Past Functional Analysis Seminar

The Cuntz semigroup of a C*-algebra is an ordered monoid consisting of equivalence classes of positive elements in the stabilization of the algebra.  It can be thought of as a generalization of the Murray-von Neumann semigroup, and records substantial information about the structure of the algebra.  Here we examine the set of positive elements having a fixed equivalence class in the Cuntz semigroup of a simple, separable, exact and Z-stable C*-algebra and show that this set is path connected when the class is non-compact, i.e., does not correspond to the class of a projection in the C*-algebra.  This generalizes a known result from the setting of real rank zero C*-algebras.

The classification of simple, unital, nuclear UCT C*-algebras with finite nuclear dimension can be achieved using an invariant derived from K-theory and tracial information. In this talk, I will present a classification theorem for certain classes of C*-algebras that rely solely on tracial deformations.  

 Structural properties of C*-Algebras such as Stable Rank One, Real Rank Zero, and radius of comparison have played an important role in classification.  Crossed product C*-Algebras are useful examples to study because knowledge of the base Algebra can be leveraged to determine properties of the crossed product.  In this talk we will discuss the permanence of various structural properties when taking crossed products of several types.  Crossed products considered will include the usual C* crossed product by a group action along with generalizations such as crossed products by a partial automorphism.  

This talk is based on joint work with Julian Buck and N. Christopher Phillips and on joint work with Maria Stella Adamo, Marzieh Forough, Magdalena Georgescu, Ja A Jeong, Karen Strung, and Maria Grazia Viola.

The Zappa–Szép (ZS) product of two groupoids is a generalization of the semi-direct product: instead of encoding one groupoid action by homomorphisms, the ZS product groupoid encodes two (non-homomorphic, but “compatible”) actions of the groupoids on each other. I will show how to construct the ZS product of two twists over such groupoids and give an example using Weyl twists from Cartan pairs arising from Kumjian--Renault theory.

 Based on joint work with Boyu Li, New Mexico State University

The set $M_n(\mathbb{R})$ of all $n \times n$ matrices over the real numbers is an example of an algebraic structure called a $C^*$-algebra. The subalgebra $D$ of diagonal matrices has special properties and is called a \emph{Cartan subalgebra} of $M_n(\mathbb{R})$. Given an arbitrary $C^*$-algebra, it can be very hard (but also very rewarding) to find a Cartan subalgebra, if one exists at all. However, if the $C^*$-algebra is generated by a cocycle $c$ and a group (or groupoid) $G$, then it is natural to look within $G$ for a subgroup (or subgroupoid) $S$ that may give rise to a Cartan subalgebra. In this talk, we identify sufficient conditions on $S$ and $c$ so that the subalgebra generated by $(S,c)$ is indeed a Cartan subalgebra of the $C^*$-algebra generated by $(G,c)$. We then apply our theorem to $C^*$-algebras generated by $k$-graphs, which are directed graphs in higher dimensions. This is joint work with J. Briones Torres, A. Duwenig, L. Gallagher, E. Gillaspy, S. Reznikoff, H. Vu, and S. Wright.

A self-similar k-graph is a pair consisting of a (discrete countable) group and a k-graph, such that the group acts on the k-graph self-similarly. For such a pair, one can associate it with a universal C*-algebra, called the self-similar k-graph C*-algebra. This class of C*-algebras embraces many important and interesting C*-algebras,  such as the higher rank graph C*-algebras of Kumjian-Pask, the Katsura algebras,  the Nekrashevych algebras constructed from self-similar groups, and the Exel-Pardo algebra. 

In this talk, we will survey some results on self-similar k-graph C*-algebras. 

A central theme in the 40-year-old Ribe program is the quest for metric invariants that characterize local properties of Banach spaces. These invariants are usually closely related to the geometry of certain sequences of finite graphs (Hamming cubes, binary trees, diamond graphs...) and provide quantitative bounds on the bi-Lipschitz distortion of those graphs.

A more recent program, deeply influenced by the late Nigel Kalton, has a similar goal but for asymptotic properties instead. In this talk, we will motivate the (asymptotic) notions of infrasup umbel convexity (introduced in collaboration with Chris Gartland (UC San Diego)) and bicone convexity. These asymptotic notions are inspired by the profound work of Lee, Mendel, Naor, and Peres on the (local) notion of Markov convexity and of Eskenazis, Mendel, and Naor on the (local) notion of diamond convexity. 

All these metric invariants share the common feature of being derived from point-configuration inequalities which generalize curvature inequalities.

If time permits we will discuss the values of these invariants for Heisenberg groups.

A group is called W*-superrigid if its group von Neumann algebra completely remembers the original group. In this talk, I will present a recent joint work with Stefaan Vaes in which we generalize W*-superrigidity for groups in two directions. Firstly, we find a class of groups for which W*-superrigidity holds in the presence of a twist by an arbitrary 2-cocycle: the twisted group von Neumann algebra completely remembers both the original group and the 2-cocycle. Secondly, for the same class of groups, the superrigidity also holds up to virtual isomorphism.

