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\(C^*\)-correspondences over Cuntz-Pimsner Algebras: a construction
We will construct a functor that maps \(C^*\)-correspondences to their Cuntz-Pimsner algebras, and allows one to define a \(C^*\)-correspondence between Cuntz-Pimsner algebras. We will use this functor to recover and generalize some well-known results in just few steps. If the time permits, we shall discuss other possible applications of this functor.
Nuclear Dimension of Graph \(C^*\)-Algebras
Nuclear dimension is a useful tool in the classification of \(C^*\)-Algebras, first described by Winter and Zacharias. Involving finite dimensional approximations and “colorings” of cpc maps of order zero, nuclear dimension is a noncommutative analogue of covering dimension from topology. Nuclear dimension is well understood in the case of simple \(C^*\)-algebras. Graph algebras make for an approachable class of non-simple \(C^*\)-algebras. In this talk, we give some results about the nuclear dimension of \(C^*(E)\), when \(E\) is a directed graph with Condition (K). Time-permitting, we will also cover an initial result for \(\mathcal{Z}\)-stability of graph algebras, a condition that is closely related to nuclear dimension. This is joint work with Christopher Schafhauser.
The Novikov conjecture, operator K theory, and diffeomorphism groups
In this talk, I will discuss some recent work on a version of the Novikov conjecture for certain subgroups of diffeomorphism groups. This talk will be about joint work with Jianchao Wu, Zhizhang Xie, and Guoliang Yu.
An optimal approximation problem for noncommutative polynomials
Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial \(f\) in \(d\) noncommuting arguments (an nc polynomial), find an nc polynomial \(p_n\), of degree at most \(n\), to minimize \(c_n := \|p_n f − 1\|^2\). (Here the norm is the \(\ell^2\) norm on coefficients.) We show that \(c_n\to 0\) if and only if \(f\) is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the \(d\)-shift.
Zappa-Szep product of groupoids and beyond
Zappa-Szep product is a way to combine two groups together using a two-way interaction, which generalizes the one-way action in a semi-direct product. Our goal is to extend the Zappa-Szep product construction to operator algebras and generalize results in crossed product of operator algebras to this new context. In this talk, I will first present a Zappa-Szep product of a Fell bundle with a groupoid and show a generalized imprimitivity theorem arising from such interactions with its applications. I will also present some recent progress on the Zappa-Szep product of twisted groupoids arising more intrinsically from Cartan subalgebras. This is a joint work with Anna Duwenig.
Hypergeometric Multiple orthogonal polynomials and free finite convolution
Some multiple orthogonal polynomials can be wrote explicitly as terminating generalized hypergeometric functions. However, extracting the information about their zeros from this fact is not trivial. In this talk, we address some recently discovered applications of the notion of the free finite convolution of polynomials (developed in the framework of the free probability theory) to the study of properties of zeros of hypergeometric polynomials. In particular, we discuss some consequences for multiple orthogonal polynomials.
Von Neumann dimension for faithful normal strictly semifinite weights
The notion of von Neumann dimension for a tracial von Neumann algebra \((M,\tau)\) has been used extensively throughout the theory, particularly in defining numerical invariants from Jones’ index of a subfactor to Connes and Shlyakhtenko’s \(\ell^2\)-Betti numbers of von Neumann algebras. The latter relies on work of Lück showing that Murray and von Neumann’s original definition could be extended to purely algebraic \(M\)-modules, and more recently Petersen further extended von Neumann dimension to pairs \((M,\tau)\) where \(\tau\) is a faithful normal semifinite tracial weight. In this talk, I will introduce a yet further extension of this theory to pairs \((M,\varphi)\) where \(\varphi\) is a faithful normal strictly semifinite weight. Here strict semifiniteness means the restriction of \(\varphi\) to the centralizer subalgebra \(M^\varphi \subset M\) is still semifinite (note this condition is automatic for faithful normal states), and by work of Takesaki this is equivalent to the existence of a \(\varphi\)-invariant faithful normal conditional expectation \(\mathcal{E}_\varphi\colon M \to M^\varphi\). Consequently, one can consider the Jones basic construction for the inclusion \(M^\varphi \subset M\), and this is the key ingredient in our definition of von Neumann dimension for the pair \((M,\varphi)\). I will discuss properties of this dimension including its (in)dependence from the choice of weight. This is based on joint work with Aldo Garcia Guinto and Matthew Lorentz.
The Rank Problem for \(C^*\)-algebras
For each normalized trace on a unital \(C^*\)-algebra we can define an associated real-valued rank for a positive operator in the algebra. Computing this rank at each trace gives a lower semicontinuous and affine map from traces into the reals, we can ask a simple question: which such functions occur? We will explain how this question is connected deeply to the theory of classification for separable nuclear \(C^*\)-algebras, and how we have, since 2016, been stuck. We’ll then discuss a road map for getting unstuck, and how this relates to the homotopy groups of positive definite matrices of constrained rank.
Spectral truncations of groups with polynomial growth
Spectral truncation is a new scope in noncommutative geometry, which aims to approximate the noncommutative geometric date from partial spectral date. In quantum physics, spectrum of Hamiltonian are possible energy levels with corresponding state vectors and physical quantities are operators acting on these state vectors. But realistically, we can not see all these state vectors through experiments if we have infinitely many possible energy levels. We have an access only to state vectors whose energy is in some fixed finite interval and the restriction of physical quantities (operators) to these state vectors. We discuss how these restricted date approximate the entire spectral triple for spectral triples coming from discrete groups.
Last update: 2024-02-29 13:10:45