[Paper Review] When a C*-algebra is a coefficient algebra for a given endomorphism

V. I. Bakhtin, А. В. Лебедев|ArXiv.org|Feb 19, 2005

Advanced Operator Algebra Research4 references10 citations

TL;DR

This paper establishes a necessary and sufficient condition for a unital C*-algebra to be a coefficient algebra associated with a given endomorphism: the existence of a complete transfer operator for the endomorphism. The authors prove that such a coefficient algebra exists if and only if there exists a faithful, non-degenerate representation of the algebra on a Hilbert space and a partial isometry satisfying specific covariance conditions with respect to the endomorphism and its associated transfer operator.

ABSTRACT

The paper presents a criterion for a C*-algebra to be a coefficient algebra associated with a given endomorphism

Motivation & Objective

• To determine when a unital C*-algebra can serve as a coefficient algebra for a given endomorphism.
• To establish a criterion based on the existence of a transfer operator compatible with the endomorphism.
• To connect the theory of coefficient algebras with the structure of crossed products via endomorphisms.
• To clarify the role of transfer operators in the construction of C*-algebras generated by endomorphisms and partial isometries.

Proposed method

• Introduce the concept of a transfer operator δ* for a given endomorphism δ on a unital C*-algebra A, satisfying δ*(δ(a)b) = aδ*(b) for all a,b ∈ A.
• Define a complete transfer operator as one for which the associated projection P = δ*(1) satisfies Pδ(a)P = δ(a) for all a ∈ A.
• Construct a Hilbert space H as a direct sum of copies of A, indexed by integers, with an inner product defined via the transfer operator.
• Define a representation π: A → L(H) by setting π(a)h_n = a h_n for n ≥ 0 and π(a)h_n = δ^{|n|}(a) h_n for n < 0.
• Define a partial isometry U on H via U h_n = δ(h_{n-1}) for n > 0 and U h_n = δ^{|n|+1}(1) h_{n-1} for n ≤ 0.
• Verify that the operators satisfy the covariance relations: Uπ(a)U* = π(δ(a)) and U*π(a)U = π(δ*(a)) for all a ∈ A, proving the existence of the required representation.

Experimental results

Research questions

• RQ1 Under what conditions is a unital C*-algebra a coefficient algebra for a given endomorphism?
• RQ2 What is the role of the transfer operator in characterizing coefficient algebras?
• RQ3 When does an endomorphism of a C*-algebra arise from a partial isometry in a C*-algebra extension?
• RQ4 How can the existence of a complete transfer operator be linked to the structure of the coefficient algebra?
• RQ5 What conditions ensure that a given endomorphism is generated by an isometry?

Key findings

• A unital C*-algebra A is a coefficient algebra for a given endomorphism δ if and only if there exists a complete transfer operator δ* for δ.
• The existence of a complete transfer operator ensures the construction of a faithful, non-degenerate representation π of A on a Hilbert space H and a partial isometry U such that the covariance relations Uπ(a)U* = π(δ(a)) and U*π(a)U = π(δ*(a)) hold for all a ∈ A.
• An endomorphism δ is generated by an isometry if and only if there exists a complete transfer operator δ* with δ*(1) = 1.
• The projection P = δ*(1) plays a central role in determining the range of the endomorphism and the structure of the associated C*-algebra.
• The construction of the Hilbert space H as a direct sum of copies of A, equipped with an inner product defined via δ*, ensures the well-definedness and adjointness of the operators U and U*.
• The proof establishes that the representation π is faithful, relying on the Gelfand-Naimark construction and the positivity of the transfer operator.