Abstracts

A remarkable pair of theorems of Grothendieck say if $p : \mathbb{C}^g \to \mathbb{C}^g$ is an injective polynomial, then $p$ is bijective and its inverse is a polynomial. We prove a free analog of this.

Recall that a free polynomial mapping in $g$ freely non-commuting variables sends $g$-tuples of matrices (of the same size) to $g$-tuples of matrices (of the same size).

Our result is as follows; if $p$ is a free polynomial mapping that is injective, then it has a free polynomial inverse. We will make use of a free version of the Jacobian Conjecture as well as results from free analysis, formal power series and skew fields.

If there is enough time we will discuss the generalization of the theorem to free rational mappings.

The Free Grothendieck Theorem is related to free analysis, automorphisms of the free algebra and tame vs. wild automorphism of the free algebra.

For every associative algebra $A$ and every class $C$ of representations of $A$ the following question (related to nullstellensatz) makes sense:

Characterize all tuples of elements $a_1,\ldots,a_n \in A$ such that vectors $\pi(a_1)v,\ldots,\pi(a_n)v$ are linearly dependent for every $\pi \in C$ and every $v$ from the representation space of $\pi$.

We answer this question for Weyl algebras and enveloping algebras of some Lie algebras. For free algebras the answer was given by Brešar and Klep in 2013.

A classical theorem of Loewner gives that a function is matrix monotone if and only if the function is real analytic and continues to an analytic self-map of the complex upper half plane. We will discuss work appearing over the last five years that extends Loewner's result and related techniques in the noncommutative setting, culminating in a recent paper of J. E. Pascoe on operator systems.  

The techniques used in the proof of Loewner’s theorem point to way to a more general approach to investigating the relationship between functional behavior on real objects (e.g. monotonicity, convexity) and analytic continuation. We will discuss work in progress on automatic analyticity and a heuristic approach to results qualitatively akin to Loewner’s theorem.

The Fejér-Riesz theorem states that a nonnegative trigonometric polynomial in one variable is the (hermitian) square of an analytic polynomial of the same degree.  This result was extended to operator valued polynomials by Rosenblum, and the speaker showed that Schur complement techniques can be used to factor (strictly) positive trigonometric operator polynomials in any finite number of variables as a sum of squares of polynomials of possibly large degrees, though this method fails for polynomials which are simply nonnegative.  Results of Scheiderer from real algebra imply that nonnegative scalar valued trigonometric polynomials can also be factored in this way.  Here we discuss an analytic approach to factoring nonnegative operator valued trigonometric polynomials in two variables.

We pose and treat a noncommutative version of the classical Waring problem for polynomials. That is, for a homogeneous noncommutative polynomial $p$, we give a condition equivalent to $p$ being expressible as sums of powers of homogeneous noncommutative polynomials.

We show that if a noncommutative polynomial $p$ has a Waring decomposition, then its coefficients must satisfy a compatibility condition. If this condition is satisfied, then we prove that $p$ has a noncommutative Waring decomposition if and only if the restriction of $p$ to commuting variables has a classical Waring decomposition.

An application of noncommutative Waring decompositions and more generally tensor decompositions is they can be used to efficiently evaluate noncommutative polynomials on tuples of matrices.

This talk is based on joint work with J. William Helton, Shiyuan Huang, and Jiawang Nie.

The talk will describe some progress over the last year on noncommunicative sets and functions. There are several lines of work to choose from. While several will be mentioned, the focus will likely be on the characterization of the composition $r$ of a convex with an analytic noncommutative rational function, $r(z)= f(q(z))$. These are free analogues of plurisubharmonic functions.

The work is joint with Meric Augat, Eric Evert, Igor Klep, Scott McCullough, Jurij Volcic.

Every nc function $f(X)$ of nonnegative real part in the row ball can be represented as an nc Herglotz integral of a positive functional $\mu$ (which we will call an nc measure) on the Cuntz-Toeplitz operator system. The vacuum state $m$ is the natural analog, in this context, of Lebesgue measure on the circle. We introduce a notion of Lebesgue decomposition of $\mu$ with respect to $m$, thus defining absolutely continuous and singular parts of $\mu$, and describe the relationship of this decomposition to nc Cauchy transforms. We then show how absolutely continuous part can be recovered from the nc function $f(X)$, giving a partial nc analog of Fatou's theorem on radial boundary values — here the notion of almost everywhere convergence is replaced by the notion of strong resolvent convergence from the theory of unbounded operators. Our main tool is the theory of (unbounded) quadratic forms, and the construction is ultimately inspired by von Neumann's $L^2$ proof of the Radon-Nikodym theorem.

