# Ohio University Analysis Seminar

This is the website for the Analysis Seminar of the Mathematics Department of Ohio University. Find below a list of past and upcoming talks. For further details, please contact Marcel Bischoff and/or Adam Fuller.

## Spring 2018

### Feb. 8th, Adam Fuller (Ohio University), Morton Hall 326, 11:00-12:00

Title: Graded C*-algebras

Abstract: A C*-algebra A is graded by a group G if it is spanned by linear subspaces A satisfying A_g A_h = A_gh. This can be seen as an analytic analogue of a graded ring. From another perspective it gives a non-commutative analogue of Fourier analysis. In this talk we will discuss a recent result of Raeburn which shows that if a C*-algebra is graded by a group G, then this grading is necessarily implemented by an action of the dual group of G on A.

### Feb. 1st, Marcel Bischoff (Ohio University), Morton Hall 326, 11:00-12:00

Title: Discrete Subfactors and Amenability

Abstract: A factor is a von Neumann algebra with trivial center and a subfactor is an unital inclusion of factors. The guiding example of discrete subfactors are the crossed products of factors by discrete groups, the fixed point under minimal actions of compact groups, and the "quantum double" inclusion associated with a rigid C*-tensor category. I will review the notion of (relative) amenability for discrete subfactors and give some examples. Then I will talk about some conjectures and partial results for discrete inclusions of conformal nets.

### Jan. 25th, Shehzad Ahmed (Ohio University), Morton Hall 326, 11:00-12:00

Title: Naimark's Problem

Abstract: Let K(H) denote the algebra of compact operators over a (not necessarily separable) Hilbert space H. Naimark showed that each K(H) has one irreducible representation up to unitary equivalence, and asked whether or not this property characterizes K(H) as a C*-algebra. That is, can we build a C*-algebra A such that A has one irrep up to unitary equivalence, but A is not isomorphic to some K(H)? Weaver was able to construct such an algebra with the use of Diamond, which can be thought of us a strengthening of the continuum hypothesis. In this talk, we will discuss obstacles (in ZFC) to building an algebra as above, what diamond is, and how Weaver used diamond to circumvent these aforementioned obstacles. No prior acquaintance with set theory proper will be assumed throughout this talk.

## Fall 2017

### Dec. 7th, Daniel Bossaller (Ohio University), Morton Hall 322, 11:00-12:00

Title: Abelian Core of Graph C*-Algebras II

This talk will concern graph C*-algebras, which are combinatorially defined algebras arising from a directed graph, E. Two main theorems govern how one can realize a graph C*-algebra in another C*-algebra A, the Gauge-Invariant Uniqueness theorem and the Cuntz-Krieger Uniqueness theorem. While the first theorem has no restrictions on the underlying graph, the second requires that no cycle in the graph can have an entry. This latter theorem was generalized by Szmanski in 2000 to a theorem which has no condition on the underlying graph. In 2012, Nagy and Reznikoff gave an alternate proof of Szymanski's result which relies on a characterization of the maximal abelian sub-algebras of graph algebras. In this talk, I will give an overview of the construction of graph C*-algebras and then give Nagy and Reznikoff's proof.

### Nov. 30th, Daniel Bossaller (Ohio University), Morton Hall 322, 11:00-12:00

Title: Abelian Core of Graph C*-Algebras

This talk will concern graph C*-algebras, which are combinatorially defined algebras arising from a directed graph, E. Two main theorems govern how one can realize a graph C*-algebra in another C*-algebra A, the Gauge-Invariant Uniqueness theorem and the Cuntz-Krieger Uniqueness theorem. While the first theorem has no restrictions on the underlying graph, the second requires that no cycle in the graph can have an entry. This latter theorem was generalized by Szmanski in 2000 to a theorem which has no condition on the underlying graph. In 2012, Nagy and Reznikoff gave an alternate proof of Szymanski's result which relies on a characterization of the maximal abelian sub-algebras of graph algebras. In this talk, I will give an overview of the construction of graph C*-algebras and then give Nagy and Reznikoff's proof.

### Nov. 16th, no seminar this week.

### Nov. 9th, no seminar this week.

### Nov. 2nd, Marcel Bischoff (Ohio University), Morton Hall 322, 11:00 - 12:00

Title: Completely positive actions on von Neumann algebras

Abstract: I will introduce a notion of completely positive actions of semigroups on von Neumann algebras which contains some sufficiency criterion to ensure that the fixed point is again a von Neumann algebra. Then I will show that certain inclusions of von Neumann factors are fixed points by a so-called hypergroup acting by completely positive maps. The talk will be loosely based on arXiv:1608.00253 [math-ph] which focuses more on applications to quantum field theory.

### Oct. 26th, Adam Fuller (Ohio University), Morton Hall 322, 11:00 - 12:00

Title: C*-algebra embeddings and twisted groupoids II.

