The Israeli Noncommutative Analysis Seminar
Orr Shalit and I organized the Israeli Noncommutative Analysis Seminar for the Winter semester (2025–26). The seminar covers a wide range of topics in operator algebras, operator theory, commutative and noncommutative function theory, quantum mathematics, and related areas.
If you have any questions about the seminar or wish to be added to the seminar mailing list, please feel free to email me at saikiah@myumanitoba.ca .
Title: A Non-commutative Boundary for the Dilation Order
Abstract:
Arveson’s hyperrigidity conjecture concerns the unique extension property (UEP) of representations of C*-algebras with respect to a generating operator system. Bilich and Dor-On showed that the conjecture does not hold in general. Nevertheless, the conjecture remains interesting for several reasons. The states that are maximal in the dilation order fully encapsulate the cyclic representations of a C*-algebra possessing the UEP. The collection of all such maximal states forms a norm-closed set that remains stable under absolute continuity. In this talk, we will discuss an equivalent characterization of the dilation-maximal states in terms of a boundary projection. Subsequently, we will present a reformulation of Arveson’s hyperrigidity conjecture in terms of the non-commutative topological properties of this boundary projection. This is joint work with Raphaël Clouâtre.
Title: The Hao-Ng isomorphism theorem for reduced crossed products
Abstract: "Beyond its conceptual impact on noncommutative Choquet theory, the counterexample to Arveson's hyperrigidity we discovered with Bilich informs us on how to prove functoriality properties for the C*-envelope in non-commutative dynamical contexts, which then leads to insight on structural and classification problems through the use of the C*-envelope. For instance, for the seemingly unrelated, long-standing Hao-Ng isomorphism problem, which asks whether the Cuntz-Pimsner C*-algebra construction commutes with taking the full or reduced crossed product by a locally compact Hausdorff group action, beyond the amenable case of Hao and Ng.
Building on ideas that emerge from the discovery of the counterexample to Arveson's hyperrigidity conjecture with Bilich, together with ideas arising from a recent amendment to Arveson's hyperrigidity conjecture by Clouatre and Thompson, we resolve the reduced version of the Hao-Ng isomorphism problem in full generality. More precisely, for a non-degenerate C*-correspondence $X$ and a generalized gauge action $G \curvearrowright X$ by a locally compact Hausdorff group $G$, we prove the commutation ${\mathcal{O}}_{X\rtimes_rG}\cong {\mathcal{O}}_X\rtimes_rG$ of the reduced crossed product with the Cuntz-Pimsner C*-algebra construction.
*Based on joint work with Ian Thompson."
Title: A C*-Cover Lattice Dichotomy.
Abstract: C*-covers of operator algebras, rst studied by Arveson about 50 years ago, remain a vibrant area of research. Recently, Adam Humeniuk and Christopher Ramsey studied the lattice of C*-covers of operator algebras, focusing on its structure and on whether this lattice uniquely determines an operator algebra up to completely isometric isomorphisms.
In my talk, I will give an introduction to operator algebras and their C*-covers, and answer the question of whether there exists nontrivial operator algebras with a one point lattice. Additionally, I will characterize the possible cardinalities of the lattice of C*-covers.
This is a joint work with Adam Humeniuk and Christopher Ramsey.
Title: Non-self-adjoint dynamics and the full Hao-Ng isomorphism
Abstract: A decade ago, Katsoulis and Ramsey began a program for studying non-self-adjoint dynamical systems and their enveloping C*-algebras. Within their program, the central problem was to determine the C*-envelope of the non-self-adjoint reduced and full crossed product operator algebras. This is of paramount importance for various classification statements as, for one, this would provide answers to the 17 year-old reduced and full Hao-Ng isomorphism problems. In past work with Adam Dor-On, we gave a complete positive answer to the reduced case. In forthcoming work, we provide a corresponding answer to the much more mysterious setting of full crossed products, with a complete answer to the full Hao-Ng isomorphism problem.
Title: On L^p approximations of groups.
Abstract: Some major open problems in group theory can be described within the framework of approximation of groups: For instance, whether all groups are sofic, hyperlinear or MF. We will present the general framework for metric approximation of groups, and particularly with respect to the L^p norm, p \in [1, \infty]. Previous results prove that certain Deligne central extension of lattices in higher rank groups are not L^p approximated for p \in (1, \infty), and we extend this result to the case p=1.
Based on joint work with Alon Dogon and Alex Lubotzky.
Title: Continuous operators and the Banach Tarski paradox
Abstract: We demonstrate continuous operators on function spaces from a probability space to a simplex such that the only fixed points of these operators is a function that cannot be measurable with respect to any finitely additive measure that extends the probability measure. We believe this situation is ubiquitous and exists in many other contexts.
Title: Symbolic Dynamics and Graph Algebras.
