[Paper Review] Stability of C^*-algebra associated with the twisted CCR

Daniil Proskurin, Yu. S. Samoĭlenko|ArXiv.org|Dec 27, 2001

Advanced Operator Algebra Research6 references3 citations

TL;DR

This paper establishes that the universal C*-algebra associated with twisted canonical commutation relations (TCCR) for any μ ∈ (−1, 1) is isomorphic to a C*-algebra generated by partial isometries satisfying specific relations. It further proves that the Fock representation of this algebra is faithful, providing a complete structural and representation-theoretic characterization of the TCCR C*-algebra across the full parameter range.

ABSTRACT

The $C^*$-algebra associated with the twisted CCR constructed by W. Pusz and S.L. Woronowicz is considered. It is proved that the $C^*$-algebras corresponding to different values of parameter $0<=μ<1$ are isomorphic. It is proved that Fock representation is faithful on both *-algebra and $C^*$-algebra levels.

Motivation & Objective

• To determine the structure of the universal C*-algebra Aμ associated with twisted canonical commutation relations (TCCR) for μ ∈ (−1,1).
• To establish isomorphism between Aμ and a C*-algebra A0 generated by partial isometries with specific relations.
• To prove that the Fock representation of Aμ is faithful, ensuring the representation captures the full algebraic structure.

Proposed method

• Constructing a polar decomposition for the generators of Aμ and defining a family of partial isometries {t̂i} via an inductive procedure.
• Proving that the partial isometries {t̂i} satisfy the relations t̂i*tj = δij(1 − ∑k<i t̂k t̂k*), and t̂j t̂i = 0 for i ≠ j.
• Using the universal property of C*-algebras to define surjective homomorphisms between A0 and Aμ, showing mutual inverses and thus isomorphism.
• Analyzing irreducible representations of A0 using tensor product decompositions involving the unilateral shift S on l2(ℕ).
• Establishing a homomorphism from the Fock representation C*-algebra to any irreducible representation C*-algebra of A0, proving faithfulness via restriction and tensor product structure.
• Leveraging the nuclearity of C*(S,S*) ≅ T(C(𝕋)) to define the required homomorphism between tensor products of C*-algebras.

Experimental results

Research questions

• RQ1 Is the universal C*-algebra Aμ for twisted CCR isomorphic to a simpler C*-algebra for all μ ∈ (−1,1)?
• RQ2 Does the Fock representation of Aμ remain faithful across the entire parameter range μ ∈ (−1,1)?
• RQ3 Can the structure of Aμ be fully described via partial isometries satisfying canonical relations independent of μ?
• RQ4 How do irreducible representations of A0 relate to the Fock representation in terms of C*-algebra embeddings?

Key findings

• For all μ ∈ (−1,1), the C*-algebra Aμ is isomorphic to A0, the universal C*-algebra generated by partial isometries satisfying t̂i*tj = δij(1 − ∑k<i t̂k t̂k*) and t̂j t̂i = 0 for i ≠ j.
• The Fock representation of Aμ is faithful, meaning it injectively maps Aμ into bounded operators on a Hilbert space.
• The isomorphism between Aμ and A0 is established via explicit homomorphisms constructed from polar decompositions and inductive definitions of partial isometries.
• Irreducible representations of A0 are classified via tensor products of shifts and projections, with the Fock representation corresponding to the case j = d.
• The faithfulness of the Fock representation is proven by constructing a C*-algebra homomorphism from the Fock C*-algebra to any irreducible representation C*-algebra, showing the kernel is trivial.
• The structure of A0 allows for a uniform description of representations using the unilateral shift S and projections, enabling the construction of the required homomorphism for faithfulness.