| Let X be a locally compact Hausdorff space along with n proper continuous maps sigma = (sigma 1,···,sigman). Then the pair (X, sigma) is called a dynamical system. To each system one can associate a universal operator algebra called the tensor algebra A (X, sigma). The central question in this theory is whether these algebras characterize dynamical systems up to some form of natural conjugacy.;In the n = 1 case, when there is only one self-map, we will show how this question has been completely determined. For n ≥ 2, isomorphism of two tensor algebras implies that the two dynamical systems are piecewise conjugate. The converse was only established for n = 2 and 3. We introduce a new construction of the unitary group U(n) that allows us to prove the algebraic characterization question in n = 2, 3 and 4 as well as translating this conjecture into a conjecture purely about the structure of the unitary group. |
