Renormalization procedures for C*-algebras




Hume, Jeremy

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Renormalization procedures for families of dynamical systems have been used to prove many interesting results. Examples of results include that the bifurcation rate for the attractors of an analytic one-parameter family of quadratic-like maps is universal for all such families, unique ergodicity for almost every interval exchange mapping, a unique ergodicity criterion for the vertical translation flow of a flat surface in terms of its ``renormalization dynamics", known as Masur's criterion, and the classification of circle diffeomorphisms up to $C^{\infty}$ conjugation. We introduce renormalization procedures for $C^{*}$-algebras and étale groupoids using the concepts of $C_{0}(X)$-algebras and Morita equivalence for the former, and groupoid bundles and groupoid equivalence, in the sense of Muhly, Renault and Williams, for the latter. We focus on proving analogs to Masur's criterion in both cases using $C^{*}$-algebraic methods. Applying our criterion to our examples of renormalization procedures provides a unique trace criterion for unital AF algebras extending the one provided by Treviño in the setting of flat surfaces and the one provided by Veech in the setting of interval exchange mappings. Also, we recover the old fact that rotation of the circle by an irrational angle is uniquely ergodic, and the new fact that interesting groupoids associated to certain iterated function systems, recently introduced by Korfanty, have unique invariant probability measures whenever they are minimal. Lastly, we show how an étale groupoid renormalization procedure arises from an étale groupoid which factors down onto a groupoid associated to its renormalization dynamics, whenever it is a local homeomorphism.



Operator Algebras, C*-algebras, Renormalization, Groupoids, Groupoid bundles