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Eigenstates of CQ*-algebras

Fabio Bagarello, Hiroshi Inoue, Camillo Trapani, Salvatore Triolo

TL;DR

This work extends the notion of eigenstates from C*-algebras to CQ*-algebras by developing an extension of continuous positive functionals to the larger algebra ${\mathfrak A}$ via $\overline{\omega}$ and by adapting the GNS construction to accommodate unbounded vectors. It defines eigenstates for elements $X\in{\mathfrak A}$ through the relation $\overline{\omega}(AX)=\alpha\overline{\omega}(A)$ and connects these states to representation-theoretic eigenvectors in the GNS picture, situating them within the left and full spectra of ${\mathfrak A}_0$. In the *-semisimple case, generalized eigenvalues are characterized using ips-forms and a weak multiplication, linking algebraic and Hilbert-space realizations via kernels of $\pi_\varphi(X)-\alpha I_\varphi$. The paper also analyzes dynamics generated by hermitian observables, showing a one-parameter group of automorphisms and a corresponding derivation, and introduces a locally convex expansion ${\mathfrak A}_1$ to guarantee bounded GNS representations. Overall, the framework provides a robust algebraic foundation for eigenstate analysis in CQ*-algebras with potential applications to quantum dynamics and generalized spectral theory, including avenues for non-Hermitian extensions.

Abstract

Motivated by some recent results, we consider the notion of eigenstate (and eigenvalue) for an element $X$ of a CQ*-algebras and the consequences on algebraic quantum dynamics and on its related derivations are investigated.

Eigenstates of CQ*-algebras

TL;DR

This work extends the notion of eigenstates from C*-algebras to CQ*-algebras by developing an extension of continuous positive functionals to the larger algebra via and by adapting the GNS construction to accommodate unbounded vectors. It defines eigenstates for elements through the relation and connects these states to representation-theoretic eigenvectors in the GNS picture, situating them within the left and full spectra of . In the *-semisimple case, generalized eigenvalues are characterized using ips-forms and a weak multiplication, linking algebraic and Hilbert-space realizations via kernels of . The paper also analyzes dynamics generated by hermitian observables, showing a one-parameter group of automorphisms and a corresponding derivation, and introduces a locally convex expansion to guarantee bounded GNS representations. Overall, the framework provides a robust algebraic foundation for eigenstate analysis in CQ*-algebras with potential applications to quantum dynamics and generalized spectral theory, including avenues for non-Hermitian extensions.

Abstract

Motivated by some recent results, we consider the notion of eigenstate (and eigenvalue) for an element of a CQ*-algebras and the consequences on algebraic quantum dynamics and on its related derivations are investigated.

Paper Structure

This paper contains 10 sections, 16 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Positive linear functionals of the CQ*-algebra ${\mathfrak A}$
  4. Eigenstate
  5. Eigenstates and spectrums
  6. *-Semisimple CQ*-algebras: a hilbertian approach
  7. Eigenstates and dynamics
  8. Ground states
  9. A brief digression: A locally convex $\ast$-algebra constructed from ${\mathfrak A}_0(\|\cdot\|)$
  10. Conclusions

Key Result

Proposition 3.3

Let $\omega$ be a $\|\cdot\|$-continuous positive linear functional on ${\mathfrak A}_0$. We can define a triple $(\pi_{\overline{\omega}},\lambda_{\overline{\omega}},\mathcal{H}_{\overline{\omega}})$ satisfying Here $\mathcal{H}_{\overline{\omega}}=\mathcal{H}_\omega$, ${\mathcal{D}}(\pi_{\overline{\omega}})=\lambda_\omega({\mathfrak A}_0)$, and $\pi_{\overline{\omega}}$ and $\lambda_{\overline{

Theorems & Definitions (49)

  • Example 2.1
  • Example 2.2
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Definition 4.3
  • ...and 39 more