Associated to every shift space, the Cuntz-Krieger algebra (C-K algebra for abbreviation) is an invariant of conjugacy defined and developed by K. Matsumoto, S. Eilers, T. Carlsen, and many of their collaborators in the last decade. In particular, Carlsen defined the C-K algebra to be the full groupoid C*-algebra of the “cover”, which is a topological system consisting of a surjective local homeomorphism on a zero-dimensional space induced by the shift space. 

In 2022, K. Brix proved that the C-K algebra of the Sturmian shift has finite nuclear dimension, where the Sturmian shift is the (unique) minimal shift space with the smallest complexity function: p_X(n)=n+1. In recent results (joint with Z. He), we show that for any minimal shift space with finitely many left special elements, its C-K algebra always have finite nuclear dimension. In fact, this can be further applied to the class of aperiodic shift spaces with non-superlinear growth complexity. 

Recently, Elliott, Li, and Niu proved a classification theorem for Villadsen-type algebras using the combination of the Elliott invariant and the radius of comparison, an invariant that was introduced by Toms in order to distinguish between certain non-isomorphic AH algebras with the same Elliott invariant. This might have raised the prospect that the Elliott classification program can be extended beyond the Z-stable case by adding the radius of comparison to the invariant. I will discuss a recent preprint in which we show that this is not the case: we construct an uncountable family of nonisomorphic AH algebras with the same Elliott and same radius of comparison. We can distinguish between them using a finer invariant, which we call the local radius of comparison. This is joint work with N. Christopher Phillips.

A $C^*$-diagonal is a certain commutative subalgebra of a $C^∗$ -algebra with a rich structure. Renault and Kumjian showed that finding a $C^*$ -diagonal of a $C^∗$-algebra is equivalent to realizing the $C^*$-algebra via a groupoid. This establishes a close connection between $C^∗$-diagonals and dynamics and allows one to relate the geometric properties of groupoids to the properties of $C^∗$ -diagonals. 

In this talk, I will explore the unique pure state extension property of an Abelian $C^*$-subalgebra of a 1-dim NCCW complex, the approximation of morphisms between two 1-dim NCCW complexes by $C^*$-diagonal preserving morphisms, and the existence of $C^*$-diagonal in inductive limits of certain 1-dim NCCW complexes.

One possible version of the Kirchberg—Phillips theorem states that simple, separable, nuclear, purely infinite C*-algebras are classified by KK-theory. In order to generalize this result to non-simple C*-algebras, Kirchberg first restricted his attention to those that absorb the Cuntz algebra O2 tensorially. C*-algebras in this class carry no KK-theoretical information in a strong sense, and they are classified by their ideal structure alone. It should be mentioned that, although this result is in Kirchberg’s work, its full proof was first published by Gabe. In joint work with Gábor Szabó, we showed a generalization of Kirchberg's O2-stable theorem that classifies G-C*-algebras up to cocycle conjugacy, where G is any second-countable, locally compact group. In our main result, we assume that actions are amenable, sufficiently outer, and absorb the trivial action on O2 up to cocycle conjugacy. In very recent work, I moreover show that the range of the classification invariant, consisting of a topological dynamical system over primitive ideals, is exhausted for any second-countable, locally compact group.

In this talk, I will recall the classification of O2-stable C*-algebras, and describe their classification invariant. Subsequently, I will give a short introduction to the C*-dynamical working framework and present the classification result for equivariant O2-stable actions. Time permitting, I will give an idea of how one can build a C*-dynamical system in the scope of our classification with a prescribed invariant. 

Product systems represent powerful contemporary tools in the study of mathematical structures. A major success in the theory came from Katsura (2007), who provided a complete description of the gauge-invariant ideals of many important C*-algebras arising from product systems over Z+. This result recaptures existing results from the literature, illustrating the versatility of product system theory. The question now becomes whether or not Katsura's result can be bolstered to product systems over semigroups other than Z+ and, if so, what applications do we obtain? An answer has been elusive, owing to the more pathological nature of product systems over general semigroups. However, recent strides by Dor-On and Kakariadis (2018) supply a more tractable subclass of product systems that still includes the important cases of C*-dynamics, row-finite higher-rank graphs, and regular product systems. 

In this talk we will build a parametrisation of the gauge-invariant ideals, starting from first principles and gradually increasing in complexity. We will pay particular attention to the higher-rank subtleties that are not witnessed in Katsura's theorem, and comment on the applications.
 

Transportation cost spaces are of high theoretical interest,  and they also are fundamental in applications in many areas of applied mathematics, engineering, physics, computer science, finance, and social sciences. 

Obtaining low distortion embeddings of transportation cost spaces into L_1 became important in the problem of finding nearest points, an important research subject in theoretical computer science. After introducing

these spaces we will present some results on upper  and lower estimates of the distortion of embeddings of Transportation Cost Spaces into L_1

The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying solidity of von Neumann algebras. We introduce the notion of biexactness for von Neumann algebras, which allows us to place many previous solidity results in a more systematic context, and naturally leads to extensions of these results. We will also discuss examples of solid factors that are not biexact. This is a joint work with Jesse Peterson.