This is joint work with Robert T. W. Martin.

Let $\mathcal{S}$ be a set of operators. We say $\mathcal{S}$ satisfies the column-row property if there exists a constant $C>0$ such that for any  sequence from $\mathcal{S}$ (finite or infinite,) $\|\sum S_iS_i^*\| \leq C\| \sum S_i^*S_i\|.$ If such a $C$ can be chosen to equal $1$, we say $\mathcal{S}$ satisfies the true column-row property. The column-row property when $\mathcal{S}$ is taken to be a space of multipliers is important in the theory of interpolating sequences. As far as the speaker knows, there is no known commutative complete Nevanlinna-Pick space for which the multipliers do not satisfy the (true) column-row property. We showed, in joint work with Augat and Jury, that the column-row property fails for the Fock space in two or more variables. Creating further discord, under a suitable model, a randomly chosen infinite sequence of multipliers satisfies $\|\sum S_iS_i^*\| \leq \| \sum S_i^*S_i\|$ for any space such that the monomials satisfy $\|z^\alpha\|\|z^\beta\| \geq \|z^{\alpha+\beta}\|$ in the Hilbert space norm. A deep question is whether or not the Drury-Arveson space satisfies the true column-row property. Our results suggest that naive random search may be unlikely to produce a counter-example, even if one exists.​

If $T$ is a $d$-tuple of operators acting on a Hilbert space $H$, then the matrix range $W(T)$ is a closed and bounded matrix convex set containing all images of $T$ under UCP maps into matrix algebras. We consider the problem of determining when $W(T)$ uniquely determines $T$, under the assumption that $T$ is minimal in an appropriate sense. In particular, since we seek to determine $T$ up to unitary equivalence even if $H$ is infinite-dimensional, what is the appropriate (strict) sense of minimality to use? For certain restricted classes of operator tuples, we prove that the following condition is sufficient: “the compression of $T$ to any proper closed subspace has a strictly smaller matrix range”. We require this condition even for subspaces which are not reducing, and this distinction is crucial even in the case of compact operators. Joint work with Orr Shalit.

Let $V$ be a noncommutative algebraic subvariety of the nc unit ball. A couple of years ago, Salomon, Shamovich and I showed that the algebra $H^\infty(V)$ of all bounded nc analytic functions on $V$ is determined up to completely isometric isomorphism by "the geometry of $V$". In other words, $H^\infty(V)$ is completely isometrically isomorphic to $H^\infty(W)$ if and only if there is an automorphism of the nc unit ball that maps $V$ onto $W$. In our latest paper, we tackled the problem of when two such algebras are boundedly isomorphic. Deep theorems of Ball-Marx-Vinnikov and Agler-McCarthy imply that the nc variety $V$ cannot be the complete invariant for classification up to bounded isomorphism. It turns out that the correct geometric invariant to consider is the similarity envelope of the variety $V$, endowed with a certain metric. Working with the similarity envelopes of varieties we ran into new problems in nc function theory and multivariable operator theory. In certain cases we could overcome these problems, and our central result is that when $V$ and $W$ are homogeneous, $H^\infty(V)$ is boundedly isomorphic to $H^\infty(W)$ if and only if the similarity envelopes of $V$ and $W$ are related by an invertible bi-Lipschitz linear map. In my talk I will explain our results with an emphasis on the challenges that the similarity envelope poses. 

Joint work with Eli Shamovich and Guy Salomon.

Free nc function theory is an extension of the theory of holomorphic functions of several complex variables to the theory of functions on matrix tuples $Z=(Z_1,\cdots,Z_d)$ where $Z_i\in M_n(\mathbb{C})$ and $n$ is allowed to vary.

