Around the Arveson-Douglas conjecture

Michael Hartz, Wednesday 10–11 a.m.

Title: Boundary behavior in the Drury-Arveson space

The Drury-Arveson space, where the Arveson-Douglas essential normality conjecture takes place, is a space of holomorphic functions on the open unit ball in ℂ^d. The behavior of functions close to the boundary is relevant for many operator theoretic and function theoretic questions.

I will discuss in which sense one can take boundary limits of functions in the Drury-Arveson space. This leads to a notion of capacity and an associated potential theory. I will also talk about the maximum principle for potentials in the Drury-Arveson space, which is relevant in the context of Carleson measures.

Lijia Ding, Wednesday 11:30 a.m.–12:30 p.m.

Title: Helton-Howe trace formulae on quotient modules

In this talk, we will discuss Helton-Howe trace formulae on Hilbert modules over Stein strongly pseudoconvex finite manifolds. For higher-dimensional smooth subvarieties that transversely intersect with the strongly pseudoconvex boundary, we establish an integral trace formula for the compression of Toeplitz operators on holomorphic Sobolev quotient modules, provided the restriction maps satisfy a q-essentially isometric condition for some q strictly less than the dimension of the subvariety. Our findings extend known results from the unit ball to the Stein strongly pseudoconvex manifolds and partially answer a problem recently proposed for the Helton-Howe trace formula on quotient modules. This talk is based on joint work with Prof. Kunyu Guo and Prof. Kai Wang.

Abhijeet Vats, Wednesday 1:30–2:30 p.m.

Title: The BDF Ext Groups, Schochet C*-algebras & Semiprojectivity

In their 1973 paper "Ext(X) from a homological point of view", (Jerome) Kaminker and (Claude) Schochet proposed an alternative approach to showing the homotopy-invariance of Ext(-) by comparison with a functor which is known to be homotopy-invariant. They prove that Ext(-) coincides with this functor on all compact metrizable spaces which satisfy the C*-Covering Homotopy Property (C*-CHP). Their proof prompted them to ask if every compact metrizable space satisfies the C*-CHP. In 1984, within Topological Methods for C*-algebras III, Schochet introduced a new class of morphisms, called cofibrations, which are important in studying the homotopical structure of the category of C*-algebras. He conjectured that, in fact, the Calkin projection on an infinite-dimensional, separable Hilbert space is not a cofibration.

Our goal is to address the relationship between the question and the conjecture, explaining that both of them speak about the same underlying notions. We will see that the conjecture has a positive answer and that, therefore, the question of Kaminker and Schochet has a negative answer. We will also introduce the notion of a Schochet C*-algebra and discuss some interesting permanence properties of this class of C*-algebras. Finally, we will relate our work to the notion of semiprojectivity and, in particular, explain a possible obstruction to Blackadar's Semiprojectivity Conjecture, as well as discuss some ongoing work in this direction. This talk is based on work done in the master's thesis of the speaker.

Matthew Kennedy, Wednesday 2:30–3:30 p.m.

Title: Essential Normality and Hyperrigidity

Hyperrigidity is an important approximation-theoretic notion for generating sets of C*-algebras that was introduced by Arveson. I will give a brief introduction to hyperrigidity, and I will discuss my result with Orr Shalit implying that the Arveson-Douglas conjecture for a polynomial ideal is equivalent to the hyperrigidity of a certain corresponding operator system. I will then discuss new joint work with James Pascoe and Eli Shamovich that provides a new characterization of hyperrigidity in terms of a noncommutative Dirichlet-type problem, along with some applications.

Jingbo Xia, Wednesday 4–5 p.m.

