Unbounded Derivations of C*-algebras and the Heisenberg Commutation Relation

Authors

Lara M. Ismert, University of Nebraska-LincolnFollow

First Advisor

Allan Donsig

Second Advisor

David Pitts

Date of this Version

5-2019

Document Type

Article

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professors Allan Donsig and David Pitts. Lincoln, Nebraska: May, 2019.

Copyright (c) 2019 Lara M. Ismert

Abstract

This dissertation investigates the properties of unbounded derivations on C*-algebras, namely the density of their analytic vectors and a property we refer to as "kernel stabilization." We focus on a weakly-defined derivation δ_D which formalizes commutators involving unbounded self-adjoint operators on a Hilbert space. These commutators naturally arise in quantum mechanics, as we briefly describe in the introduction.

A first application of kernel stabilization for δ_D shows that a large class of abstract derivations on unbounded C*-algebras, defined by O. Bratteli and D. Robinson, also have kernel stabilization. A second application of kernel stabilization provides a sufficient condition for when a pair of self-adjoint operators which satisfy the Heisenberg Commutation Relation on a Hilbert space must both be unbounded.

A directly related classification program is of pairs of unitary group representations which satisfy the Weyl Commutation Relation on a Hilbert space. The famous Stone-von Neumann Theorem classifies these pairs when the group is locally compact abelian. In collaboration with L. Huang, we extend the Stone-von Neumann Theorem to a uniqueness statement for representations of C*-dynamical systems on Hilbert K(H)-modules.

Advisors: A.P. Donsig and D.R. Pitts