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Unbounded Derivations of C*-Algebras and the Heisenberg Commutation Relation
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 δD 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 δD 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.
Ismert, Lara, "Unbounded Derivations of C*-Algebras and the Heisenberg Commutation Relation" (2019). ETD collection for University of Nebraska - Lincoln. AAI13862300.