arXiv subjects
Publications and source records attributed to Ian F. Putnam.
We show that every minimal, free action of the group Z^2 on the Cantor set is orbit equivalent to an AF-relation. As a consequence, this extends the classification of minimal systems on the Cantor set up to orbit equivalence to include AF-relations, Z-actions and Z^2-actions.
We prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being `small' in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case--when Y is a finite set--this result is highly non-trivial. The result itself--called the absorption theorem--is a powerful and crucial tool for the study of the orbit structure of minimal Z^n-actions on the Cantor set [GMPS]. The absorption theorem is a significant generalization of the main theorem proved in [GPS2]. However, we shall need a few key results from [GPS2] in order to prove the absorption theorem.
The Bost-Connes Hecke C^*-algebra can be regarded as a direct limit of subalgebras involving finite sets of primes. Each of these finite-prime analogues of the Bost-Connes algebra is a crossed product by a semigroup N^F, where F is finite. We describe composition series of ideals for these semigroup crossed products in which the intermediate subquotients are finite direct sums of classifiable AT-algebras. We then use similar techniques to identify subquotients of the Bost-Connes algebra as classifiable AT-algebras.