Vol. 281, No. 2, 2016

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Good traces for not necessarily simple dimension groups

David Handelman

Vol. 281 (2016), No. 2, 365–419
Abstract

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
Mathematical Subject Classification 2010
Primary: 19K14
Secondary: 06F20, 46L80, 37A55, 37A35, 46A55, 28E99
Milestones
Received: 4 September 2014
Revised: 2 August 2015
Accepted: 3 August 2015
Published: 16 February 2016
Authors
David Handelman
Mathematics Department
University of Ottawa
585 King Edward
Ottawa K1N 6N5 ON
Canada