ×, with fibers X_m spanned by elements Z^k 1_m.
• Define a Toeplitz representation ψ of X using a unitary u and isometries w_m satisfying relations (B1)–(B4), including u^*u = 1 and w_m w_m^* = u^m w_m w_m^* u^{*-m}.
• Prove that the representation ψ is Nica covariant by verifying the condition for all m,n ∈ ℕ× using the structure of the compact operators on X_m.
• Use the universal property of the Nica-Toeplitz algebra T_cov(X) to define a surjective *-homomorphism from T_cov(X) onto the algebra generated by u and w_m.
• Identify the Cuntz-Nica-Pimsner algebra NO(X) as the quotient of T_cov(X) by the ideal generated by differences i(f) − i^{(m)}(φ_m(f)) for f ∈ A and m ∈ ℕ×.
• Establish that NO(X) satisfies the relation ∑_{k=0}^{m−1} u^k w_m w_m^* u^{−k} = 1 for all m ∈ ℕ×, which characterizes Cuntz’s algebra 𝒬_ℕ.

Experimental results

Research questions

• RQ1 How can the Cuntz algebra 𝒬_ℕ be realized as a Cuntz-Nica-Pimsner algebra of a product system over ℕ×?
• RQ2 What is the precise presentation of the Nica-Toeplitz algebra associated with the product system over ℕ×?
• RQ3 How does the Nica covariance condition constrain the structure of the universal C*-algebra generated by the product system?
• RQ4 What is the relationship between the Cuntz-Nica-Pimsner algebra of this product system and the Laca-Raeburn Toeplitz algebra of ℕ ⋊ ℕ×?
• RQ5 Can the isomorphism between the Cuntz-Nica-Pimsner algebra and 𝒬_ℕ be established directly via universal properties without relying on classification theorems?

Key findings

• The Nica-Toeplitz algebra T_cov(X) associated with the product system X over ℕ× admits a presentation in terms of generators u (unitary) and w_m (isometries) satisfying relations (B1)–(B4), including u^*u = 1 and w_m w_m^* = u^m w_m w_m^* u^{*-m}.
• The Cuntz-Nica-Pimsner algebra NO(X) of the product system satisfies the relation ∑_{k=0}^{m−1} u^k w_m w_m^* u^{−k} = 1 for all m ∈ ℕ×, which is a defining relation of Cuntz’s algebra 𝒬_ℕ.
• The Cuntz-Nica-Pimsner algebra NO(X) is isomorphic to Cuntz’s algebra 𝒬_ℕ, as both are simple, purely infinite C*-algebras with free abelian K₁-group and satisfy the same universal relations.
• The isomorphism between NO(X) and 𝒬_ℕ is established via the universal property of 𝒬_ℕ and the simplicity of 𝒬_ℕ, ensuring the homomorphism is injective and surjective.
• The construction provides a direct realization of 𝒬_ℕ as a Cuntz-Nica-Pimsner algebra, offering a new perspective on its structure beyond graph C*-algebras or crossed products.
• The results confirm and extend Yamashita’s construction of a ∞-graph whose C*-algebra is 𝒬_ℕ, by identifying the underlying product system and verifying the universal relations explicitly.