Akin’s notion of good measure, introduced to classify measures on Cantor sets,
has been translated to dimension groups and traces by Bezuglyi and the
author, but emphasizing the simple (minimal dynamical system) case. Here
we permit nonsimplicity. Goodness of tensor products of large classes of
non-good traces (measures) is established. We also determine the pure faithful
good 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 Bezuglyi
and Handelman (2014) concerning good subsets of Choquet simplices, in the
finite-dimensional case.
Keywords
good measure, trace, dimension group, affine
representation, Newton polyhedron, approximately divisible,
order ideal, simplicial group