Title: Geometric Arveson-Douglas conjecture for the Drury-Arveson space: the case of one-dimensional variety

We consider a class of analytic subsets M̃ of an open neighborhood of the closed unit ball in ℂⁿ. Such an M̃ gives rise to a submodule R and a quotient module Q of the Drury-Arveson module H_n² in n variables. The geometric Arveson-Douglas conjecture predicts that the quotient module Q is p-essentially normal for p > d = dim_ℂ M̃. We prove this conjecture for the case of dimension d = 1. In fact, we prove that if d = 1, then Q is 1-essentially normal, which is a stronger result than the original prediction.

Ian Thompson, Wednesday 5–6 p.m.

Title: From Arveson to Arveson, and back to Arveson

By work of Kennedy and Shalit, Arveson's essential normality conjecture can be reformulated in terms of a rigidity property for a particular operator subsystem of a Toeplitz algebra. This rigidity property, known as hyperrigidity, was conjectured by Arveson in a separate work to always be able to be captured at the level of the noncommutative Choquet boundary. Arveson's hyperrigidity conjecture is now known to fail due to a construction of Bilich and Dor-On and, since then, several variations on this construction have permeated the literature. In response to this, within joint work with Clouâtre, we established a positive answer to a variation on Arveson's hyperrigidity conjecture. The ideas found within had subsequently inspired joint work with Dor-On, in which we determine the boundary of a C*-correspondence dynamical system. In addition, the work of Clouâtre also evokes several natural questions that surround the Arveson-Douglas conjecture, and return us back to the original work of Kennedy and Shalit.

Chris Bruce, Thursday 9–10 a.m.

Title: Toeplitz C*-algebras and boundary quotients

I will give an overview of Toeplitz C*-algebras arising from left cancellative monoids and, more generally, left cancellative small categories. Such C*-algebras possess notions of a “boundary quotient”, and the classification problem for these boundary quotients within a given class of such C*-algebras is often important and difficult. I will survey some results in this setting, including a theorem with Kubota and Takeishi in which we complete the classification problem for the boundary quotients of semigroup C*-algebras arising from integral dynamics. The Arveson–Douglas conjecture is analogous to such classification problems, and I aim to make this analogy clear during the talk.

Yi Wang, Thursday 10–11 a.m.

Title: Operator model via compact embedding

We extend several function-theoretic and geometric constructions to the realm of multi-variable operator theory. Our subject of study is a commuting tuple of bounded linear operators on a Hilbert space, equipped with a cyclic vector. We encode the complete information of such a tuple into a single positive compact operator on the Fock space.

Drawing parallels with spectral geometry, we investigate how the spectral data of this positive compact operator—its eigenvalues and eigenfunctions—reflect fundamental properties of the operator tuple.

Our main results are threefold. First, we establish a Weyl-type approximation formula for certain operator tuples, demonstrating that the asymptotic behavior of eigenvalues carries geometric information about the vanishing variety. Second, we construct two kernel functions derived from the eigenvalues and eigenfunctions: the first extends the Bergman kernel, while the second extends the Fourier-Laplace transformation. We prove that the Fourier-Laplace kernel defines a reproducing kernel Hilbert space (RKHS) on which the coordinate differential operators are unitarily equivalent to the adjoint tuple. Consequently, we show that the convergence points of the Bergman-type kernel characterize the joint eigenvalues of the adjoint tuple. Finally, we obtain a Paley-Wiener-Schwartz type theorem for cyclic commuting tuples, characterizing cyclic commuting tuples whose associated Agler's linear functional is a distribution. For tuples consisting of matrices, we obtain a more explicit characterization. In the end, we explain some potential application to the Arveson-Douglas conjecture.

Camila Sehnem, Thursday 11:30 a.m.–12:30 p.m.

Title: Injective envelopes of partial C*-dynamical systems

In a series of papers starting in the late 70s Hamana introduced injective envelopes of objects in various categories of operator systems and C*-algebras. Notably, Hamana established the existence of injective envelopes of C*-dynamical systems, and these have been a crucial tool in the analysis of simplicity of reduced group C*-algebras and, more generally, of the ideal structure of reduced crossed products. In this talk I will discuss joint work with M. Kennedy and L. Kroell in which we extend Hamana’s theory of injective envelopes to the realm of partial C*-dynamical systems.