We introduce the notion of a tensorially absorbing inclusion of C*-algebras, i.e., when a unital inclusion absorbs a strongly self-absorbing C*-algebra. This is a strong condition that ensures certain properties of both algebras (and their intermediate subalgebras) in a very strong sense. We discuss such inclusions, their non-triviality, and how often these inclusions appear.

In the world of von Neumann algebras, the factors that do not have a trace, the so-called type III factors, are the most difficult to study. Some of their key structural properties are still not well-understood. In this talk, I will give a gentle introduction to Connes's Bicentralizer Problem, which is the most important open problem in the theory of type III factors. I will then present some recent progress on this problem and its applications.

Results from a few years ago of Kennedy and Schafhauser attempt to characterize the simplicity of reduced crossed products, under an assumption which they call vanishing obstruction. 

However, this is a strong condition that often fails, even in cases of finite groups acting on finite dimensional C*-algebras. In this work, we give complete C*-dynamical characterization, of when the crossed product is simple, in the setting of FC-hypercentral groups. 

This is a large class of amenable groups that, in the finitely-generated setting, is known to coincide with the set of groups with polynomial growth.

We discuss Connes’ quantized calculus on quantum tori and Euclidean spaces, as applications of the recent development of noncommutative analysis.
This talk is based on a joint work in progress with Xiao Xiong and Kai Zeng.
 

Often, one tries to understand the behaviour of non-commutative random variables or of von Neumann algebras through matricial approximations. In some cases, such as when appealing to the determinant conjecture or investigating the soficity of a group, it is important to find approximations by matrices with good algebraic conditions on their entries (e.g., being integers). On the other hand, the most common tool for generating asymptotic independence -- conjugating with random unitaries -- often destroys such delicate structure.

 I will speak on recent joint work with de Santiago, Hayes, Jekel, Kunnawalkam Elayavalli, and Nelson, where we investigate graph products (an interpolation between free and tensor products) and conjugation of matrix models by large structured random permutations. We show that with careful control of how the permutation matrices are chosen, we can achieve asymptotic graph independence with amalgamation over the diagonal matrices. We are able to use this fine structure to prove that strong $1$-boundedness for a large class of graph product von Neumann algebras follows from the vanishing of the corresponding first $L^2$-Betti number. The main idea here is to show that a version of the determinant conjecture holds as long as the individual algebras have generators with approximations by matrices with entries in the ring of integers of some finite extension of Q satisfying some conditions strongly reminiscent of soficity for groups.

I will give an introduction to quasidiagonality of group actions wherein an action on a C^*-algebra is approximated by actions on matrix algebras.  This has implications for crossed product C^*-algebras, especially as pertains to finite dimensional approximation.  I'll sketch the proof that all isometric actions are quasidiagonal, which we can view as a dynamical Petr-Weyl theorem.  Then I will discuss an interplay between quasidiagonal actions and semiprojectivity of C^*-algebras, a property that allows "almost representations" to be perturbed to honest ones.  

To every action of a discrete group on a compact Hausdorff space one can canonically associate a C*-algebra, called the crossed product. The crossed product construction is an extremely popular one, and there are numerous results in the literature that describe the structure of this C* algebra in terms of the dynamical system. In this talk, we will focus on one of the central notions in the realm of the classification of simple, nuclear C*-algebras, namely Jiang-Su stability. We will review the existing results and report on the most recent progress in this direction, going beyond the case of free actions both for amenable and nonamenable groups. 

Parts of this talk are joint works with Geffen, Kranz, and Naryshkin, and with Geffen, Gesing, Kopsacheilis, and Naryshkin. 

The (finite) Hilbert matrix is arguably one of the single most well-known matrices in mathematics. The infinite Hilbert matrix H was introduced by David Hilbert around 120 years ago in connection with his double series theorem. It can be interpreted as a linear operator on spaces of analytic functions by its action on their Taylor coefficients. The boundedness of H on the Hardy spaces H^p for 1 < p < ∞ and Bergman spaces A^p for 2 < p < ∞ was established by Diamantopoulos and Siskakis. The exact value of the operator norm of H acting on the Bergman spaces A^p for 4 ≤ p < ∞ was shown to be π /sin(2π/p) by Dostanic, Jevtic and Vukotic in 2008. The case 2 < p < 4 was an open problem until in 2018 it was shown by Bozin and Karapetrovic that the norm has the same value also on the scale2 < p < 4. In this talk, we introduce some background, review some of the old results, and consider the still partly open problem regarding the value of the norm on weighted Bergman spaces. We also consider a generalised Hilbert matrix operator and its (essential) norm. The talk is partly based on a joint work with Mikael Lindström, David Norrbo, and Niklas Wikman (Åbo Akademi University).
 

I will present novel freeness results in ultraproducts of tracial von Neumann algebras. As a particular case, I will show that if a and b are the generators of the free group F_2, then the relative commutants of a and b in the ultraproduct of the free group factor are free with respect to the ultraproduct trace. The proof is based on a surprising application of Lp-boundedness results of Fourier multipliers in free group factors for p > 2. I will describe applications of these results to absorption and model theory of II_1 factors. This is joint work with Adrian Ioana.