Groupoid Actions on Fractafolds

We define a bundle over a totally disconnected set such that each fiber is homeomorphic to a fractal blowup. We prove that there is a natural action of a Renault-Deaconu groupoid on our fractafold bundle and that the resulting action groupoid is a Renault-Deaconu groupoid itself. We also show that when the bundle is locally compact the associated $C^*$-algebra is primitive and has a densely defined lower-semicontinuous trace.Comment: SIGMA 10 (2014), 068, 14 page

Generalised morphisms of k-graphs: k-morphs

In a number of recent papers, (k+l)-graphs have been constructed from k-graphs by inserting new edges in the last l dimensions. These constructions have been motivated by C*-algebraic considerations, so they have not been treated systematically at the level of higher-rank graphs themselves. Here we introduce k-morphs, which provide a systematic unifying framework for these various constructions. We think of k-morphs as the analogue, at the level of k-graphs, of C*-correspondences between C*-algebras. To make this analogy explicit, we introduce a category whose objects are k-graphs and whose morphisms are isomorphism classes of k-morphs. We show how to extend the assignment \Lambda \mapsto C*(\Lambda) to a functor from this category to the category whose objects are C*-algebras and whose morphisms are isomorphism classes of C*-correspondences.Comment: 27 pages, four pictures drawn with Tikz. Version 2: title changed and numerous minor corrections and improvements. This version to appear in Trans. Amer. Math. So