I’ve been pretty busy the past semester and change, so I figured I might as well discuss a little bit about what I’m actually doing for my honors thesis, and how it plays into potential future research I’ll hopefully be doing in graduate school.

The outline for my thesis is as follows.

  • Categorical Constructions: here, I detail some categorical constructions, such as free groups, free vector spaces, and free *-algebras that I’ll be using in later chapters of the honors thesis. I also include a discussion of tensor products and norms on the tensor products, in order to prepare for the discussion in the final chapter.
  • Paradoxical decompositions: my honors thesis is centered around the idea of amenability; what it is, how we can determine it, and a little bit of why we care about it. I discuss a little bit about the Banach–Tarski paradox, and offer some insight into where it comes from. Ultimately, the paradox hinges on our ability to create a special decomposition of a subgroup of the group $\text{SO}(3)$, which is the group of 3x3 matrices acting on $\mathbb{R}^3$ that are orthogonal and have determinant one. This allows us to effectively decompose the unit sphere (and then the unit ball, then any bounded set with nonempty interior) into five or so finite pieces, apply a series of isometries, and then create the sphere twice over. This is also known as the “pea to the sun” paradox, and it shows us that there is no finitely additive measure on $\mathbb{R}^3$ that is invariant under isometries, something which does exist for $\mathbb{R}$ and $\mathbb{R}^2$.
  • Tarski’s Theorem: one of the first proofs related to amenability I discuss is Tarski’s theorem, which shows that if a group does not admit a paradoxical decomposition, then the group is what is known as amenable, or that it admits a finitely additive measure that is invariant under translations. This is a proof that is honestly not particularly pretty, and I’m probably going to move some stuff around so the proof of Tarski’s theorem comes after I define amenability and invariant states, but stylistically I like that there’s a proof related to paradoxicality right after an elaboration on paradoxicality.
  • Amenability and Invariant States: we elaborate on amenability, including the true definition of a mean. Specifically, a mean on a discrete group $G$ is a finitely additive probability measure $\mu\colon P(G)\rightarrow [0,1]$ such that $\mu(A) = \mu(tA)$ for each $A\in P(G)$ and $t\in G$. We show that this is equivalent to the existence of an state $\rho\colon \ell_{\infty}(G)\rightarrow \mathbb{C}$ such that $\rho\left(\lambda_s(f)\right) = \rho(f)$, where $\lambda_s$ is a translation operator on the space $\ell_{\infty}$. Equipped with this definition, we eventually show that all abelian groups and all solvable groups are amenable. In a set of collected remarks, I discuss a little bit more about the Tits alternative and explaining what we discussed at the beginning of the chapter on paradoxical decompositions, and show that there is a mean on the group of isometries for $\mathbb{R}^2$ and $\mathbb{R}$.
  • Følner’s Condition: we discuss a combinatorial characterization for establishing amenability in discrete groups, and prove that the Følner condition, the existence of what is known as an approximate mean, and the existence of an invariant state (or mean) on the group are equivalent. We apply this to establish the amenability of a class of groups known as the groups of subexponential growth. In the final section on remarks, I discuss a section of Kate Juschenko’s Amenability of Discrete Groups by Examples that discuss a weakened version of the Følner condition, which also includes a little bit of exposition on filters and ultrafilters.
  • The Left-Regular Representation, Kesten’s Criterion, and Hulanicki’s Criterion: we move towards representing groups as elements of $\mathcal{U}\left(\mathbb{B}\left(\ell_2\left(G\right)\right)\right)$, which is the group of unitary operators acting on the space of square-summable sequences with domain $G$. This enables us to once again use tools from functional analysis, as we did in the chapter on invariant states, to establish equivalent criteria for amenability. I’m harmonize material from Juschenko’s Amenability of Discrete Groups by Examples as well as my professor’s textbook in order to provide insight into both the left-regular representation and how it can be used to establish amenability. Specifically, there are a few criteria criteria — the existence of an almost-invariant vector (akin to an approximate mean), weak containment of the trivial representation, Kesten’s criterion, and Hulanicki’s criterion — all of which we will show are equivalent to one of the previously established equivalent definitions of mean, approximate mean, and the Følner condition.

This semester is focused on finalizing the last chapter.

  • Amenability in C*-Algebras: from any group, we can construct what is known as the group *-algebra, where we construct a multiplication on $\mathbb{C}][\Gamma]$, the free vector space over $\mathbb{C}$ with basis $\Gamma$. Then, we can apply a norm (in fact, many norms) on the group *-algebra, from which we can create a C*-algebra, and explore properties of the group through properties of the group C*-algebra, such as amenability. There are a number of results (Brown and Ozawa claim $10^{10^{10}}$, but I’m doubtful) relating to amenability in the context of operator algebras, but I’m going to discuss the few that are brought up in the book C*-algebras and Finite-Dimensional Approximations, relating to certain analytical properties, such as nuclearity, a nonempty character space, and equivalence between the universal and reduced group C*-algebras.

I also have a lot of appendices, largely written for my own recollection and edification on the general subjects that I discuss throughout the thesis — specifically, algebra/linear algebra, point-set topology, measure theory, functional analysis, and operator algebras.

This is an incredibly ambitious honors thesis, and I’m honestly shocked it’s gotten this far. I’m glad it has worked out thus far, and I hope it continues to work out, because it is a really fun topic that I would absolutely enjoy studying in graduate school and beyond. Feel free to check out the notes page.