Schedule for: 25w5371 - NC function theory: The non-Commutative Frontier of Analysis and Algebra

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

A holomorphic function can be thought of as a generalized polynomial in various ways. A non-commutative holomorphic function is a generalized non-commutative polynomial. I will try to explain the analogy and discuss why non-commutative function theory is useful.

A quiver Q is just a directed graph. If we attach vector spaces to vertices and linear maps to the arrows then we get a representation of Q. By applying Invariant Theory to this setup one can construct moduli spaces of quiver representations, and more generally moduli spaces for finite dimensional modules over an associative algebra. There is an interesting connection between stability (in the sense of Geometric Invariant Theory) of quiver representations and ranks of matrices with entries in the free skew field. I will give an overview of Invariant Theory for quiver representations and its connection to ranks of matrices over free skew fields.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Realizations of functions have a long and storied history in analysis, algebra, and operator theory stretching back at least five decades. Briefly, a realization of a function is a representation in terms of $(I - Az)^{-1}$ for some explicit operator $A$. Many properties of the represented function are witnessed by the chosen operator $A$, e.g. if $A$ is a nilpotent matrix, then the function it represents is a polynomial. In this talk we present a theory of realizations for noncommutative analytic functions on some neighborhood of $0$ -- all these results readily apply to the single variable commutative case. We show that every "familiar" analytic function defined in a neighborhood about $0$ has a compact realization, and that every noncommutative uniformly entire function has a compact and (jointly) quasinilpotent realization. Moreover, we show that any class of operators with a few reasonable closure properties corresponds with an algebra of analytic functions that naturally embeds in a (skew) field. This project is joint work with Robert Martin and Eli Shamovich.

The study of algebras of bounded NC functions on subvarieties of NC unit balls has led us to associate a spectral radius function $\rho_E$ with every finite-dimensional operator space $E$. Concretely, if $A$ is a tuple of matrices, then $\rho_E(A)$ 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, $\rho_E$ coincides with the joint spectral radius studied by Bunce, Popescu, and others. In a recent preprint with Eli Shamovich, we introduced $\rho_E$ and proved that $\rho_E(A) < 1$ if and only if $A$ is jointly similar to a tuple that lies in the NC ball corresponding to $E$. For example, when $E$ is the row operator space, this means that $A$ is jointly similar to a strict row contraction. In this talk, based on joint work with Jeet Sampat and with Eli Shamovich, I will explain why we were led to this notion and how the proof evolved, present some examples, and describe possible applications.

Suppose a given tuple of matrices A_1,...,A_n "almost" satisfies a polynomial equation f(X_1,...,X_n), in the sense that f(A_1,...,A_n) has small rank. When can we perturb A_1,...,A_n by small-rank matrices to obtain a genuine solution of f? We explore this stability question, along with related problems, using representation-theoretic techniques and tools from asymptotic metric algebra. Connections to amenability and soficity of associative and Lie algebras are also discussed.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The free skew field or the skew field of noncommutative rational functions is the free noncommutative (nc, for short) analogue of the field of rational functions: it is the universal skew field of fractions of the ring of noncommutative polynomials, i.e., of the free algebra. It was first introduced by S. Amitsur in 1966 in his fundamental study of rational identities and then studied in great depth by P.M. Cohn as a part of a general study of localization in noncommutative rings with an emphasis on the free setting. NC rational functions are closely related to nc rational power series that originated in the work of S.C. Kleene and M.P. Schuetzenberger in the late 1950s and the early 1960s in automata theory, leading to the important technique of realizations (aka linearizations). NC functions are functions on tuples of square matrices of all sizes that are graded (respect matrix size) and respect direct sums and similarities (or equivalently, intertwinings). NC function theory was pioneered by J.L. Taylor in the early 1970s in his study of noncommutative spectral theory and was further developed by D.-V. Voiculsecu with a view towards free probability. NC rational functions play a central role in nc function theory much as their commutative sisters and brothers do in the usual commutative complex analysis in one and several variables. In this guided walk, or maybe a hike, through the free skew field, I will introduce nc rational functions from the nc function theory perspective. I will discuss difference-differential calculus and Taylor--Taylor series expansions, and the realization theory. Time permitting I will also describe the results of M. Fliess, of M. Porat and the speaker, and of G. Duchamps and C. Reutenauer characterizing nc rational power series, and the results of T. Mai, R. Speicher, and S. Yin on the absence of nc rational identities for wide classes of noncommutative random variables.

We give an overview of some of the recent developments in factorization theory in noncommutative domains.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A (formal) power series in one variable is called rational if it is the Taylor series expansion of a rational function at the origin. Noncommutative rational series are a natural noncommutative, multivariate generalization of this notion. They admit various algebraic and combinatorial characterizations. Noncommutative rational series appear in theoretical computer science as generating functions of weighted automata, and in number theory in connection with k-automatic and k-regular sequences. After introducing noncommutative rational series, I will discuss characterizations of interesting subclasses by their growth and by arithmetic properties.

