Operator-valued Jacobi parameters

Michael Anshelevich

Multivariable tensorial formal power series and a correspondence-based Schur-Agler class: preliminary results

Joseph Ball

Muhly-Solel introduced the Hardy algebra H^∞(E) of a W^*-correspondence E over a W^*-algebra M. Given a normal *-representation σ : M → B(H_σ) of M, they furthermore defined a left Hilbert W^*-module over the commutant von Neumann algebra σ(M)', denoted as (E^σ)^*, consisting of Hilbert-space operators η from E⊗_σ H_σ to H_σ which intertwine the respective σ(M)'-actions on H_σ and E⊗_σ H_σ. They then introduced a Berezin transform to identify elements of the Hardy algebra H^∞(E) as B(H_σ)-valued functions on 𝔻( (E_σ)^*) (elements η of (E_σ)^* with operator norm ∥η∥<1). Such functions F : η → F(η) have a tensorial power series representation and are characterized by complete positivity of the associated de Branges-Rovnyak kernel K^F: 𝔻 ((E^σ)^*) × 𝔻((E^σ)^*) → B(σ(M)', B(H_σ)) as well as by existence of a linear-fractional transfer-function realization with contractive (or coisometric) system matrix U. The goal of this talk is to describe an analogous class of functions defined on the Muhly-Solel polydisk 𝔻((E^σ)^*)^d = 𝔻((E^σ)^*) × ··· × 𝔻((E^σ)^*) (the d-variable Schur-Agler class 𝒜𝒮_E,d on 𝔻((E^σ)^*)^d) consisting of those functions with a d-variable tensorial power series representation not only with contractive values when considered as functions on ( 𝔻((E^σ)^*)^d but also with contractive values when extended to n × n-matrices over 𝔻((E^σ)^*)^d (or equivalently, d-tuples of matricial points from (𝔻((E^σ)^*)^{n × n}) for n=1,2,.... Preliminary calculations suggest that such functions F are characterized via a suitable d-fold completely-positive kernel decomposition for the defect operator I - F(η) F(ζ)^* (or more generally, for b - F(η) b F(ζ)^* where b ∈ σ(M)'), as well as via a multivariable version of the linear-fractional transfer-function realization formula, analogous to the original version due to Jim Agler in 1990 for the case of the classical commutative polydisk (M = ℂ, E = ℂ).

• J. A. Ball, G. Marx, and V. Vinnikov, Noncommutative reproducing kernel Hilbert spaces} arXiv:1602.00760
• J. A. Ball, G. Marx, and V. Vinnikov, Interpolation and transfer-function realization for the noncommutative Schur-Agler class, arXiv:1602.00762
• P. S. Muhly and B. Solel, Hardy algebras, $W^{*}$-correspondences and interpolation theory, Math. Ann. 330 (2004), 353--415
• P. S. Muhly and B. Solel, Tensorial function theory: from Berezin transforms to Taylor's Taylor series and back, Integr. Equ. Oper. Theory 76 (2013), 463--508

Outlying eigenvalues for polynomial functions of two random matrices

Hari Bercovici

We discuss the location of eigenvalues outside the "bulk" for a matrix model of the form P(A(N),B(N)), where A(N) and B(N) are unitarily invariant selfadjoint random matrices of size N. The results depend on the noncommutative subordination functions of free probability and on a classical linearization technique. This joint work with S. Belinschi and M. Capitaine.
SLIDES

Statistics of Farey fractions and distribution of eigenvalues in large sieve matrices

Florin Boca

We will show how existing results on correlations of Farey fractions can be used to study the distribution of eigenvalues in large sieve matrices. The objects that we will discuss are elementary to define. This is joint work with Maksym Radziwill and Alexandru Zaharescu.

Traces arising from regular inclusions

Danny Crytser

We study the problem of extending a state on an abelian C*-subalgebra to a tracial state on the ambient C*-algebra. We propose an approach that is well-suited to the case of regular inclusions, in which there is a large supply of normalizers of the subalgebra. Conditional expectations onto the subalgebra give natural extensions of a state to the ambient C*-algebra; we prove that these extensions are tracial states if and only if certain invariance properties of both the state and conditional expectations are satisfi ed. In the example of a groupoid C*-algebra, these invariance properties correspond to invariance of associated measures on the unit space under the action of bisections. Using our frame- work, we are able to completely describe the tracial state space of a Cuntz-Krieger graph algebra. Along the way we introduce certain operations called graph tightenings, which both streamline our description and provides connections to related fi niteness questions in graph C*-algebras. Our investigation has close connections with the so-called unique state extension property and its variants.
SLIDES

Haagerup-Schultz projections, upper triangular forms and spectrality of Dixmier traces

Ken Dykema

Brown measure is a sort of spectral distribution for all operators (including non-selfadjoint ones) in II₁-factors. Haagerup and Schultz proved existence of hyperinvariant projections for operators in II₁-factors, that decompose 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 II₁-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).
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Obstructions to lifting cocycles on groupoids and the associated C*-algebras

Marius Ionescu

Given an etale groupoid Γ and an abelian group A, the set Z_Γ(A) of continuous cocycles from Γ to A forms an abelian group and it is a functor. A short exact sequence of abelian groups induces a left exact sequence of the group of cocycles. The obstruction to lifting a cocycle is given by a twist over Γ. In this talk that is based on join work with Alex Kumjian I will present some properties of these groupoid twists. The main results implies that the C*-algebra of such a twist is *-isomorphic to the induced algebra as described in work by Raeburn, Rosenberg, and Williams.
SLIDES

