## Graduate Studies

## 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

## 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 *O*_{2} and multiplies it by a unitary operator in the diagonal MASA of the representation. This results in a new "perturbed" representation of *O*_{2} 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 (*C*_{1},*D*_{1}) generated by the representation and appeal to the twisted groupoid structure for (*C*_{1},*D*_{1}) 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 *G*_{1} for (*C*_{1},*D*_{1}).

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

## Comments

Copyright 2024, Catherine Zimmitti. Used by permission