The seminar meets on Mondays, 16:00-17:00, in -101

This Week

Arel Pinhas (BGU)

de Branges Spaces on Compact Riemann Surfaces

It is a well-known fact that 1D systems and non-selfadjoint operators are closely related via the notion of operator colligation. The study of the characteristic function of a colligation is related to the study of de Branges spaces of analytic functions on an open set in the Riemann sphere. It allows us, for instance, to give an alternative proof for the Beurling’s Theorem using Livšic Colligations and de Branges spaces.

In this talk, I will characterize de Branges spaces, i.e. reproducing kernel Hilbert spaces of analytic sections defined on a real compact Riemann surface, rather than on the Riemann sphere. This is done through the vessel theory, a generalization of the colligation theory to the case of -tuple commuting non-selfadjoint operators. The characteristic function of a vessel is then a bundle mapping defined on a compact Riemann surfaces and which also carries the input-output relation of a 2D system. As a consequence, I will introduce a Beurling-type Theorem on finite bordered Riemann surfaces.

It is a well-known fact that 1D systems and non-selfadjoint operators are closely related via the notion of operator colligation. The study of the characteristic function of a colligation is related to the study of de Branges spaces of analytic functions on an open set in the Riemann sphere. It allows us, for instance, to give an alternative proof for the Beurling’s Theorem using Livšic Colligations and de Branges spaces.

In this talk, I will characterize de Branges spaces, i.e. reproducing kernel Hilbert spaces of analytic sections defined on a real compact Riemann surface, rather than on the Riemann sphere. This is done through the vessel theory, a generalization of the colligation theory to the case of -tuple commuting non-selfadjoint operators. The characteristic function of a vessel is then a bundle mapping defined on a compact Riemann surfaces and which also carries the input-output relation of a 2D system. As a consequence, I will introduce a Beurling-type Theorem on finite bordered Riemann surfaces.

By a result of Glimm, we know that classifying
representations of non-type-I -algebras up to unitary equivalence is essentially impossible (at least with countable structures). Instead of this, one either restricts to a tractable subclass or weakens the invariant.

In the theory of free semigroup algebras, the latter is done for Toeplitz-Cuntz algebras, and is achieved via two key results in the theory: the first is a theorem of Davidson, Katsoulis and Pitts on the structure of free semigroup algebras, and the second, a Lebesuge-von Neumann-Wold decomposition theorem of Kennedy.

This talk is about joint work with Ken Davidson and Boyu Li, where we generalize this theory to representations of Toeplitz-Cuntz-Krieger algebras associated to a directed graph We prove a structure theorem akin to that of Davidson, Katsoulis and Pitts, and provide a Lebesuge-von Neumann Wold decomposition using Kennedy’s theorem. We discuss some of the difficulties and similarities when passing to the more general context of operator algebras associated to directed graphs.

It is well known that a function (where ) is the set of all bounded linear operators on a Hilbert space) being (1) a positive kernel in the sense of Aronszajn (i.e. for all , , and ) is equivalent to (2) being the reproducing kernel for a reproducing kernel Hilbert space , and (3) having a Kolmogorov decomposition for an operator-valued function where is an auxiliary Hilbert space.

In work with Joe Ball and Victor Vinnikov, we extend this result to the setting of free noncommutative functions with the target set of replaced by where is a -algebra. In my talk, I will give a sketch of our proof and discuss some well-known results (e.g. Stinespring’s dilation theorem for completely positive maps) which follow as corollaries. With any remaining time, I will talk about applications and more recent related results.

It is well known that a function (where ) is the set of all bounded linear operators on a Hilbert space ) being (1) a positive kernel in the sense of Aronszajn (i.e. for all , , and ) is equivalent to (2) being the reproducing kernel for a reproducing kernel Hilbert space , and (3) having a Kolmogorov decomposition for an operator-valued function where is an auxiliary Hilbert space.

Last time, I introduced free noncommutative function theory and wrote down the analogue of the result above for noncommutative kernels. In part two, I will give a sketch of our proof and discuss some well-known results (e.g. Stinespring’s dilation theorem for completely positive maps) which follow as corollaries. With any remaining time, I will talk about applications and more recent related results.

I will briefly discuss the general things that Magdalena Georgescu, Joav Orovitz, and I determined one needs to take into consideration for constructing inverse sequences of groupoids with Haar systems such that the pullback morphism induce a directed sequence of groupoid C*-algebras (to be clear, the groupoid C*-algebra of the inverse limit groupoid is the direct limit of the induced directed system of groupoid C*-algebras). Then I will proceed to discuss a variety of examples of how to create, in a simple way, groupoids whose groupoid C*-algebras are matrix algebras, UHF-algebras, infinite tensor powers of direct sums of such things, and dimension drop algebras where and are natural or even supernatural numbers. I will briefly discuss my work with Atish Mitra on our current project for making the Jiang-Su algebra as a groupoid C*-algebra of an inverse limit groupoid (which, I believe is much more understandable and geometric than other groupoids which have Jiang-Su algebra as groupoid C*-algebra that show up in the literature). I will also discuss my project with Magdalena Georgescu on taking inverse limits of sigma-compact groupoids by second countable groupoids as a way to bootstrap known results about second countable groupoids to sigma-compact groupoids.