Non-commutative inequalitiesCopyright (c) 2021 Washington University in St. Louis All rights reserved.
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities
Recent Events in Non-commutative inequalitiesen-usThu, 15 Apr 2021 09:32:17 PDT3600TBA
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/14
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/14Thu, 21 Jul 2016 15:00:00 PDT
TBA
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Christoph HanselkaSchur-type decompositions in II-1 factors
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/13
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/13Fri, 22 Jul 2016 17:30:00 PDT
Brown measure is a sort of spectral distribution for arbitrary operators (including non-selfadjoint ones) in II_1-factors. Haagerup and Schultz proved existence of hyperinvariant projections for operators in II_1-factors, that split the operator according to the Brown measure. With Sukochev and Zanin, we used these to prove a sort of upper trianguler decomposition result for such elements, analogous to Schur's famous result for matrices. More recently, we have partially extended these results to certain unbounded operators affiliated with II_1-factors. One application is to show that every trace is spectral (i.e., the value of the trace depends only on the Brown measure of the operator) for traces on certain bimodules of affiliated operators (these are often called Dixmier traces). Time permitting, we will mention some results about the nature of the spectrally trivial parts in these upper triangular forms.
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Ken DykemaTBA
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/12
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/12Fri, 22 Jul 2016 17:00:00 PDT
TBA
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Ryan Tully-DoylePower series expansions and realization theory for noncommutative rational functions around a matrix point
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/11
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/11Fri, 22 Jul 2016 16:00:00 PDT
A noncommutative rational function which is regular at 0 can be expanded into a noncommutative formal power series. In fact, by the results of Kleene, Sch\" utzenberger, and Fliess, a noncommutative formal power series is rational (i.e., belongs to the smallest subring containing the noncommutative polynomials and closed under the inversion of invertible power series) if and only if the corresponding Hankel matrix has finite rank, and the image of the Hankel matrix can be used to construct the unique minimal (equivalently, controllable and observable) state space realization of a noncommutative rational function which is regular at 0. We use the Taylor--Taylor expansion around an arbitrary matrix point coming from noncommutative function theory to generalize these results to noncommutative rational function regular at an arbitrary matrix point, covering thereby all noncommutative rational functions.
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Victor VinnikovCircular Free Spectrahedra
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/10
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/10Fri, 22 Jul 2016 15:30:00 PDT
This talk considers matrix convex sets invariant under several types of rotations. It is known that matrix convex sets that are free semialgebraic are solution sets of Linear Matrix Inequalities (LMIs); they are called free spectrahedra. The talk will classify all free spectrahedra that are circular, that is, closed under multiplication by $e^{i t}$: up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal.\looseness=-1 This talk also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix. Up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials. This talk is based on joint work with Bill Helton, Igor Klep, and Scott McCullough.
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Eric EvertA noncommutative Bishop-de Leeuw theorem
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/9
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/9Fri, 22 Jul 2016 15:00:00 PDT
The Bishop-de Leeuw theorem asserts the equivalence of various sort of peaking phenomena for function spaces in $C(X)$. We discuss a noncommutative version of this theorem for an operator system $S$ in $B(H)$ in terms of either the representations of $C*(S)$ or of $C^*_e(S)$. Under certain conditions on $S$, $C^*(S)$, or $C^*_e(S)$, we exhibit connections between Choquet points and noncommutative peak points.
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Craig KleskiMonotonicity in several non-commuting variables
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/8
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/8Fri, 22 Jul 2016 14:30:00 PDT
Let $f : (a,b) → \mathbb{R}.$ The function f is said to be matrix monotone if $A \leq B$ implies $f(A) \leq f(B)$ for all pairs of like- sized self-adjoint matrices with spectrum in $(a,b)$. Classically, Charles Loewner showed that a bounded Borel function is matrix monotone if and only if it is analytic and extends to be a self-map of the upper half plane. The theory of matrix montonicity has profound consequences for any general theory of matrix inequalities. For example, it might seem surprising that $X \leq Y$ does not imply that $X^2 \leq Y^2$, which is a consequence of Loewner’s theorem. We will discuss commutative and noncommutative generalizations to several variables of Loewner’s theorem
