Bram Mesland (Max Planck Institute for Mathematics Bonn)
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In this talk I will discuss several examples of spectral triples for C*-algebras that arise as "noncommutative boundaries" of commutative dynamical systems. The examples concern Cuntz-Krieger algebras, crossed products of Kleinian groups by their limit sets and transverse groupoid C*-algebras of a Delone sets. The common theme for these spectral triples is that they carry non-trivial K-homological content, they are not finitely summable, and their operators formally resemble the logarithm of the Laplacian on R^n. The latter admits a representation as a singular integral operator, the expression for which makes sense in the context of metric measure spaces.
The talk selects from joint works with C. Bourne, M. Goffeng and M.H. Sengun
Spectral triples and noncommutative boundaries Sponsored by the Meyer Fund
Oct. 11, 2018 3pm (MATH 220)
Functional Analysis
Elizabeth Gillaspy (University of Montana)
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Higher-rank graphs (-graphs) are category theoretic objects which can also be viewed as generalizations of directed graphs. In order to better understand the C*-algebras associated to -graphs, Kumjian, Pask, and Sims introduced two cohomology theories for -graphs. Using ad hoc methods, Kumjian, Pask, and Sims showed that the th categorical and cubical cohomology groups of a -graph are isomorphic for .
This talk presents recent joint work with Jianchao Wu, in which we show that for all $i \in \Z$ the th cubical and categorical homology and cohomology groups of any -graph are isomorphic. This proves a conjecture posed by Kumjian, Pask, and Sims in 2015, and also uncovers more structural information about the categorical (co-)homology groups.
Our first proof of this result relies on the topological realization of a -graph (as defined by Kaliszewski, Kumjian, Quigg, and Sims) and the reformulation of categorical cohomology using -modules, as introduced by Gillaspy and Kumjian. Time permitting, we will also present our more computational second proof, which gives an explicit formula for passing between cubical and categorical (co-)cycles.
Cubical and categorical (co-)homology for k-graphs Sponsored by the Meyer Fund