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An obstruction to isomorphism of tensor algebras of multivariable dynamical systems

Boris Bilich

TL;DR

The paper disprove the conjecture that tensor algebras of multivariable dynamical systems are isomorphic exactly when the systems are piecewise conjugate. It develops homotopy invariants for paths in spaces of unitary matrices and uses them to establish a topological obstruction to the existence of admissible maps on the $2$-skeleton of the $S_n$-simplex, showing the simplex conjecture fails for $n=4$. This obstruction is then leveraged to construct two piecewise conjugate $4$-variable systems with nonisomorphic tensor algebras, and a rigidification technique is introduced to extend the non-equivalence to non-conjugacy, yielding a definitive counterexample to the piecewise conjugacy conjecture. The results emphasize that piecewise conjugacy is not a complete invariant for tensor algebra isomorphism and reveal deep connections between topological obstructions and operator-algebraic classifications.

Abstract

In their paper on multivariable dynamics, Davidson and Katsoulis conjectured that two multivariable dynamical systems have isomorphic tensor algebras if and only if they are piecewise conjugate. We disprove the conjecture by constructing two piecewise conjugate multivariable dynamical systems with four maps on a two-dimensional space, whose tensor algebras are not isomorphic.

An obstruction to isomorphism of tensor algebras of multivariable dynamical systems

TL;DR

The paper disprove the conjecture that tensor algebras of multivariable dynamical systems are isomorphic exactly when the systems are piecewise conjugate. It develops homotopy invariants for paths in spaces of unitary matrices and uses them to establish a topological obstruction to the existence of admissible maps on the -skeleton of the -simplex, showing the simplex conjecture fails for . This obstruction is then leveraged to construct two piecewise conjugate -variable systems with nonisomorphic tensor algebras, and a rigidification technique is introduced to extend the non-equivalence to non-conjugacy, yielding a definitive counterexample to the piecewise conjugacy conjecture. The results emphasize that piecewise conjugacy is not a complete invariant for tensor algebra isomorphism and reveal deep connections between topological obstructions and operator-algebraic classifications.

Abstract

In their paper on multivariable dynamics, Davidson and Katsoulis conjectured that two multivariable dynamical systems have isomorphic tensor algebras if and only if they are piecewise conjugate. We disprove the conjecture by constructing two piecewise conjugate multivariable dynamical systems with four maps on a two-dimensional space, whose tensor algebras are not isomorphic.
Paper Structure (14 sections, 25 theorems, 45 equations, 2 figures)

This paper contains 14 sections, 25 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Homotopy invariants of paths in spaces of unitary matrices
  4. Homotopy of paths in U(1)
  5. Homotopy of paths in U(P)
  6. Refutation of the simplex conjecture
  7. Admissible maps
  8. Homotopy of admissible maps on the 1-skeleton
  9. The obstruction
  10. Multivariable dynamical systems
  11. Tensor and covariance algebras
  12. Unitary equivalence for systems without fixed points
  13. Construction of piecewise but not unitarily equivalent systems
  14. Rigidification of non-equivalence to non-conjugacy

Key Result

Theorem 1

There exist two piecewise conjugate $4$-variable dynamical systems on a two-dimensional compact metrizable space such that their tensor and covariance algebras are not algebraically isomorphic.

Figures (2)

  • Figure 1: $\gamma^{g_0, g_1, g_2}$ as a concatenation of paths.
  • Figure 2: $D$-domains

Theorems & Definitions (54)

  • Theorem : \ref{['t:pc-not-iso']}
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 44 more