#### Date of Award

Fall 2009

#### Project Type

Dissertation

#### Program or Major

Mathematics

#### Degree Name

Doctor of Philosophy

#### First Advisor

Liming Ge

#### Abstract

I develop the theory of Kadison-Singer algebras, introduced recently by Ge and Yuan. I prove basic structure theorems, construct several new examples and explore connections to other areas of operator algebras. In chapter 1, I survey those aspects of the theory of non-selfadjoint algebras that are relevant to this work. In chapter 2, I define Kadison-Singer algebras and give different proofs of results of Ge-Yuan, which will be further extended in the last chapter. In chapter 3, I analyse in detail a class of elementary Kadison-Singer algebras that contain Hinfinity and describe their lattices of projections. In chapter 4, I use ideas from free probability theory to construct Kadison-Singer algebras with core the free group factors L(Fr) for r < 2. I then introduce two constructions that yield new Kadison-Singer algebras - The maximal join and the minimal join. In chapter 5, I analyse tensor products of Kadison-Singer algebras, showing that they are never Kadison-Singer. I then show how under certain conditions, one may construct a Kadison-Singer algebra with core the tensor product of the core of two given Kadison-Singer algebras.

#### Recommended Citation

Ravichandran, Mohan, "Kadison -Singer algebras with applications to von Neumann algebras" (2009). *Doctoral Dissertations*. 501.

https://scholars.unh.edu/dissertation/501