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Admissibility of C*-Covers for Operator Algebra Dynamical Systems

Mitch Hamidi

Abstract

We characterize when a C*-cover admits a C*-dynamical extension of dynamics on an operator algebra in terms of the boundary ideal structure for the operator algebra in its maximal representation and show that the C*-covers that admit such an extension form a complete lattice. We study dynamical systems arising from groups acting via inner automorphisms in a C*-cover and produce an example of a C*-cover that admits no extension of dynamics on a finite-dimensional non-self-adjoint operator algebra. We construct a partial action on a class of C*-covers that recovers the crossed product of an operator algebra as a subalgebra of the partial crossed product, even when the C*-cover admits no dynamical extension.

Admissibility of C*-Covers for Operator Algebra Dynamical Systems

Abstract

We characterize when a C*-cover admits a C*-dynamical extension of dynamics on an operator algebra in terms of the boundary ideal structure for the operator algebra in its maximal representation and show that the C*-covers that admit such an extension form a complete lattice. We study dynamical systems arising from groups acting via inner automorphisms in a C*-cover and produce an example of a C*-cover that admits no extension of dynamics on a finite-dimensional non-self-adjoint operator algebra. We construct a partial action on a class of C*-covers that recovers the crossed product of an operator algebra as a subalgebra of the partial crossed product, even when the C*-cover admits no dynamical extension.
Paper Structure (10 sections, 20 theorems, 36 equations)

This paper contains 10 sections, 20 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator Algebras and C*-Covers
  4. $\alpha$-Admissibility and Crossed Products of Operator Algebras
  5. Structure of C*-Covers
  6. The Complete Lattice of C*-Covers
  7. $\alpha$-Admissibility and the Complete Lattice of C*-Covers
  8. Inner Dynamical Systems
  9. A New Example of a Non-Admissible C*-Cover
  10. Decomposition of Complete Isometries and Recovering Dynamics Using Partial Actions

Key Result

Theorem \ref{thm:admisscorrespondence}

Let $(\mathcal{A}, G, \alpha)$ be a dynamical system and let $(\mathcal{D}, i)$ be an $\alpha$-admissible C*-cover for $\mathcal{A}$. If $(\mathcal{C},j)$ is any C*-cover for $\mathcal{A}$ such that there exists a $*$-homomorphism $\pi$ of $\mathcal{D}$ onto $\mathcal{C}$ satisfying $\pi\circ i = j$

Theorems & Definitions (53)

  • Theorem \ref{thm:admisscorrespondence}
  • Theorem \ref{thm:partial-action}
  • Remark \ref{thm:partial-action}
  • Remark \ref{thm:partial-action}
  • Definition \ref{thm:partial-action}
  • Proposition \ref{thm:partial-action}
  • proof
  • Proposition \ref{thm:partial-action}
  • proof
  • Definition \ref{thm:partial-action}
  • ...and 43 more