Past Functional Analysis Seminar

In this talk, I will discuss the notion of quantum limits from different viewpoints: Cordes' work on the Gelfand theory for pseudo-differential operators dating from the 70’s as well as the micro-local defect measures and semi-classical measures of the 90’s. I will also explain my motivation and strategy to obtain similar notions in subRiemannian or subelliptic settings. 

I will introduce a K-theoretic complete invariant of inductive limits of finite dimensional actions of fusion categories on unital AF-algebras. This framework encompasses all such actions by finite groups on AF-algebras. Our classification result essentially follows from applying Elliott's Intertwining Argument adapted to this equivariant context, combined with tensor categorical techniques.

Our invariant roughly consists of a finite list of pre-ordered abelian groups and positive homomorphisms, which can be computed in principle. Under certain conditions this can be done in full detail. For example, using our classification theorem, we can show torsion-free fusion categories admit a unique AF-action on certain AF-algebras.

Connecting with subfactors, inspired by Popa’s classification of finite-depth hyperfinite subfactors by their standard invariant, we study unital inclusions of AF-algebras with trivial centers, as natural analogues of hyperfinite II_1 subfactors. We introduce the notion of strongly AF-inclusions and an Extended Standard Invariant, which characterizes them up to equivalence.

Since the completion of the Elliott classification programme it is an important question to ask which C*-algebras satisfy the assumptions of the classification theorem. We will ask this question for the case of crossed-product C*-algebras associated to actions of nonamenable groups and focus on two extreme cases: Actions on commutative C*-algebras and actions on simple C*-algebras. It turns out that for a large class of nonamenable groups, classifiability of the crossed product is automatic under the minimal assumptions on the action. This is joint work with E. Gardella, S. Geffen, P. Naryshkin and A. Vaccaro. 

One of the most important constructions in operator algebras is the tracial ultrapower for a tracial state on a C*-algebra. This tracial ultrapower is a finite von Neumann algebra, and it appears in seminal work of McDuff, Connes, and more recently by Matui-Sato and many others for studying the structure and classification of nuclear C*-algebras. I will talk about how to generalise this to unbounded traces (such as the standard trace on B(H)). Here the induced tracial ultrapower is not a finite von Neumann algebra, but its multiplier algebra is a semifinite von Neumann algebra.

 I will discuss ongoing work with Toke Carlsen and Aidan Sims on ideal structure of C*-algebras of commuting local homeomorphisms. This is one aspect of a general attempt to bridge C*-algebras with multidimensional (symbolic) dynamics.

Hirschman-Widder densities may be viewed as the probability density functions of positive linear combinations of independent and identically distributed exponential random variables. They also arise naturally in the study of Pólya frequency functions, which are integrable functions that give rise to totally positive Toeplitz kernels. This talk will introduce the class of Hirschman-Widder densities and discuss some of its properties. We will demonstrate connections to Schur polynomials and to orbital integrals. We will conclude by describing the rigidity of this class under composition with polynomial functions.

 This is joint work with Dominique Guillot (University of Delaware), Apoorva Khare (Indian Institute of Science, Bangalore) and Mihai Putinar (University of California at Santa Barbara and Newcastle University).

The string 2-group is a fundamental object in string geometry, which is a refinement of spin geometry required to describe the spinning string. While many models for the string 2-group exist, the construction of a representation for it is new. In this talk, I will recall the notion of strict 2-group, and then give two examples: the automorphism 2-group of a von Neumann algebra, and the string 2-group. I will then describe the representation of the string 2-group on the hyperfinite III_1 factor, which is a functor from the string 2-group to the automorphism 2-group of the hyperfinite III_1 factor.

Groupoid C*-algebras and twisted groupoid C*-algebras are introduced by Renault in the late ’70. Twisted groupoid C*-algebras have since proved extremely important in the study of structural properties for large classes of C*-algebras. On the other hand, Steinberg algebras are introduced independently by Steinberg and Clark, Farthing, Sims and Tomforde around 2010 which are a purely algebraic analogue of groupoid C*-algebras. Steinberg algebras provide useful insight into the analytic theory of groupoid C*-algebras and give rise to interesting examples of *-algebras. In this talk, I will first recall some relevant background on topological groupoids and twisted groupoid C*-algebras, then I will introduce twisted Steinberg algebras which generalise the Steinberg algebras and provide a purely algebraic analogue of twisted groupoid C*-algebras. If I have enough time, I will further introduce pair of algebras which consist of a Steinberg algebra and an algebra of locally constant functions on the unit space, it is an algebraic analogue of Cartan pairs

A few years back, Smale spaces were shown to exhibit noncommutative Poincaré duality (with Jerry Kaminker and Ian Putnam). The fundamental class was represented as an extension by the compacts. In current work we describe a Fredholm module representation of the fundamental class. The proof uses delicate approximations of the Smale space arising from a refining sequence of (open) Markov partition covers. I hope to explain all these notions in an elementary manner. This is joint work with Dimitris Gerontogiannis and Joachim Zacharias.