One can view such a tuple as a representation of $\mathbb{C}^d$ (viewed as a bimodule over  $\mathbb{C}$). Here, a representation of a bimodule $X$ over an algebra $A$ is a pair $(\sigma,T)$ where $\sigma$ is a representation of $A$ and $T$ is a bimodule map $T(axb)=\sigma(a)T(x)\sigma(b)$.

An nc function is a function defined on such tuples $Z$ and takes values in $\cup_n M_n(\mathbb{C})$ which is graded and respects direct sums and similarity (equivalently, respects intertwiners).

In previous works we studied functions that are defined on the space of representations of a bimodule $E$ (more precisely, a correspondence) over a $W^*$-algebra $M$ (instead of the algebra $\mathbb{C}$), are graded and respect direct sums and similarities. We referred to these functions as matricial functions.

Note that, while the representations of $\mathbb{C}$ are parameterized by $\mathbb{N}\cup \{\infty\}$, those of a general $W^*$-algebra form a more complicated category.

The classical correspondence between positive kernels and Hilbert spaces of functions has been recently extended by Ball, Marx and Vinnikov to nc completely positive kernels and Hilbert spaces of nc functions.

In this talk I will discuss a similar correspondence in our matricial context. In place of a Hilbert space of functions we will get a $W^*$-correspondence whose elements are matricial functions.

I will also discuss what we get for an important class of kernels.

This is a joint work with Paul Muhly.

This talk addresses the local theory of noncommutative functions, which branches in two directions. First we will see that the ring of noncommutative functions analytic about a scalar point admit a universal skew field of fractions, whose elements are called meromorphic germs. On the other hand, if $Y$ is a semisimple (non-scalar) point, then there exist nilpotent analytic noncommutative functions about $Y$. Nevertheless, the ring of germs about $Y$ is described as the completion of the free algebra with respect to the vanishing ideal at $Y$. This is a consequence of our second main result, a free Hermite interpolation theorem: if $f$ is a noncommutative function, then for any finite set of semisimple points and a natural number $L$ there exists a noncommutative polynomial that agrees with $f$ at the chosen points up to differentials of order $L$.

This is joint work with Igor Klep and Victor Vinnikov.

Kippenhahn's Theorem asserts that the numerical range of a matrix is the convex hull of an algebraic curve. A generalization of the numerical range is the joint numerical range (JNR) of finitely many hermitian matrices. Chien and Nakazato (Linear Algebra Appl 432, 173–179, 2010) have shown that the analogous assertion, the JNR is the convex hull of an affine variety, fails for three hermitian matrices.

Here, we show that the JNR is the closed convex hull of a semi-algebraic set. First, we discuss a known statement regarding the dual convex cone to a hyperbolicity cone (Sinn, Mathematical Sciences 2:3, 2015, doi:10.1186/s40687-015-0022-0). Secondly, we prove that the class of convex bases of these dual cones is closed under linear operations. 

We discuss applications in quantum mechanics, namely singularities of Wigner distributions (Schwonnek and Werner, arXiv:1802.08343 [quant-ph], 2018) and local Hamiltonians. A research opportunity is finding the non-commutative counterpart to the toric variety, which describes implicitly the log-linear model of commutative local Hamiltonians in statistics (Geiger et al., Ann Stat 34, 1463–1492, 2006).

This is joint work with Daniel Plaumann (TU Dortmund) and Rainer Sinn (FU Berlin).

The tracial truncated moment problem asks to characterize when a finite sequence of real numbers indexes by words in non-commuting variables can be represented with tracial moments of matrices. In the talk we will study the bivariate quartic case, i.e., indices run over words in two variables of degree at most four. By the result of Burgdorf and Klep every such sequence with a corresponding moment matrix of size 7 which is positive definite has a representing measure. We will present what can be said if the moment matrix is singular. First observation is that it must be of rank 4 and then we will look at each of ranks 4, 5 and 6 separately. Ranks 4 and 5 can be completely solved, i.e., the existence and the uniqueness questions of the measures can be answered. In rank 6 the existence is equivalent to the feasibility problem of certain linear matrix inequalities. Flat extensions of moment matrix, which is the most powerful tool for solving the classical problem, are mostly not a necessary condition for the existence of a measure in the tracial case.