[Paper Review] Crossed product by an arbitrary endomorphism

B. K. Kwaśniewski, А. В. Лебедев|ArXiv.org|Mar 27, 2007

Analytical Chemistry and Chromatography3 citations

TL;DR

This paper introduces a generalized crossed product construction for a unital C*-algebra equipped with an arbitrary endomorphism, showing that the universal crossed product depends not only on the algebra and endomorphism but also on a chosen ideal orthogonal to the kernel of the endomorphism. The key contribution is a matrix calculus for the algebraic structure and a norm evaluation that enables the definition of a $C^*$-algebraic crossed product $C^*(\mathcal{A}, \delta, J)$, with an isomorphism theorem proving faithfulness of representations under a specific covariance condition.

ABSTRACT

Starting from an arbitrary endomorphism δof a unital C*-algebra A we construct a crossed product. It is shown that the natural construction depends not only on the C*-dynamical system (A,δ) but also on the choice of an ideal J orthogonal to Ker δ.

Motivation & Objective

• To generalize the crossed product construction from automorphisms to arbitrary endomorphisms of unital C*-algebras.
• To resolve foundational issues in defining crossed products by endomorphisms, particularly the lack of an inverse and the choice of relations for the implementing unitary.
• To show that the universal crossed product depends on an auxiliary choice of an ideal $J$ orthogonal to $\ker \delta$, not just $\mathcal{A}$ and $\delta$.
• To provide a matrix calculus and norm evaluation for the algebraic structure of the crossed product.
• To establish a universal property via an isomorphism theorem that characterizes faithful representations of the crossed product.

Proposed method

• Define a covariant representation of a $C^*$-dynamical system $(\mathcal{A}, \delta)$ as a triple $(\pi, U, H)$, where $\pi$ is a non-degenerate faithful representation and $U$ is a partial isometry satisfying $U\pi(a)U^* = \pi(\delta(a))$ and $U^*U \in \pi(\mathcal{A})'$.
• Introduce a notion of orthogonal ideals and classify covariant representations by the ideal $J$ they determine, which is orthogonal to $\ker \delta$.
• Construct a $*$-algebra $\mathcal{M}(\mathcal{A})$ of formal Laurent series with coefficients in $\mathcal{A}$, equipped with a convolution product $\star$ defined via $a^{(k)} \star b^{(l)} = \Phi_{(\pi,U)}(a^{(k)}) \cdot \Phi_{(\pi,U)}(b^{(l)})$.
• Define a seminorm $||\cdot||_J$ on $\mathcal{M}(\mathcal{A})$ using the representation $\Phi_{(\pi,U)}$, and take the completion to form the crossed product $C^*(\mathcal{A}, \delta, J)$ as the enveloping $C^*$-algebra of the quotient $\mathcal{M}(\mathcal{A}) / ||\cdot||_J$.
• Establish a matrix calculus for the algebraic structure of $C^*(\mathcal{A}, \delta, J)$, expressing elements as sums over integer degrees with coefficients in $\mathcal{A}$, and prove norm submultiplicativity.
• Prove the Isomorphism Theorem (Theorem 5.1), showing that two covariant representations associated with the same ideal $J$ and satisfying a specific covariance condition $(*)$ induce isomorphic $C^*$-algebras generated by their images.

Experimental results

Research questions

• RQ1 How can a crossed product be universally defined for a $C^*$-algebra and an arbitrary endomorphism, given the absence of an inverse operation?
• RQ2 Why does the universal construction of the crossed product depend not only on the $C^*$-algebra and endomorphism but also on an auxiliary ideal $J$ orthogonal to $\ker \delta$?
• RQ3 What algebraic and normative structure underlies the crossed product when the endomorphism is not an automorphism?
• RQ4 How can one characterize faithful representations of the crossed product in terms of the underlying covariant representations?
• RQ5 What role does the spectral decomposition into $\overline{B}_k$ subspaces play in the structure of the crossed product?

Key findings

• The crossed product $C^*(\mathcal{A}, \delta, J)$ is defined as the enveloping $C^*$-algebra of the quotient $\mathcal{M}(\mathcal{A}) / ||\cdot||_J$, where $||\cdot||_J$ is a seminorm derived from a covariant representation associated with an ideal $J$ orthogonal to $\ker \delta$.
• The construction depends on the choice of $J$, showing that multiple non-isomorphic crossed products can arise from the same $(\mathcal{A}, \delta)$, depending on $J$.
• A matrix calculus is developed that describes elements of the crossed product as formal Laurent series with coefficients in $\mathcal{A}$, and the convolution product $\star$ is defined via the representation $\Phi_{(\pi,U)}$.
• The norm of a product $a \star b$ satisfies $||a \star b|| \leq ||a|| \cdot ||b||$, proving that the seminorm $||\cdot||_J$ is submultiplicative and thus the quotient algebra is a $*$-algebra with a well-defined $C^*$-norm.
• The Isomorphism Theorem (Theorem 5.1) establishes that two covariant representations associated with the same ideal $J$ and satisfying property $(*)$ generate isomorphic $C^*$-algebras, providing a universal characterization.
• The spectral subspaces $\overline{B}_k$ of the crossed product correspond to the $k$-th Fourier coefficients, and $\overline{B}_0$ is the fixed point algebra for the $S^1$-action $\alpha_z(U) = zU$, with $\alpha_z$ extending to an automorphism of the crossed product.