Kai Wang, Thursday 1:30–2:30 p.m.

Title: Essential Normality of K-Invariant Quotient Modules on Bounded Symmetric Domains

In this talk, I will survey recent progress on the problem of essential normality for K-invariant quotient modules on bounded symmetric domains.

Dimitris Gerontogiannis, Thursday 2:30–3:30 p.m.

Title: Veronese powers of subproduct systems

Subproduct systems of C*-correspondences were introduced by Shalit–Solel as a natural framework for studying CP-semigroups, around the same time as the closely related work of Bhat–Mukherjee. In recent years, subproduct systems have also appeared in the representation theory of quantum groups, dynamical systems, and as a source of new examples of C*-algebras. One way to understand and expand these examples, in the spirit of noncommutative geometry, is to introduce natural operations on subproduct systems. In this talk, I will discuss one such operation: taking Veronese powers of a subproduct system, a construction coming from ring theory. Applications include an example of a Toeplitz algebra that is not KK-equivalent to the complex numbers, and new examples of singular affine homogeneous varieties for which the Arveson–Douglas conjecture holds. This is joint work with Francesca Arici and Yufan Ge.

Xiang Tang, Thursday 4–5 p.m.

Title: A Scheme Approach to the Arveson-Douglas Conjecture

The Arveson-Douglas conjecture was introduced by Arveson and Douglas in the early 2000s. This is a fundamental and beautiful problem connected to many different branches of Mathematics. In this talk, we will discuss a proposal to study the Arveson-Douglas conjecture through the analytic scheme associated to an ideal.

Ada Masters, Friday 9–10 a.m.

Title: Unbounded Kasparov modules for fissured Fell bundles over locally compact groups

I will discuss a framework for the building of spectral triples for locally compact group C*-algebras, encompassing a number of constructions in the literature. This framework, most generally, involves the building of unbounded Kasparov modules for ‘fissured’ Fell bundles. In many cases, one can use ‘directed length functions’, giving rise to nontrivial index theory. Furthermore, given an extension G of a l.c. group H by a l.c. group N, it is natural to ask whether a spectral triple for C^*(G) can be built from corresponding data on H and N. As it turns out, this is a microcosm of the constructive unbounded Kasparov product, which can very often be achieved using functional calculus. As a special case, we obtain a constructive expression for the dual Dirac element (and so γ-element) for any almost connected group.

This includes joint work with Adam Rennie and Anne Thomas.

Eli Shamovich, Friday 10–11 a.m.

Title: Determinantal representations with connections to the Arveson-Douglas conjecture

Determinantal representations and LMI domains are important objects of study in real algebraic geometry. In particular, Beauville described necessary and sufficient conditions on a projective hypersurface to admit a determinant representation. In general, determinantal representations follow the general strategy of replacing a complicated polynomial object with linear pencils of matrices. In this talk, I am going to describe a class of determinantal representations of projective varieties of pure dimension k in the d-dimensional complex projective space. I will explain how these give rise to a perhaps particularly amenable class of varieties for the Arveson-Douglas conjecture.

This talk is based on joint works with Vinnikov, Kummer, and Kennedy and Pascoe.

Harald Upmeier, Friday 11:30 a.m.–12:30 p.m.

Title: Essentially normal Hilbert modules and K-theory over bounded symmetric domains

Essentially normal Hilbert modules and index theory for bounded symmetric domains

Bounded symmetric domains, generalizing the unit ball, play a central role in complex and harmonic analysis. For higher rank these domains are not strongly pseudo-convex but there exist interesting (singular) subvarieties with strongly pseudo-convex boundary. We show that these varieties give rise to essentially normal Hilbert quotient modules and prove index formulas related to K-theory. We also generalize to families of Fredholm operators of Toeplitz type.