Abstract: Symbolic dynamics provides simple yet surprisingly rich models for dynamical systems, though many fundamental classification problems remain unresolved. A central example is the conjugacy problem for subshifts of finite type (SFTs), whose complexity is reflected in deep links to combinatorics and matrix theory. Building on the seminal work of Cuntz and Krieger, one can associate graph C*-algebras and Leavitt path algebras to SFTs in a way that recovers classical invariants while also producing new (operator) algebraic obstructions to conjugacy
In this talk, I will present recent advances in understanding the relationship between symbolic dynamics and graph algebras, including a resolution of a question of Eilers showing that eventual conjugacy of SFTs coincides with graded homotopy equivalence of the associated graph C*-algebras. These results illustrate how graph algebras, in both the analytic and purely algebraic settings, not only shed light on longstanding problems in dynamics but also open new avenues at the interface of functional analysis, pure algebra, non-commutative geometry, and quantum mathematics.
Title: Constrained sphere packing and the interpolation problem for positive definite functions
Abstract: Delsarte's method provides a framework for analysis of sphere packing problems in terms of positive definite functions. Constrained packing, where only a prespecified set of angles between points may occur, is, in turn, naturally governed by the interpolation theory for positive definite functions. In this talk, we discuss the general theory of positive definite functions on the sphere, Delsarte's method and its geometric kernel embedding interpretation, related interpolation problems, and other topics.
Based on joining work with Sujit Sakharam Damase.
Title: Noncommutative meromorphic functions
Abstract: A classical result in complex analysis is that a meromorphic function on the complex plane is a ratio of two entire functions. A very simple example of a meromorphic function is a rational function. In the noncommutative setting rational functions are a more complicated object. The free skew field (the skew field of all noncommutative rational functions) was constructed by Amitsur and by Cohn. A very concrete way to look at these objects is via realizations. Realizations allow us to deal with linear objects and a single inversion for the price of dealing with (perhaps very big) matrices and vectors. In this talk, I will discuss two constructions of noncommutative meromorphic functions, one arising from realizations and one from a purely algebraic construction. The first one gives interesting results even in the single variable case whereas the second one provides a set of new examples of topological semi-free ideal rings. If time will permit, I will discuss an application to free probability. The talk is based on joint work with Meric Augat and Rob Martin.
Title: Rigidity for Graph Product von Neumann Algebras.
Abstract: I will discuss joint work with Ionut Chifan and Daniel Drimbe on various rigidity aspects of von Neumann algebras arising from graph product groups whose underlying graph is a certain cycle of cliques and whose vertex groups are wreath-like product property (T) groups. In particular, I will describe all symmetries of these von Neumann algebras by establishing formulas in the spirit of Genevois and Martin’s results on automorphisms of graph product groups. In doing so, I will highlight the methods used from Popa’s deformation/rigidity theory as well as new techniques pertaining to graph product algebras.
Title: Twisted representations of product systems of C^*-correspondences: Wold decomposition and unitary extensions
Abstract: We investigate Wold-type decompositions and unitary extension problems for multivariable isometric covariant representations associated with product systems of C*-correspondences. First, we establish an operator-theoretic characterization for the existence of a Wold decomposition for the k-tuple consisting of isometric covariant representations of a C*-correspondences. We then introduce twisted and doubly twisted covariant representations of product systems. For doubly twisted isometric representations, we prove the existence of a Wold decomposition, recovering earlier results for doubly commuting representations as special cases. We further obtain explicit descriptions of the resulting Wold summands and develop concrete Fock-type models realizing each component. We present non-trivial examples of these families. Finally, we construct unitary extensions via a direct-limit procedure. As applications, we obtain unitary extensions for several previously studied classes of operator tuples, including doubly twisted, doubly non-commuting, and doubly commuting isometries, and for a special class of doubly twisted representations of a product system.
Title: A joint spectral radius for tuples of operators
Abstract: The study of algebras of bounded noncommutative analytic functions on subvarieties of noncommutative unit balls has led us to associate a spectral radius function with every finite-dimensional operator space E. Concretely, given a d-tuple of operators, the spectral radius is defined via a certain tensor power limit formula, which reduces to Gelfand’s spectral radius formula when E is one-dimensional. When E is the row operator space, for example, then our spectral radius coincides with the joint spectral radius studied by Bunce, Popescu and others. For every operator space E, the value of the spectral radius associated to E on given a tuple is strictly smaller than 1 if and only if the tuple is jointly similar to a tuple that lies in the open noncommutative unit ball corresponding to E. In this talk, based on joint work with Eli Shamovich, I will explain why we were led to this notion, present some examples, and describe possible applications. Finally, I will discuss recent progress on the case of commuting tuples, where several alternative notions of joint spectrum coincide, and pose some open problems in the noncommutative case.