The well-known classical Denjoy-Wolff Theorem states that the iterations of a nontrivial holomorphic self-map of the unit disk in the complex plane converge uniformly on compacts to a constant map that sends the disk to one point belonging to its closure. This result follows (indirectly) from the contractivity of analytic self-maps with respect to the (pseudo)hyperbolic metric. Noncommutative domains sometimes benefit from their noncommutative structure to permit the recovery of a hyperbolic-type pseudometric through algebraic means, with respect to which free noncommutative self-maps of such a domain are automatically contractions. In this talk, we will discuss some special cases of such domains and show how to prove Denjoy-Wolff type theorems, and, where needed, extensions, on them. We will outline possible consequences and applications of such results. The talk is based on joint work with Eli Shamovich and with Victor Vinnikov.

We show that any complete Nevanlinna–Pick space whose multiplier algebra has isometric Gelfand transform (or commutative C*-envelope) is essentially the Hardy space on the open unit disk. Here, C*-envelope is the ``noncommutative Shilov boundary".

For an open NC unit ball corresponding to a finite dimensional operator space $E$, we discuss the problem of classifying algebras of bounded NC functions over subvarieties of the NC unit ball. In our previous work, joint with Orr Shalit, we classified algebras of NC functions that are continuous up to the boundary of the NC unit ball, and for which the variety is homogoneous. It turns out that two such algebras are completely isometrically isomorphic to each other if and only if the subvarieties are biholomorphic to one another. Moreover, we showed that under natural assumptions on the subvariety, the biholomorphism can be chosen to be a restriction of a biholomorphism between the ambient NC unit balls. In this talk, also based on recent joint work with Orr Shailt, I will introduce the notion of a canonical weak-* topology on the algebra of bounded NC functions. Using this notion, we are able to extend our previous results to the case when the algebra consists of all bounded NC functions and the subvariety need not be homogeneous but contains a scalar point. In this case, we obtain classification up to weak-* continuous and completely isometric isomorphisms. If time permits, I will also mention a few words about the difficulties that arise when we drop the weak-* continuity assumption.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

TBA

Firstly, I shall provide a gentle brief introduction to the field of noncommutative geometry (NCG), covering C*-algebras, operator K-theory, and K-homology. I shall then explain how quadratic algebras provide a natural framework for studying quantum spaces and deformations arising from the theory of quantum groups, in line with Manin’s programme, and how those algebras can be studied using the tools of NCG. I will also discuss recent results concerning the extension of various operations on quadratic algebras to their C*-counterparts, such as free products and Veronese powers.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this survey talk, we explore the vibrant interplay between noncommutative inequalities and noncommutative real algebraic geometry through a series of fundamental theorems, conjectures, and emerging applications. We begin with an examination of Bill Helton's and Scott McCullough's Sum-of-Squares (SOS) theorem for positive noncommutative polynomials, followed by several other Positivstellensätze, which provide a robust framework for addressing positivity conditions in a noncommutative setting. Included is the Procesi-Schacher Positivstellensatz, which certifies positivity in fixed-dimension (nxn) matrices. These results serve as a gateway to applications in quantum information theory, where we illustrate how the so-called state polynomials play a critical role in certifying nonlinear Bell inequalities. Our second main topic is noncommutative convexity, with an emphasis on Linear Matrix Inequalities (LMIs). We will discuss aspects of duality, extreme points, and bianalytic maps, as well as recent developments in partial convexity. Throughout the talk, we will also highlight a number of compelling open problems that invite further exploration and discussion in this rapidly evolving field.

The main object of study of classical commutative invariant theory are subalgebras of multivariate polynomial rings consisting of polynomials fixed by some group of linear substitutions of the variables. So the elements in these algebras are polynomial functions on a vector space that are constant along the orbits of a group of linear transformations, and hence they play a role in the problem of classifying the orbits. To develop noncommutative invariant theory one has to replace the commutative polynomial algebra by an appropriate noncommutative algebra, and investigate the subalgebras of fixed elements under a group of automorphisms of the given algebra. In order to have a theory with a sufficient supply of interesting examples, we need algebras with a rich automorphism group. Natural candidates are algebras defined by universal properties, like free algebras, or universal enveloping algebras of Lie algebras. Moreover, another possibility to bring noncommutativity into the picture is to consider equivariant polynomial maps from Lie algebras into matrix algebras (starting from a finite dimensional representation of the Lie algebra). In the talk we shall give a survey on some research directions and results along the above lines.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