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James PascoeBeurling-Lax representation for weighted Bergman-shift
invariant subspaces
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/7
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/7Thu, 21 Jul 2016 18:00:00 PDT
The Beurling-Lax-Halmos theorem tells us that any invariant subspace ${\mathcal M}$ for the shift operator $S \colon f(z) \mapsto z f(z)$ on the vectorial Hardy space over the unit disk $H^{2}_{\mathcal Y} = \{f(z) = \sum_{j=0}^{\infty} f_{j} z^{j} \colon \| f \|^{2} =\sum_{j \ge 0} \| f_{j} \|^{2} < \infty\}$ (the Reproducing Kernel Hilbert Space with reproducing kernel $K(z,w) = (1 - z \overline{w})^{-1} I_{\mathcal Y}$) can be represented as ${\mathcal M} = M_{\Theta} H^{2}_{\mathcal U}$ where $M_{\Theta} \colon H^{2}_{\mathcal U} \to H^{2}_{\mathcal Y}$ is an isometric multiplication operator $M_{\Theta} \colon u(z) \mapsto \Theta(z) u(z)$. We focus on three constructions of $\Theta(z)$: \smallskip \noindent (1) \textbf{the wandering subspace construction of Halmos}: ${\mathcal M} = \bigoplus_{j \ge 0} S^{j} ({\mathcal M} \ominus S {\mathcal M}) = M_{\Theta} H^{2}_{\mathcal U}$ where $M_{\Theta}$ is constructed so that $M_{\Theta} \colon \cU \to {\mathcal M} \ominus S {\mathcal M}$ is unitary; \smallskip \noindent (2) as the \textbf{Sz.-Nagy--Foias characteristic function} $\Theta_{T}(z)$ of the $C_{\cdot 0}$ contraction operator $T = P_{{\mathcal M}^{\perp}} S|_{{\mathcal M}^{\perp}}$, and \smallskip \noindent (3) via the \textbf{functional-model realization formula} $\Theta(z) = D + z C (I - z A)^{-1} B$ where $\left[ \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \right] = \left[ \begin{smallmatrix} S^{*} & S^{*} M_{\Theta} \\ \mathbf{ev}_{0} & \Theta(0) \end{smallmatrix} \right] \colon \left[ \begin{smallmatrix} {\mathcal M}^{\perp} \\ {\mathcal U} \end{smallmatrix} \right] \to \left[ \begin{smallmatrix} {\mathcal M}^{\perp} \\ {\mathcal Y} \end{smallmatrix} \right]$ with $\mathbf{ev}_{0} \colon f \mapsto f(0)$ is unitary. \smallskip \noindent We discuss analogues of these results for the weighted Bergman-shift operator $S_{n} \colon f(z) \mapsto z f(z)$ acting on the weighted Bergman space ${\mathcal A}_{n, {\mathcal Y}} = \{f(z) = \sum_{j \ge 0} f_{j} z^{j} :$ $\|f\|^{2} = \sum_{j \ge 0} \mu_{n,j} \| f_{j} \|^{2}_{\mathcal Y} < \infty\}$ (where $\mu_{n,j} = 1/\left( \begin{smallmatrix} j+n-1 \\ j \end{smallmatrix} \right)$ are reciprocal binomial coefficients), or ${\mathcal A}_{n, {\mathcal Y}}$ is the Reproducing Kernel Hilbert Space with reproducing kernel equal to $K_{n,{\mathcal Y}}(z,w) = (1 - z \overline{w})^{-n} I_{\mathcal Y}$, as well as for the freely noncommutative weighted Bergman shift-tuple ${\mathbf S}_{n} = (S_{n,1}, \dots, S_{n,d})$ where $S_{n,j} \colon f(z) \mapsto f(z) z_{j}$ ($j = 1, \dots, d$) on the weighted Bergman-Fock space ${\mathcal A}_{n,{\mathcal Y}}({\mathbb F}_{d})$ consisting of formal power series $f(z) = \sum_{{\mathfrak a} \in {\mathbb F}_{d}} f_{\mathfrak a} z^{\mathfrak a}$ for which $\| f \|^{2} = \sum_{{\mathfrak a} \in {\mathbb F}_{d}} \mu_{n,|{\mathfrak a}|} \| f_{\mathfrak a} \|^{2} < \infty$ (${\mathbb F}_{d}$ is the free semigroup (also called the monoid) on $d$ generators $1, \dots, d$, $z = (z_1, \dots, z_d)$ is a $d$-tuple of freely noncommuting indeterminates, $z^{\mathfrak a}$ is the noncommutative monomial $z^{\mathfrak a} = z_{i_{1}} \cdots z_{i_{N}}$ if ${\mathfrak a} = i_{1} \cdots i_{N}$, $|{\mathfrak a}|$ is the \textbf{length} of ${\mathfrak a}$ (equal to $N$ if ${\mathfrak a} = i_{1} \cdots i_{N}$). This talk reports on joint work with Vladimir Bolonikov of the College of William and Mary.
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Joseph BallNon-Commutative Functions on the Non-Commutative Ball
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/6
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/6Thu, 21 Jul 2016 17:30:00 PDT
In this talk we will discuss nc-functions on the unit nc-ball \mathfrak{B}_d. The focus of the talk will be the algebra H^{\infty}(\mathfrak{B}_d) of multipliers of the nc-RKHS on the unit ball obtained from the non-commutative Szego kernel. We will give a new proof for the fact that the non-commutative Szego kernel is completely Pick. Then we will consider subvarieties of \mathfrak{B}_d and quotients of H^{\infty}(\mathfrak{B}_d) arising as multipliers on those varieties. We are interested in determining when the multiplier algebras of two varieties are completely isometrically isomorphic. It is natural to conjecture that two such algebras are completely isometrically isomorphic if and only if there is an automorphism of the nc ball that maps one variety onto the other. We present several partial results in this direction.
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Eli ShamovichThe noncommutative Poulsen simplex
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/5
https://openscholarship.wustl.edu/iwota2016/special/NCinequalities/5Thu, 21 Jul 2016 17:00:00 PDT
The Poulsen simplex is a canonical object in the theory of Choquet simplices. It can be characterized as the unique metrizable Choquet simplex with dense extreme boundary. In my talk I will explain what is the noncommutative Poulsen simplex, and why it plays a similar canonical role in the theory of noncommutative simplices. This is joint work with Ken Davidson and Matt Kennedy.
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Martino Lupini