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A unified approach for classifying simple nuclear \(C^*\)-algebras
In this talk, I give an overview of a new proof of the Kirchberg–Phillips theorem, obtained by adapting the framework laid out by Carrión, Gabe, Schafhauser, Tikuisis and White for classifying separable simple unital nuclear stably finite Z-stable \(C^*\)-algebras satisfying the UCT. Not only does this give a unified approach to classifying stably finite and purely infinite \(C^*\)-algebras, in contrast to the other proofs of the Kirchberg–Phillips theorem, this proof does not rely on Kirchberg’s Geneva Theorems, but instead implies them as corollaries. This is joint work with Jamie Gabe.
Orbit breaking in Deaconu-Renault groupoid \(C^*\)-algebras
The standard orbit breaking construction of Putnam applies to a homeomorphism on a compact metric space. It produces certain subalgebras of the crossed product algebra. As a result, the standard orbit breaking construction only produces stably finite \(C^*\)-algebras. I will discuss joint work with Ian Putnam and Karen Strung where we generalize the orbit breaking method to a local homeomorphism. This produces certain subalgebras of Deaconu-Renault groupoid \(C^*\)-algebras, such algebras (and their subalgebras) can be purely infinite.
\(C^*\)-algebras and group representations
I’ll talk about \(C^*\)-algebras generated by unitary representations of polycyclic groups. Polycyclic groups are the class of groups containing all cyclic groups and closed under taking cyclic extensions. I’ll focus most of my time on twisted group \(C^*\)-algebras of torsion free nilpotent groups (a special class of polycyclic groups). I’ll try to convince you that they are worthy of further study. I’ll show you how the group structure simplifies the \(C^*\)-algebraic structure and trace space to convince you they are a somewhat tractable class of \(C^*\)-algebras. I’m a fan of these algebras for the same reason I’m a fan of noncommutative tori—one can apply big hammer theorems to them but often one doesn’t need to. I’ll explain that and then tell you what we know about them and what we’d like to know.
\(\mathcal{Z}\)-Stability of Graph \(C^*\)-Algebras
Graph \(C^*\)-algebras form a well-understood class of examples that form a natural test case for many conjectures. Recent work (F.-Schafhauser, Evington-Ng-Sims-White) has examined the nuclear dimension of graph \(C^*\)-algebras. Tensorial absorption of the Jiang-Su algebra \(\mathcal{Z}\) is closely related to finite nuclear dimension, as it is conjectured that they are equivalent for \(C^*\)-algebras without elementary subquotients. In this talk, we discuss a graphical condition that guarantees \(mathcal{Z}\)-stability of the graph algebra when it has finitely many ideals. As a corollary, we obtain the well-known fact that AF-algebras are \(\mathcal{Z}\)-stable when they do not have an elementary subquotient.
Classifiability of crossed products
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.
When Can Cohomology Obstruct Un-normalized Schatten p-Stability?
A group is said to be stable in the unnormalized Schatten p-norm if a map from the group to unitary matrices that is “almost multiplicative” in the point p-norm topology is “close” to an honest representation in the same topology. A result of Dadarlat shows that even cohomology obstructs operator norm stability. In this talk I will explain why that many topological invariants that arise in his argument vanish for maps that are almost multiplicative in the Schatten p-norm. The obstruction in his proof can be realized as follows. To each almost-representation, we may associate a vector bundle. This vector bundle has topological invariants, called Chern characters which lie in the even cohomology of the group. If any of these invariants are nonzero, the almost-representation is far (in operator norm) from a genuine representation. This talk will explain that if a map is sufficiently multiplicative in the Schatten p-norm then for \(k\geq p\) the kth Chern character vanishes. The upshot is that \(2k\) cohomology should not obstruct Schatten p-stability for \(k\geq p\).
Commutation, Approximation, and Obstruction for Unitary Matrices
We’ll present recent work in progress concerning the question of whether almost commuting matrices are nearly commuting. We’ll discuss exact and approximate representations from geometry as bounded operators and report a quantitative bound on appropriate approximate representations of the torus.
Discrete Inclusions of \(C^*\)-algebras
Property (SI) and approximations using pure states
The property (SI) was firstly introduced by Matui and Sato in 2012 to show that the only remaining piece of the Toms-Winter conjecture, which is the implication from strict comparison to \(\mathcal{Z}\)-stability, holds under certain assumptions on the trace space. The property is then generalized to maps, with domains that are not necessarily simple, for instance, in [BBSTWW, 2015].
