[Paper Review] A New Look at The Crossed-Product of a C*-algebra by an Endomorphism

Ruy Exel|ArXiv.org|Dec 12, 2000

Advanced Operator Algebra Research12 references4 citations

TL;DR

This paper introduces a new crossed-product construction for C*-algebras by endomorphisms, incorporating a transfer operator as an additional structure. It generalizes earlier constructions, particularly for monomorphisms with hereditary ranges, and shows that the Cuntz–Krieger algebra $\mathcal{O}_A$ arises naturally as the quotient of a universal C*-algebra defined by generators and relations involving the endomorphism and transfer operator.

ABSTRACT

We give a new definition for the crossed-product of a C*-algebra A by a *-endomorphism α, which depends not only on the pair (A,α) but also on the choice of a transfer operator (defined in the paper). With this we generalize some of the earlier constructions in the situations in which they behave best (e.g. for monomorphisms with hereditary range), but we get a different and perhaps more natural outcome in other situations. For example, we show that the Cuntz-Krieger algebra O_A arises as the result of our construction when applied to the corresponding Markov subshift and a very natural transfer operator.

Motivation & Objective

• To generalize existing crossed-product constructions for C*-algebras by endomorphisms, especially in cases where earlier methods fail or are unnatural.
• To incorporate the concept of a transfer operator as a fundamental ingredient in the crossed-product construction, motivated by classical dynamics and operator algebras.
• To provide a unified framework that recovers known examples—such as the Cuntz–Krieger algebra $\mathcal{O}_A$—as quotients of a universal C*-algebra generated by a C*-algebra, an endomorphism, and a transfer operator.
• To resolve limitations of prior constructions, such as the requirement of hereditary range or isometric implementers, by using a Hilbert bimodule structure and redundancy relations.

Proposed method

• Define a universal C*-algebra $\mathbb{T}(A,\alpha,\mathcal{L})$ generated by a C*-algebra $A$, an endomorphism $\alpha$, and a transfer operator $\mathcal{L}$ satisfying $\mathcal{L}(\alpha(a)b) = a\mathcal{L}(b)$ for all $a,b \in A$.
• Introduce a Hilbert bimodule $M$ as the closure of $AS$, where $S$ is a generator satisfying $Sa = \alpha(a)S$ and $S^*aS = \mathcal{L}(a)$.
• Construct the crossed-product as a quotient of $\mathbb{T}(A,\alpha,\mathcal{L})$ by an ideal $\mathcal{I}$ generated by 'redundancies'—relations of the form $abS = kbS$ for $a,b \in A$, $k \in \overline{ASS^*A}$.
• Use the universal property of $\mathbb{T}(A,\alpha,\mathcal{L})$ to define a *-homomorphism into $\mathcal{O}_A$, and prove that it factors through the quotient by showing the homomorphism vanishes on $\mathcal{I}$.
• Leverage the graded structure of $\mathcal{O}_A$ over the free group $\mathbf{F}(\mathcal{G})$ to define a canonical element $s = \alpha(Q)^{-1/2} \sum_{x \in \mathcal{G}} s_x$ satisfying the defining relations of $S$.
• Establish an isomorphism between the quotient $C(\Omega_A) \mathop{\square}_{\alpha,\mathcal{L}} \mathbb{N}$ and $\mathcal{O}_A$ by constructing inverse *-homomorphisms using the universal property and verifying vanishing on $\mathcal{I}$.

Experimental results

Research questions

• RQ1 Can a new crossed-product construction for C*-algebras by endomorphisms be defined that incorporates a transfer operator and generalizes existing constructions?
• RQ2 Does this new construction yield the Cuntz–Krieger algebra $\mathcal{O}_A$ when applied to the associated Markov subshift and a natural transfer operator?
• RQ3 How can the ideal $\mathcal{I}$ of redundancies be characterized in the universal algebra $\mathbb{T}(A,\alpha,\mathcal{L})$ to ensure the quotient yields a well-defined crossed-product?
• RQ4 What is the role of the Hilbert bimodule structure $M = \overline{AS}$ in constructing the crossed-product and ensuring the quotient is well-behaved?
• RQ5 Under what conditions does the universal algebra $\mathbb{T}(A,\alpha,\mathcal{L})$ map surjectively onto $\mathcal{O}_A$ via a quotient by $\mathcal{I}$, and when is this map an isomorphism?

Key findings

• The Cuntz–Krieger algebra $\mathcal{O}_A$ is isomorphic to the quotient $C(\Omega_A) \mathop{\square}_{\alpha,\mathcal{L}} \mathbb{N}$, where $\Omega_A$ is the spectrum of the C*-algebra of continuous functions on the path space of the Markov subshift, $\alpha$ is the shift endomorphism, and $\mathcal{L}$ is the transfer operator defined by averaging over preimages.
• The construction of the crossed-product as a quotient of the universal algebra $\mathbb{T}(A,\alpha,\mathcal{L})$ by the ideal $\mathcal{I}$ of redundancies yields a C*-algebra that is isomorphic to $\mathcal{O}_A$, thus providing a new realization of $\mathcal{O}_A$.
• The *-homomorphism $\psi: \mathbb{T}(C(\Omega_A),\alpha,\mathcal{L}) \to \mathcal{O}_A$ defined by sending $S$ to $s = \alpha(Q)^{-1/2} \sum_{x \in \mathcal{G}} s_x$ vanishes on the ideal $\mathcal{I}$, ensuring that $\psi$ factors through the quotient.
• The inverse homomorphism $\phi: \mathcal{O}_A \to C(\Omega_A) \mathop{\square}_{\alpha,\mathcal{L}} \mathbb{N}$ is well-defined and is shown to be the inverse of the induced map from $\psi$, proving the isomorphism.
• The proof demonstrates that for any redundancy $(a,k)$, the image $\psi(k)$ satisfies $\psi(k) = a$ when $\alpha(1) = 1$, which implies $\psi(a - k) = 0$, confirming that $\mathcal{I}$ is contained in the kernel.
• The construction shows that the ideal $\mathcal{I}$ generated by redundancies is precisely the kernel of the map $\psi$, and thus the quotient $\mathbb{T}(A,\alpha,\mathcal{L}) / \mathcal{I}$ is isomorphic to $\mathcal{O}_A$.