13 research outputs found
Groupoid Crossed Products
We present a number of findings concerning groupoid dynamical systems and
groupoid crossed products. The primary result is an identification of the
spectrum of the groupoid crossed product when the groupoid has continuously
varying abelian stabilizers and a well behaved orbit space. In this case, the
spectrum of the crossed product is homeomorphic, via an induction map, to a
quotient of the spectrum of the crossed product by the stabilizer group bundle.
The main theorem is also generalized in the groupoid algebra case to an
identification of the primitive ideal space. This generalization replaces the
assumption that the orbit space is well behaved with an amenability hypothesis.
We then use induction to show that the primitive ideal space of the groupoid
algebra is homeomorphic to a quotient of the dual of the stabilizer group
bundle. In both cases the identification is topological. We then apply these
theorems in a number of examples, and examine when a groupoid algebra has
Hausdorff spectrum. As a separate result, we also develop a theory of principal
groupoid group bundles and locally unitary groupoid actions. We prove that such
actions are characterized, up to exterior equivalence, by a cohomology class
which arises from a principal bundle. Furthermore, we also demonstrate how to
construct a locally unitary action from a given principal bundle. This last
result uses a duality theorem for abelian group bundles which is also included
as part of this thesis.Comment: Author's PhD Thesis, 337 page
Free Entropy Minimizing Persuasion in a Predictor-Corrector Dynamic
Persuasion is the process of changing an agent's belief distribution from a
given (or estimated) prior to a desired posterior. A common assumption in the
acceptance of information or misinformation as fact is that the
(mis)information must be consistent with or familiar to the individual who
accepts it. We model the process as a control problem in which the state is
given by a (time-varying) belief distribution following a predictor-corrector
dynamic. Persuasion is modeled as the corrector control signal with the
performance index defined using the Fisher-Rao information metric, reflecting a
fundamental cost associated to altering the agent's belief distribution. To
compensate for the fact that information production arises naturally from the
predictor dynamic (i.e., expected beliefs change) we modify the Fisher-Rao
metric to account just for information generated by the control signal. The
resulting optimal control problem produces non-geodesic paths through
distribution space that are compared to the geodesic paths found using the
standard free entropy minimizing Fisher metric in several example belief
models: a Kalman Filter, a Boltzmann distribution and a joint Kalman/Boltzmann
belief system.Comment: 16 pages, 7 figure
Groupoid equivalence and the associated iterated crossed product
Given groupoids and and a -equivalence we may form the
transformation groupoid . Given a separable groupoid
dynamical system we may restrict to
an action of on and form the crossed product . We show that there is an action of on
and that the iterated crossed product is
naturally isomorphic to the crossed product .Comment: 18 pages; changed typo in titl
THE MACKEY MACHINE FOR CROSSED PRODUCTS BY REGULAR GROUPOIDS. I
Abstract. We first describe a Rieffel induction system for groupoid crossed products. We then use this induction system to show that, given a regular groupoid G and a dynamical system (A, G, Ξ±), every irreducible representation of AβG is induced from a representation of the group crossed product A(u)βSu where u β G (0) , A(u) is a fibre of A, and Su is a stabilizer subgroup of G