Abstract: Last week we got sidetracked by Cartan subalgebras in von Neumann algebra, and their description via extensions of inverse semigroups. In this talk we will proceed with the original plan. Let C(X) be a Cartan subalgebra of a C*-algebra A. We will discuss how A determines (partial) dynamics on X via the Weyl pseudogroup. This, in turn, gives rise to a twisted groupoid. We will discuss Renault's work on reconstructing Cartan embeddings from groupoid twists.

### Oct. 19th, Adam Fuller (Ohio University), Morton Hall 322, 11:00 - 12:00

Title: C*-algebra embeddings and twisted groupoids.

Abstract: Let A be a C*-algebra, and let D be an abelian C*-algebra contained in A. We will discuss how to construct a an extension of groupoids, called a groupoid twist, from such an embedding. Conversely, we will discuss how to construct abelian embeddings of C*-algebras from groupoid twists via the reduced C*-algebra of the twist. We will concentrate on the case when D is a Cartan embedding. In this case, the groupoid construction gives back the original pair A and D. This is based on work by Kumjian and Renault. Little to no background on C*-algebras will be needed if you are willing to take my word on a few things; no background on groupoids will be assumed.

### Oct. 12th, no seminar this week.

### Oct. 5th, Shehzad Ahmed (Ohio University), Morton Hall 322, 11:00 - 12:00

Title: Continuous Model Theory and C*-Algebras IV

Abstract: We say that a C*-algebra *D* is strongly self-absorbing if it is not isomorphic to $\mathhbb{C}$ , $D\otimes_{\min}D\simeq D$, and the map $\mathrm{id}_D\otimes1:D\to D\otimes_{\min} D$ is approximately unitarily equivalent to the the isomorphism of $D$ and $D\otimes_{\min}D$. This condition is rather restrictive, and there are few known examples of strongly self-absorbing algebras, one of which is the Jiang-Su algebra, $\mathcal{Z}$. In this talk, we will survey a result of Farah, Hart, R\o{}rdam, Tikuisis which gives several model-theoretic characterizations of C*-algebras $A$ for which $D\otimes_{min}A\simeq A$ where $D$ is a strongly self-absorbing algebra. In particular, we will show that this is equivalent to the asking that $D$ embed into the commutant of $A$ inside its ultrapower $A^{\mathcal{U}}$ where $\mathcal U$ is a non-principal ultrafilter over $\mathbb{N}$.

### Sept. 28th, Shehzad Ahmed (Ohio University), Morton Hall 322, 11:00 - 12:00

Title: Continuous Model Theory and C*-Algebras III

Abstract: In the previous talk, I introduced the notion of an L-formula, and linked L-formulas up to metric structures by way of variable assignments. In this talk, we will use this machinery to tackle the ultraproduct construction. The advantage of working with ultraproducts in the framework of continuous model theory is that it allows us to prove Łoś' theorem. This result tells us (roughly) that properties of metric structures that can be expressed in a first order way pass through ultraproducts. With that in mind, the goal of this talk will be to furnish a proof of Łoś' theorem, and use it to motivate the question: which properties of C*-algebras can we express in a first order way?

### Sept. 21st, No seminar this week.

### Sept. 14th, Shehzad Ahmed (Ohio University), Morton Hall 322, 11:00 - 12:00

Title: Continuous Model Theory and C*-algebras II

Abstract: In the previous talk, I laid out the general framework of languages and metric structures. In this talk, we will use this new framework to tackle the ultraproduct construction. Ultraproducts allow us to build new structures out of old ones in a way that preserves a certain class of properties: those expressible in first order (continuous) logic. This is the content of Łoś's theorem. In order to understand Łoś's theorem though, we first need the notion of a first order formula in (continuous) logic. Once we have that, we will work through the ultraproduct construction in detail, and sketch a proof of Łoś's theorem. Time permitting, I would like to briefly discuss some examples of how ultraproducts of metric structures are useful.

The notes can be found here: https://www.math.uh.edu/analysis/2017conference/Notes-Lupini.pdf

### Sept. 7th, Shehzad Ahmed (Ohio University), Morton Hall 126, 14:00 - 14:55

Title: Continuous Model Theory and C*-Algebras

Abstract: This is a series of talks intended to provide an introduction to continuous model theory and its applications to the study of C*-algebras. Continuous model theory and continuous logic are generalizations of classical model theory and logic, which have proven useful in studying (discrete) algebraic objects. My goal for these talks is to introduce some important model-theoretic machinery (e.g. ultraproducts), and discuss several examples where these tools are useful in studying C*-algebras. I will not be assuming any familiarity with the classical theory, nor will I assume knowledge of C*-algebras beyond a course in functional analysis. These talks are based on Martino Lupini's notes for his talks at the Applications of Model Theory to Operator Algebras conference in Houston this past July.

The notes can be found here: https://www.math.uh.edu/analysis/2017conference/Notes-Lupini.pdf