A Hausdorff and etale groupoid is said to be C*-simple if its reduced groupoid C*-algebra is simple. Work on C*-simplicity goes back to the work of Kalantar and Kennedy in 2014, who classified the C*-simplicity of discrete groups by associating to the group a dynamical system. Since then, the study of C*-simplicity has received interest from group theorists and operator algebraists alike. More recently, the works of Kawabe and Borys demonstrate that the groupoid case may be tractible to such dynamical characterizations. In this talk, we present the dynamical characterization of when a groupoid is C*-simple and work out some basic examples. This is joint work with Xin Li, Matt Kennedy, Sven Raum, and Dan Ursu. No previous knowledge of groupoids will be assumed.

In the late 1980s, Berger and Coburn showed that the Hankel operator $H_f$ on the Segal-Bargmann space of Gaussian square-integrable entire functions is compact if and only if $H_{\bar f}$ is compact using C*-algebra and Hilbert space techniques. I will briefly discuss this and three other proofs, and then consider the question of whether an analogous phenomenon holds for Schatten class Hankel operators. 

Scattered topological spaces and their C*-analogs, known as scattered
C*-algebras, have been studied since the 70's and admit a number of
interesting characterizations. In this talk, I will define nowhere
scattered C*-algebras as, informally, those C*-algebras that are very
far from being scattered. I will then characterize this property in
various ways, such as the absence of nonzero elementary ideal-quotients,
topological properties of the spectrum, and divisibility properties in
the Cuntz semigroup. Further, I will also show that these divisibility
properties can be strengthened in the real rank zero or the stable rank
one case.

The talk is based on joint work with Hannes Thiel.

Q-systems were introduced by Longo to describe the canonical endomorphism of a finite Jones-index inclusion of infinite von Neumann factors. From our viewpoint, a Q-system is a unitary  version of a Frobenius algebra object in a tensor category or a C* 2-category. Following work of Douglass-Reutter, a Q-system is also a unitary version of a higher idempotent, and we will describe a higher unitary idempotent completion for C* 2-categories called Q-system completion. 

We will focus on the C* 2-category C*Alg with objects unital C*-algebras, 1-morphisms right Hilbert C*-correspondences, and 2-morphisms adjointable intertwiners. By adapting a subfactor reconstruction technique called realization, and using the graphical calculus available for C* 2-categories, we will show that C*Alg is Q-system complete.

This result allows for the straightforward adaptation of subfactor results to C*-algebras, characterizing finite Watatani-index extensions of unital C*-algebras equipped with a faithful conditional expectation in terms of the Q-systems in C*Alg. Q-system completion can also be used to induce new symmetries of C*-algebras from old. 

This is joint work with Quan Chen, Corey Jones and Dave Penneys (arXiv: 2105.12010).

I will report on recent progress concerning eigenvalues of Schrödinger operators with complex potentials. We are interested in the magnitude and distribution of eigenvalues, and we seek bounds that only depend on an L^p norm of the potential.

These questions are well understood for real potentials, but completely new phenomena arise for complex potentials. I will explain how techniques from harmonic analysis, particularly those related to Fourier restriction theory, can be used to prove upper and lower bounds. We will also discuss some open problems. The talk is based on recent joint work with Sabine Bögli (Durham).

Here we present the findings of our summer research project: an 8-week investigation of C*-Algebras. Our aim was to explore when a group is uniquely determined by its reduced group C*-algebra; i.e. when the group is C*-rigid. In particular, we applied techniques from topology, algebra, and analysis to prove C*-rigidity for a certain class of Bieberbach groups.

