Cohomologous 2-cocycles are Homotopic 2-cocycles: k-graphs and C*-algebras

Undergraduate Honors Thesis

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Citeable URL: https://scholar.colorado.edu/concern/undergraduate_honors_theses/gm80hv72j

Abstract

k-graphs equipped with a defined 2-cocycle allow one to construct examples of C^∗-algebras. For some k-graph, Λ, there does not necessarily exist a unique choice of 2-cocycle, such that for Λ there may exist multiple C^∗-algebras depending on one’s choice of 2-cocycle. There exist relations between two defined 2-cocycles on a k-graph, cohomology and homotopy, that imply features of the C^∗-algebras generated by the 2-cocycles and Λ. Cohomology implies that the two C^∗-algebras are isomorphic, and homotopy implies the two share the same invariant, K-theory. It is shown here the result that any two 2-cocycles defined on a k-graph which are cohomologous are then homotopic. Also included is the method by which one may construct a matrix equation,Ψx^→ = z^→, that encodes the information of a k-graph and 2-cocycle, where the existence of an integer solution to the equation Ψx^→= z^→implies any two 2-cocycles are homotopic.

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