Operator Algebras Seminar Ical Atom

In this talk I will give an overview of Popa’s deformation/rigidity theory for von Neumann algebra factors. The main motivation for this theory is the question of when an isomorphism of factors arising from group actions comes from an isomorphism of the groups. After providing some background on early results, examples of using deformation/rigidity to prove structural results will be given. Topics discussed in this talk include deformations, Cartan subalgebras, and intertwining of subalgebras.

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In this talk, I will describe some ring theoretic properties of certain rings of noncommutative functions. In particular, I will show that these topological rings are good analogs of the classical rings of analytic functions on discs in the plane. Our rings turn out to be semi-free ideal rings. Namely, every finitely generated right (equivalently, left) ideal is free as a module. In turn, this implies that they admit an embedding into a division ring with a certain universal property (a universal localization). I will explain how this result is a blend of techniques from ring theory and operator algebras and show an application to free probability.

This talk is based on joint work with Meric Augat and Rob Martin.

A $C^\ast$-probability space is a pair $(A,\tau)$ consisting of a $C^\ast$-algebra and a tracial state $\tau$ on $A$. For any two $C^\ast$-probability spaces, there’s a definition of a reduced free product $C^\ast$-algebra $(A,\tau) \ast_r (B,\sigma)$. This is a generalization of the case of reduced group $C^\ast$-algebras: if $G$ and $H$ are discrete groups, then the reduced free product of $C^\ast_r(G)$ and $C^\ast_r(H)$ is the reduced group $C^\ast$-algebra of the free product $G \ast H$. We show that if $A$ decomposes as a nontrivial reduced free power of infinitely many copies of separable $C^\ast$-probability spaces, then $C([0,1]) \ast_r A$ is isomorphic to $A$. Several other related isomorphism theorems are obtained as well. I will review some background and outline the proof. This is joint work with N. Christopher Phillips.

Given a natural number $n \geq 1$, the odometer semigroup $O_n$, also known as the adding machine or the Baumslag–Solitar monoid with two generators, is a well-known object in group theory. This talk will examine the odometer semigroup in relation to representations of bounded linear operators. We will focus on noncommutative operators and show that contractive representations of $O_n$ always admit nicer representations. A complete description of representations of $O_n$ on the Fock space will be presented, along with connections to odometer lifting and subrepresentations. Along the way, we will also classify Nica–covariant representations of $O_n$.

$C^\ast$-covers of operator algebras, first studied by Arveson about 50 years ago, remain a vibrant area of research. Recently, Adam Humeniuk and Christopher Ramsey studied the lattice of $C^\ast$-covers of operator algebras, focusing on its structure and on whether this lattice uniquely determines an operator algebra up to completely isometric isomorphisms.

In my talk, I will give an introduction to operator algebras and their $C^\ast$-covers, and answer the question of whether there exists nontrivial operator algebras with a one point lattice. Additionally, I will characterize the possible cardinalities of the lattice of $C^\ast$-covers. This is a joint work with Adam Humeniuk and Christopher Ramsey.