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Eli Bashwinger
Similarity Structure Groups and Their W* and C*-algebras
I will first introduce the Higman-Thompson group Vd acting on the so-called d-ary Cantor space (for d ≥ 2). This will serve to motivate the definition of a similarity structure on a compact ultrametric space and the resulting similarity structure group one can associate to this setup, which ought to be thought of as generalizing the Higman-Thompson groups. Then I will state some group-theoretic and analytic properties of these groups used to obtain structural results about their W* and C*-algebras. This is joint work with Patrick Debonis.
Jon Brown, University of Dayton
Regular Ideals, Ideal Intersections, and Quotients
Let B ⊆ A be an inclusion of C*-algebras. We study the relationship between regular ideals of B and regular ideals of A. We show that if B ⊆ A is a regular inclusion and there is a faithful invariant conditional expectation from A onto B then there is an isomorphism between the lattice of regular ideals of A and invariant regular ideals of B. We study the properties of inclusions preserved under quotients by regular ideals. This includes showing that if B ⊆ A is a Cartan inclusion and J is a regular ideal in A then B/(J ∩ B) is a Cartan subalgebra of A/J. We provide a description of regular ideals in the reduced crossed product of a C* -algebra by a discrete group. This work is joint with A. Fuller, D. Pitts, and S. Reznikoff.
Ken Davidson, University of Waterloo
Recollections of working with David, and Large Perturbations of Nest Algebras
I have known and worked with David since 1979. I will spend part of my talk on a few memories.
Let ℳ and 𝒩 be nests on separable Hilbert space. If the two nest algebras are distance less than 1, then the nests are distance less than 1. If the nests are distance less than 1 apart, then the nest algebras are similar, i.e. there is an invertible S such that Sℳ = 𝒩, so that S𝒯(ℳ)S−1 = 𝒯(𝒩). However there are examples of nests closer than 1 for which the nest algebras are distance 1 apart.
Derek DeSantis, Los Alamos National Laboratory
Learning Operators from Finite Observations
A central goal of machine learning is to approximate functions from finite collections of input-output pairs. Recently, scientific machine learning has moved beyond learning maps between finite-dimensional spaces towards learning operators between Banach spaces. Naturally, the shift to infinite dimensional learning comes with its own set of limitations and challenges. Key among them is the following question: which properties of a finite-dimensional learned approximation reflect meaningful properties of the underlying infinite-dimensional operator?
This talk will give an introduction to operator learning. I will discuss neural operators such as learned approximations to PDE solution maps, and Koopman learning as an attempt to approximate linear evolution operators associated with nonlinear dynamical systems. The central theme will be the gap between finite-dimensional computational surrogates and the continuum operators they are meant to represent. I will also discuss how physically meaningful structure can sometimes be extracted or imposed despite this gap. Time permitting, we will discuss open questions that may be of interest to operator theorists.
Ruy Exel, Universidade Federal de Santa Catarina
Regular inclusions and induction
We extend Steinberg's theory of induction of modules over Steinberg algebras to the case of \emph {regular inclusions} of algebras. Among our main results, we show that, under appropriate conditions, every irreducible module is induced by an irreducible module over a certain abstractly defined \emph {isotropy algebra}. We also describe a process of \emph{disintegration} of modules and use it to prove a version of the Effros-Hahn conjecture, showing that every primitive ideal coincides with the annihilator of a module induced from isotropy. This is based on joint work with Misha Dokuchaev and Hector Pinedo. Some of the main techniques employed are in turn based on a 2022 Springer Lecture Notes by the speaker and the honoree.
Adam Fuller, Ohio University
Coordinitization and Spectral Synthesis
Let A be an operator algebra and let B be a subalgebra of A. In certain settings, the inclusion of B inside A determines the structure of A. Examples of this phenomenon include inclusions of Cartan masas (in both the C* and von Neumann settings). In this talk we will discuss when and how the inclusion can be used to describe the B bimodules inside A. In particular we will look at whether we can describe the intermediate algebras B ⊆ C ⊆ A.
The talk will touch on work in von Neumann algebras (joint w/ Pitts and Donsig-Pitts); and C*-algebras (joint w/ Brown-Exel-Pitts-Reznikoff and Brown-Clark).
David Milan, University of Texas-Tyler
Combinatorial structures from inverse semigroups
There are many well-known constructions of C*-algebras from combinatorial objects. Often an inverse semigroup is used as an intermediate structure. In this talk we focus on recovering a combinatorial object from an inverse semigroup in a way that preserves the Morita equivalence class of the inverse semigroup.
We characterize the inverse semigroups that are Morita equivalent to graph inverse semigroups and discuss a collection of inverse semigroups associated with labeled graphs.
David Pitts, University of Nebraska–Lincoln
A Few of My Favorite Things
In this mostly expository talk, I will describe a few ideas and results which I've enjoyed over the years. The topics will mostly follow the path of my interests during my career: I'll start by discussing some classes of non-selfadjoint operator algebras, then move to self-adjoint topics.
Chris Schafhauser, University of Nebraska-Lincoln
Products of Exponentials
Let A be a unital C*-algebra and let u be a unitary in A. It is a standard fact that u is in the connected component of the identity if and only if u is a finite product of exponentials. I’ll discuss some new examples, including various reduced group C*-algebras, where one can provide a uniform bound on the number of exponentials needed in such a decomposition. This is based on joint work with Sri Kunnawalkam Elayavalli.
Jack Spielberg, Arizona State University
Effros-Shen algebras for rational numbers, and not Ext
Nonstandard presentations of the continued fraction AF algebras of Effros and Shen can be defined using certain categories of paths. The same construction gives analogous algebras for rational numbers. We will describe these algebras, and their relation with certain C*-algebra extensions.
Vrej Zarikian, United States Naval Academy
Unique Pseudo-Expectations for $C^*$-Inclusions
Pitts introduced \emph{pseudo-expectations} in his 2011 GPOTS lecture at Arizona State, which I was fortunate to attend. A pseudo-expectation for the (unital) $C^*$-inclusion $\A \subseteq \B$ is a ucp (unital completely positive) map $\theta:\B \to I(\A)$ such that $\theta(a)=a$ for all $a \in \A$. Here $I(\A)$ is the \emph{injective envelope} of $\A$. Every conditional expectation is a pseudo-expectation, but there are $C^*$-inclusions with no conditional expectations, while pseudo-expectations always exist by injectivity. In that same talk, Pitts announced that if $\A \subseteq \B$ is a regular MASA inclusion (for example if $\A \subseteq \B$ is Cartan), then there exists a \ul{unique} pseudo-expectation. In this talk I will survey 15 years of research on the \emph{unique pseudo-expectation property}, in particular the (still elusive) search for a structural characterization that applies to all $C^*$-inclusions.
Catherine Zimmiti, Nebraska Wesleyan University
Perturbations of Representations of Cartan Inclusions
In 2005, Charles Read constructed an example of a free semigroup algebra that is a von Neumann algebra. This construction, which was later simplified by Ken Davidson, involves taking a standard representation of O2 and multiplying it by a certain unitary operator in the diagonal MASA of the representation, creating a new “perturbed” representation of O2. We explore this notion of perturbed representations in the more general setting of Cartan inclusions.