Good traces for not necessarily simple dimension groups

Abstract

Akin's notion of good measure, introduced to classify measures on Cantor sets has been translated to dimension groups and corresponding traces by Bezuglyi and the author, but emphasizing the simple (minimal dynamical system) case. Here we deal with non-simple (non-minimal) dimension groups. In particular, goodness of tensor products of large classes of non-good traces (measures) is established. We also determine the pure faithful traces on the dimension groups associated to xerox type actions on AF C*-algebras; the criteria turn out to involve algebraic geometry and number theory. We also deal with a coproduct of dimension groups, wherein, despite expectations, goodness of direct sums is nontrivial. In addition, we verify a conjecture of [BeH] concerning good subsets of Choquet simplices, in the finite-dimensional case.