An obstruction to isomorphism of tensor algebras of multivariable dynamical systems

A multivariable dynamical system (MDS) consists of a compact Hausdorff space $X$ together with a finite family of continuous self-maps $\sigma_i\colon X\to X$. To each such system one naturally associates a non-selfadjoint operator algebra known as the tensor algebra, denoted by $A(X,\sigma)$, which encodes the dynamical information algebraically. In the single-variable case ($n=1$), Davidson and Katsoulis established that two tensor algebras are isomorphic if and only if the corresponding dynamical systems are conjugate. For $n\ge 2$, however, conjugacy is too strong to capture algebraic isomorphism, leading Davidson and Katsoulis to introduce the weaker notion of piecewise conjugacy, conjecturing it to be the correct criterion for classification.
In this talk, we disprove their conjecture in general. By identifying a previously unnoticed topological obstruction to the existence of certain admissible maps into spaces of unitary matrices, we produce an explicit counterexample consisting of two piecewise conjugate 4-variable systems on a two-dimensional compact space whose tensor algebras are not isomorphic. This result leaves the classification problem wide open and highlights the necessity of more refined invariants for tensor-algebra isomorphism.