TBA

Matrix convex sets are dimension free generalizations of classical convex sets which extend classical convex sets to include tuples of self-adjoint $n \times n$ matrices of all sizes $n$. As it turns out, given a classical convex set $C$, the extension of $C$ to a matrix convex set is not unique. In fact there are typically infinitely many matrix convex sets which agree with $C$ when restricted to their first level. Of particular note are the minimal and maximal matrix convex sets, $\mathcal{W}^{\min}(C)$ and $\mathcal{W}^{\max}(C)$, generated by $C$. Given a convex set $C$, a major direction of research in matrix convexity is to determine the smallest constant $s$ such that $\mathcal{W}^{\max}(C) \subseteq s \cdot \mathcal{W}^{\min}(C)$. For particular choices of $C$, e.g., when $C$ is the matrix diamond, this question is closely connected to studying the joint measurability of measurements in quantum information. By exploiting connections to extreme points of matrix convex sets, we present a variety of results and conjectures related to inclusion constants for these settings of interest. \\ This talk is based on joint work with Andreas Bluhm, Igor Klep, Victor Magron, and Ion Nechita.

Swap operators act on the space $(\mathbb{C}^d)^{\otimes{n}}$ of $n$ qudits by exchanging tensor factors of $(\mathbb{C}^d)^{\otimes{n}}.$ The algebra they generate (called the $d$-swap matrix algebra) is a subalgebra of $M_{d^n}(\mathbb{C}).$ Classically, in physics literature, the case $d=2$ of qubits has received the most attention. However, in this talk we discuss the properties of the $d$-swap matrix algebra for the case of a general $d.$ This algebra is semisimple by Maschke's theorem and is in fact identified as a quotient of a free algebra modulo symmetric group relations and a single additional relation of degree $d$. As an application, we introduce and discuss the Quantum Max $d$-Cut ($d$-QMC) problem. It is a higher-dimensional analog of the QMC (Quantum Max $d$-Cut with $d=2$) that has emerged as a test-problem for designing approximation algorithms in quantum physics. For fixed $n$ and a graph $G$ on $n$ vertices, the objective function, the $d$-QMC Hamiltonian, is defined as a linear expression in the swap operators on $(\mathbb{C}^d)^{\otimes{n}}.$ Using the block decomposition of the swap operators, we compute the maximum eigenvalue of the $d$-QMC Hamiltonian for a clique. Moreover, using a suitable clique decomposition we solve the $d$-QMC problem for a larger class of graphs, including star graphs. Lastly, we address a refined $d$-QMC problem focused on finding the largest eigenvalue within each isotypic component (irreducible block) of the graph Hamiltonian. We show that the spectrum of the star graph Hamiltonian distinguishes between isotypic components of the $3$-QMC problem. We also present low-degree relations for separating isotypic components for general $d.$ This is joint work with Igor Klep and Jurij Volčič.

Gamma convexity is a general framework for studying free sets and functions that posses some partial convexity. As an example, given two classes of variables x and y, a set is convex in x if for each fixed matrix tuple y the slice at y is convex. This talk features recent results in Gamma convexity, including a Krein-Milman theorem in the style of Davidson and Kennedy and Categorical duality in the spirit of Webster and Winkler, preceded by a sampling of the general theory. It reports on joint work with Igor Klep, Mike Jury, Mark Mancuso, James Pascoe and Tea Strekelj.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Laguerre–P´olya class is a special class of entire functions that are locally the limit of sequences of real univariate hyperbolic polynomials. We present some necessary and sufficient conditions for entire functions to belong to the Laguerre– P´olya class, or to have no complex roots. These conditions involve only their Taylor coefficients and, therefore, are easy to apply. For an entire function $f(z) = \sum_{k=0}^{\infty} a_k z^k$, we define the second quotients of Taylor coefficients as $q_n(f) := \frac{a_{n-1}^2}{a_{n-2} a_n}$, $n \geq 2$, and formulate the conditions in terms of $q_n(f)$. We also discuss the operators that preserve real-rootedness. This is joint work with Anna Vishnyakova.

A correspondence over a C*-algebra $A$ is a right Hilbert $A$-module equipped with a nondegenerate left action of $A$ by adjointable operators. A correspondence may be viewed as an action of $\mathbb{N}$ by generalized endomorphisms of $A$. The analogue of a correspondence in the context of general semigroups is called a product system. In this talk I will consider Toeplitz algebras associated to product systems over group-embeddable monoids and discuss nuclearity and exactness for these algebras in relation to the coefficient algebra beyond the case of single correspondences and compactly aligned product systems over right LCM monoids. We show that nuclearity of the Toeplitz algebra is equivalent to nuclearity of the coefficient algebra for every full product system of Hilbert bimodules over abelian monoids and over Baumslag–Solitar monoids. This is joint work with E. Katsoulis and M. Laca.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.