Algebry operatorów i ich zastosowania w fizyce

C*-algebras and minimal dynamical systems have a long shared history. Giordano, Putnam and Skau showed that C*-algebraic K-theory, via the crossed product construction, could distinguish minimal Cantor systems up to strong topological orbit equivalence. Putnam subsequently showed that the crossed products were AT algebras, and hence also classified by K-theory. At the other end of the scale, the crossed products associated to minimal diffeomorphisms never have nontrivial projections and always have identical K-theory. This makes classification of the C*-algebras in question more difficult. However, in recent work, I show that by looking at a related system on a product of the sphere and a Cantor set, one may deduce these C*-algebras are distinguished by their tracial state spaces (or equivalent, the space of invariant Borel measures of the dynamical system). In this case, we see that *-isomorphisms of C*-algebras do not necessarily tell us much about equivalence of the underlying dynamical systems.