Perturbations of Representations of Cartan Inclusions

Authors

Catherine Zimmitti, University of Nebraska-LincolnFollow

First Advisor

David Pitts

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

Date of this Version

8-2024

Document Type

Dissertation

Citation

A dissertation presented to the faculty of the Graduate College of the University of Nebraska in partial fulfillment of requirements for the degree of Doctor of Philosophy

Major: Mathematics

Under the supervision of Professor David Pitts

Lincoln, Nebraska, August 2024

Comments

Copyright 2024, Catherine Zimmitti. Used by permission

Abstract

A free semigroup algebra is the unital, weak operator topology closed algebra generated by a collection of Cuntz-Toeplitz isometries in B(H). Ken Davidson and David Pitts asked in [9] if a self-adjoint free semigroup algebra exists; Charles Read answered this question in [28] by constructing such an example, which Ken Davidson later simplified in [8]. The construction takes a standard representation of O₂ and multiplies it by a unitary operator in the diagonal MASA of the representation. This results in a new "perturbed" representation of O₂ generating a self-adjoint free semigroup algebra.

In this thesis, we generalize this notion of perturbing representations to a class of pairs of C*-algebras known as Cartan inclusions. Given an inclusion (C,D), we consider *-subsemigroups of normalizers satisfying certain density conditions, which we call semi-skeletons for the inclusion. Given a semi-skeleton M for a Cartan inclusion (C,D), we introduce the concept of a multiplier cocycle on M, which is a map [delta] from M into the local multiplier algebra for D satisfying certain cocycle-like conditions arising from the dynamics on (C,D). We show that, given a faithful representation [pi] : C [to] B(H) satisfying certain hypotheses, we may perturb the representation by a multiplier cocycle on M to obtain a new faithful representation extending the map n [maps to] [delta](n)[pi](n) for all n [in] M. We do this by considering a larger inclusion (C₁,D₁) generated by the representation and appeal to the twisted groupoid structure for (C₁,D₁) due to Kumjian [22] and Renault [30]. We show that there is a *-semigroup monomorphism from the *-semigroup of multiplier cocycles on a semi-skeleton M into the group of [T set]-valued 1-cocycles on the Weyl groupoid G₁ for (C₁,D₁).

We conclude by investigating the special case that the Cartan inclusion is a discrete reduced crossed product.

Advisor: David Pitts

Recommended Citation

Zimmitti, Catherine, "Perturbations of Representations of Cartan Inclusions" (2024). Dissertations and Doctoral Documents from University of Nebraska-Lincoln, 2023–. 167.
https://digitalcommons.unl.edu/dissunl/167