Disputas: Rui Miguel Coutinho Palma
M. Sc. Rui Miguel Coutinho Palma ved Matematisk institutt vil forsvare sin avhandling for graden ph.d.:
C*-Completions of Hecke algebras and crossed products by Hecke pairs.
Rui Miguel Coutinho Palma
Tid og sted for prøveforelesning
Professor John Quigg, School of Mathematical and Statistical Sciences, Arizona State University
Professor Dana P. Williams, Dartmouth College
- Professor Erik Christopher Bedos, Matematisk institutt, Universitet i Oslo
Leder av disputas
Professor Arne Huseby, Matematisk institutt, Universitet i Oslo
- Førsteamanuensis Nadia S. Larsen, Matematisk institutt, Universitet i Oslo
- Professor Sergey Neshveyev, Matematisk institutt, Universitet i Oslo
The interplay between C*-algebras, groups and dynamical systems is a central theme in the field of operator algebras which reflects how this field is intertwined with other areas of mathematics.
Groups can be studied via their group C*-algebras and dynamical systems are often studied via their corresponding crossed products, which are C*-algebras that encode information about the dynamics. The operation of taking the quotient of a group by a normal subgroup yields a new group which can be studied via its own group algebra and via dynamical systems involving its actions. If the subgroup is not normal, one is instead lead to consider Hecke pairs and Hecke algebras, which essentially play the role of quotient groups and their respective group algebras.
In the first part of this thesis we study C* completions of Hecke algebras and provide sufficient conditions for certain of these completions to coincide. Our techniques are shown to applicable for a large class of Hecke pairs and are used to obtain information about the representation theory of the original groups, by establishing what is known as Hall's equivalence. One of our main results states that Hall's equivalence holds for all Hecke pairs arising from nilpotent groups.
In the second part of this thesis we develop a theory of C* crossed products by actions of Hecke pairs which is intended for applications in non-abelian C* duality. One of the main consequences of this work is the establishment of a Stone-von Neumann theorem for Hecke pairs that explains certain results in the literature in the language of crossed products. Our theory is developed so that it can, in future works, bring new insight on the emerging theory of crossed products by coactions of homogeneous spaces.
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