Twisted K-theory is an enrichment of topological K-theory that allows local coefficient systems called twists. For spaces and twists equipped with an action by a group, equivariant twisted K-theory provides an even finer invariant. Equivariant twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the corresponding K-groups to the Verlinde ring of the associated loop group. From the point of view of homotopy theory only a small subgroup of all possible twists is considered in classical treatments. In this talk I will discuss a construction that is joint work with David Evans and produces interesting examples of non-classical twists over the Lie groups SU(n) and over tori constructed from exponential functors. They arise naturally as Fell bundles and are equivariant with respect to the conjugation action of the group on itself. For the determinant functor our construction reproduces the basic gerbe over SU(n) used by Freed, Hopkins and Teleman.

In this talk I will discuss a number of topics at the interface between C* algebras and Geometric Group Theory, with an emphasis on Kazhdan projections, various versions of amenability and their connection to the geometry of groups. This is based on joint work with P. Nowak and J. Mackay.

We showcase some newly emerging connections between the theory of random walks and operator algebras, obtained by associating concrete algebras of operators to random walks. The C*-algebras we obtain give rise to new and interesting notions of ratio limit space and boundary, which are computed by appealing to various works on the asymptotic behavior of transition probabilities for random walks. Our results are leveraged to shed light on a question of Viselter on symmetry-unique quotients of Toeplitz C*-algebras of subproduct systems arising from random walks.

The graph C*-algebra construction associates a unital C*-algebra to any directed graph with finitely many vertices and countably many edges in a way which generalizes the fundamental construction by Cuntz and Krieger. We say that two such graphs are C*-equivalent when they define isomorphic C*-algebras, and give a description of this relation as the smallest equivalence relation generated by a number of "moves" on the graph that leave the C*-algebras unchanged. The talk is based on recent work with Arklint and Ruiz, but most of these moves have a long history that I intend to present in some detail.

In this talk I will explain how a subfactor (ie an inclusion of type II_1 factors) give rise to a diagrammatic algebra called the Temperley-Lieb-Jones algebra. We will observe the connection between the index of the subfactor, and the TLJ algebra. In the TLJ algebra setting, we will observe that indices below four are discrete, while any number above four can be an index.

n this talk we consider a p-isometrisability property of discrete groups. If p=2 this property is equivalent to unitarisability. We prove that any group containing a non-abelian free subgroup is not p-isometrisable for any p∈(1,∞). We also discuss some open questions and possible relations of p-isometrisability with the recently introduced Littlewood exponent Lit(Γ).

The observation that the Dirac operator on a spin manifold encodes both the Riemannian metric as well as the fundamental class in K-homology leads to the paradigm of noncommutative geometry: the viewpoint that spectral triples generalise Riemannian manifolds. To encode maps between Riemannian manifolds, one is naturally led to consider the unbounded picture of Kasparov's KK-theory. In this talk I will explain how smooth cycles in KK-theory give a natural notion of noncommutative fibration, encoding morphisms noncommutative geometry in manner compatible with index theory.

Continuous product systems were introduced and studied by Arveson in the late 1980s. The study of their discrete analogues started with the work of Dinh in the 1990’s and it was formalized by Fowler in 2002. Discrete product systems are semigroup versions of C*-correspondences, that allow for a joint study of many fundamental C*-algebras, including those which come from C*-correspondences, higher rank graphs and elsewhere.
Katsura’s covariant relations have been proven to give the correct Cuntz-type C*-algebra for a single C*-correspondence X. One of the great advantages of Katsura's Cuntz-Pimsner C*-algebra is its co-universality for the class of gauge-compatible injective representations of X. In the late 2000s Carlsen-Larsen-Sims-Vittadello raised the question of existence of such a co-universal object in the context of product systems. In their work, Carlsen-Larsen-Sims-Vittadello provided an affirmative answer for quasi-lattices, with additional injectivity assumptions on X. The general case has remained open and will be addressed in these talk using tools from non-selfadjoint operator algebra theory.

The recent work on nc convex sets of Davidson, Kennedy, and Shamovich show that there is a rich interplay between the category of operator systems and the category of compact nc convex sets, leading to new insights even in the case of C*-algebras. The category of nc convex sets are a generalization of the usual notion of a compact convex set that provides meaningful connections between convex theoretic notions and notions in operator system theory. In this talk, we present a duality theorem for norm closed self-adjoint subspaces of B(H) that do not necessarily contain the unit. Using this duality, we will describe various C*-algebraic and operator system theoretic notions such as simplicity and subkernels in terms of their convex structure. This is joint work with Matthew Kennedy and Nicholas Manor.

TBA