Contractive determinantal representations of stable polynomials on a matrix polyball

Dmitry Kaliuzhnyi-Verbovetskyi

We show that a polynomial p with no zeros on the closure of a matrix unit polyball, a.k.a. a cartesian product of Cartan domains of type I, and such that p(0)=1, admits a strictly contractive determinantal representation, i.e., p=det(I-KZ_n), where n=(n₁,...,n_k) is a k-tuple of nonnegative integers, Z_n=⊕^k_r=1(Z^(r)⊗ I_nr), Z^(r)=[z^(r)_ij] are complex matrices, p is a polynomial in the matrix entries z^(r)_ij and K is a strictly contractive matrix. This result is obtained via a noncommutative lifting and a theorem on the singularities of minimal noncommutative structured system realizations. The talk is based on a joint work with A. Grinshpan, V. Vinnikov, and H. J. Woerdeman.
SLIDES

Exotic crossed products and coaction functors

Steve Kaliszewski

When a locally compact group G acts on a C*-algebra A, we have both full and reduced crossed products, each carries a dual coaction of G, and each has its own version of crossed-product duality. Inspired by work of Brown and Guentner on new C*-completions of group algebras, we have begun to understand what we call ``exotic" crossed products --- C*-algebras that lie between the familiar full and reduced crossed products --- and more generally, ``exotic coactions". Some of these coactions satisfy a corresponding exotic crossed product duality, intermediate between full and reduced duality, and this leads us to introduce and study ``coaction functors" induced by ideals of the Fourier-Stieltjes algebra of G. These functors are also related to the crossed-product functors used recently by Baum, Guentner, and Willett in a new approach to the Baum-Connes conjecture. (This is joint work with Magnus Landstad and John Quigg.)
SLIDES

Simple C*-inclusions via state extensions

Gabriel Nagy

Call a C*-algebra inclusion A⊂ B simple, if there is no non-trivial closed two sided ideal J⊂ B, such that J∩ A=0. In this talk we overview various results (joint works with Brown, Reznikoff, Sims and Williams) that identify such inclusions using the space of states on A that extend uniquely to states on B. Conceptually, this is the only technique that works, and we will explain why.
SLIDES

Convergence properties and upper-triangular forms in finite von Neumann algebras

Joseph Noles

It was shown by Haagerup and Schultz that for any operator T in a finite von Neumann algebra, the sequence |Tⁿ| ^1⁄_n converges in the strong operator topology. In this talk we will use upper-triangular decompositions first described by Dykema, Sukochev and Zanin to explore when the sequence converges in norm.
SLIDES

Regular and positive noncommutative rational functions

James Pascoe

Call a noncommutative rational function r regular if it has no singularities, i.e., r(X) is defined for all tuples of self-adjoint matrices X. In this talk, regular noncommutative rational functions r are characterized via the properties of their (minimal size) linear systems realizations. We show that a linear pencil L is invertible everywhere if, perhaps after some transformation, the coefficients satisfy certain positivity properties.The second main result is a solution to a noncommutative version of Hilbert's 17th problem: a positive regular noncommutative rational function is a sum of hermitian squares. Notably, our result generalizes Helton's theorem, which says that a positive noncommutative polynomial is a sum of hermitian squares.
SLIDES

Model theory of C*-algebras as operator systems

Thomas Sinclair

I will discuss various aspects of the model-theoretic structure of the class of C*-algebras within the class of operator systems. This is based on joint work with Isaac Goldbring.

Higher order non-commutative functions

Leonard Stevenson

NC functions are mappings on matrices of all sizes which respect direct sums and similarities. Differentiating nc functions iteratively, one arrives at functions of increasingly more matrix variables, whose values are multilinear forms, which also respect direct sums and similarities, in a certain way. We call them nc functions of higher order. In my talk, I will discuss the Taylor-Taylor formula and other results for higher order nc functions.
SLIDES

Almost-periodic functions and representations of the free group on two generators

Stefanie Wang

Characters of finite-dimensional complex representations of the free group on two generators R, L are given by almost-periodic functions on the infinite tree of valency 4. For a unital ring S, an S-linear quasigroup is a unital S-module, with automorphisms R and L, carrying a (nonassociative) multiplication * given by x*y=xR+yL. If S is the field of complex numbers, the almost-periodic characters provide a complete isomorphism invariant for S-linear quasigroups. Over other rings, most notably the ring Z of integers, it is an open problem to determine good isomorphism invariants. One approach is to tensor with the complex numbers and study the resolving power of the almost-periodic characters. This talk will introduce Z-linear representations of the free group on two generators, and the relationship with the study of quasigroups. We exhibit non-isomorphic Z-linear quasigroups that have the same almost-periodic characters, and discuss some consequences of these examples.
SLIDES

Sampling in de Branges spaces of entire functions

Eric Weber

We consider the problem of sampling and recovery of functions in the Paley-Wiener space and its generalizations to de Branges spaces. We consider both necessary and sufficient conditions for a sequence to be a sampling sequence. We also consider the problem of sampling in the context of dilation theory--embedding the de Branges space into a larger de Branges space while embedding the kernel functions associated with a sampling sequence into a Riesz basis for the larger space.
SLIDES