One of the key steps in these proofs involves an approximation property by pure states. We will show that this property actually holds for every separable nuclear \(C^*\)-algebras. As a consequence, we prove that full unital maps from nuclear domains to codomains with strict comparison have property (SI). This is generalized further to nuclear maps.
Von Neumann Orbit Equivalence
Recently, Peterson, Ruth, and myself introduced the notion of von Neumann equivalence which is an equivalence relation on the class of countable discrete groups, and it generalizes both measure equivalence and \(W^*\)-equivalence. Another important equivalence relation on groups that has been studied for a long time is Orbit Equivalence and it is closely related to measure equivalence. In this talk, I will introduce the notion of von Neumann orbit equivalence which is a non-commutative generalization of orbit equivalence. We will discuss the stability of this equivalence relation under taking free products and graph products. This is based on a joint work with Aoran Wu.
Ordinal graphs and their \(C^*\)-algebras
We introduce a class of left cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we find an associated \(C^*\)-correspondence. These allow us to apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.
Fourier convergence and Rapid Decay in groupoid \(C^*\)-algebras
Strict comparison in \(C^*\) algebras
Contractible Cuntz Classes
There is a rich history of the study of topological invariants of distinguished subsets in operator algebras. Some results include the computation of the homotopy groups of the invertible group in a Banach algebra, the unitary group in II\(_1\) factors, and the sets of projections and unitaries in various classes of \(C^*\)-algebras. The Cuntz semigroup of a \(C^*\)-algebra is a sensitive invariant which can be thought of as a generalization of the Murray-von Neumann semigroup. It consists of Cuntz equivalence classes of positive elements in the stabilization of A, and these classes come in two flavors: compact (classes of projections) and non-compact (the rest). We show that if A is unital, simple, and \(\mathcal{Z}\)-stable, then the set of positive elements in A belonging to a fixed non-compact Cuntz class is contractible. Combined with work of Jiang and Hua for compact classes, this completes the calculation of the homotopy groups of Cuntz classes for these algebras.
This is joint work with Andrew Toms.
Perturbations by nilpotent operators in a multiplier algebra
Let B be a separable simple stable purely infinite \(C^*\)-algebra, and let \(M(B)\) be the multiplier algebra of \(B\). Generalizing a result of Brown, Pearcy and Salinas, we prove that for all \(X \in M(B)\), there exists a nilpotent operator \(N \in M(B)\) for which \(X + N\) is invertible in \(M(B)\) if and only if \(N \not\in B\). Related to the above, we also have multiplier analogues of results of Dyer–Porcelli–Rosenfeld and Aiken. Interesting new questions arise in the general \(C^*\)-algebra setting.
KMS bundles with multi-parameters on the Jiang-Su algebra
In order to realize all possible KMS-bundles with multi-parameters on the Jiang-Su algebra, we construct a free group action on a rational approximately finite dimensional algebra.
TBA
Relative biexactness of amalgamated free product with injective amalgam
It was shown by Ozawa that the amalgamated free product of two exact groups over an amenable group is biexact relative to the two generating groups. In this talk, I will show that the amalgamated free product of weakly exact von Neumann algebras over an injective von Neumann subalgebra is relatively biexact, generalizing the case of groups. The proof involves a universal property of Toeplitz-Pimsner algebras and a locally convex topology on bimodules of von Neumann algebras, which is used to characterize biexactness for von Neumann algebras.
This is joint work with Zhiyuan Zhang.
**Representation stability and classification of *-homomorphisms**
Representation stability asks if approximate representations of some object (groups, \(C^*\)-algebras, … ) must be close to actual representations. For example, if \(A\) is a \(C^*\)-algebra, one can ask if any approximately multiplicative ucp map from \(A\) to a matrix algebra is close to an honest *-homomorphism. There are obstructions to this: for example, \(A\) might not even have any finite-dimensional representations; and there are also more subtle obstructions coming from \(K\)-theory. One can still ask, however, for representation stability in the case that all the known obstructions vanish. In this talk I’ll explain how to use \(C^*\)-classification type results to answer this in some cases, focusing on the case of groups and their \(C^*\)-algebras, and some associated ideas from topology.
Last update: 2025-06-06 16:02:29.181307