# Operator Algebras/Operator Theory[ Edit ]

## Moderator: Orr Shalit

*Abstract:*

The classical results of Atiyah (1967) and Atiyah, Singer (1969) provide the homotopy types of the space FB(H) of Fredholm bounded operators on a Hilbert space H and of its subspace FBsa(H) consisting of self-adjoint operators. Namely, FB(H) is a classifying space for the functor K^0, while FBsa(H) is a classifying space for the functor K^1.

However, in many applications (e.g. to differential operators) one deals with unbounded operators rather than bounded. The space B(H) of bounded operators is then replaced by the space R(H) of regular (that is, closed and densely defined) operators. The homotopy types of the spaces FR(H) and FRsa(H) were unknown for a long time. Finally, an analog, for regular operators, of results of Atiyah and Singer was obtained in 2003 by Joachim. His proof is based on the theory of C*-algebras and Kasparov KK-theory.

I will describe in the talk how this result of Joachim can be included into broader picture. In particular, I will show connections of the spaces FR(H) and FRsa(H) with other classical spaces. I will also give a simple definition of the family index for unbounded operators. All terminology will be explained during the talk.

*Abstract:*

Abstract is available here: https://noncommutativeanalysis.files.wordpress.com/2018/06/abstract.pdf

*Abstract:*

This will be the fourth talk in Adam's lecture series.

*Abstract:*

The third talk in Adam's lecture series, presenting his joint work with Davidson and Li

https://arxiv.org/abs/1709.06637

*Abstract:*

(This is the second lecture in a series of lectures)

By a result of Glimm, we know that classifying representations of non-type-I $C^*$-algebras up to unitary equivalence is a difficult problem. Instead of this, one either restricts to a tractable subclass or weakens the invariant. In the theory of free semigroup algebras, initiated by Davidson and Pitts, classification within the subclasses of atomic and finitely correlated representations of Toeplitz-Cuntz algebras can be achieved.In this talk we will sketch the proof of a classification theorem for atomic representations for Toeplitz-Cuntz-\emph{Krieger} algebras, generalizing the one by Davidson and Pitts. Furthermore, we will explain how the famous road coloring theorem, proved by Trahtman, gives us a large class of directed graphs for which the free semigroupoid algebra is in fact self-adjoint. Time permitting, we will start working our way towards classification of free semigroupoid algebras.

*Abstract:*

(This is the first in a series of several talks)

By a result of Glimm, we know that classifying representations of non-type-I $C^*$-algebras up to unitary equivalence is a difficult problem. Instead of this, one either restricts to a tractable subclass or weakens the invariant. In the theory of free semigroup algebras, initiated by Davidson and Pitts, classification of atomic and finitely correlated representations of Toeplitz-Cuntz algebras can achieved.

In this first talk, we introduce free semigroupoid algebras and discuss generalizations of the above results to representations of Toeplitz-Cuntz-*Krieger* algebras associated to a directed graph $G$. We prove a classification theorem for atomic representations and explain a classification theorem for finitely correlated representations due to Fuller. Time permitting, we will explain how the famous road coloring theorem, proved by Trahtman, gives us a large class of directed graphs for which the free semigroupoid algebra is in fact self-adjoint.

*Abstract:*

In non-commutative probability there are several well known notions of independence. In 2003, Muraki's classification, which states that there are exactly five independences coming from universal (natural) products, seemingly settled the question of what independences can be considered. But after Voiculescu's invention of bi-free independence in 2014, the question came up again. The key idea that allows to define a new notion of independence with all the features of the universal independences that appear in Muraki's classification is to consider ``two-faced'' (i.e. pairs of) random variables.In the talk, we define bi-monotone independence, a new example of an independence for two-faced random variables. We establish a corresponding central limit theorem and use it to describe the joint distribution of monotone and antimonotone Brownian motion on monotone Fock space, which yields a canonical example of a quantum stochastic process with bi-monotonely independent increments.

*Abstract:*

All talks will take place in Amado 814.

Schedule:

13:30-14:20 Ami Viselter (Haifa University)

Convolution semigroups on quantum groups and non-commutative Dirichlet forms

14:30-15:20 Michael Skeide (University of Molise)

Interacting Fock Spaces and Subproduct Systems (joint with Malte Gerhold)

15:20-15:50 Coffee break

15:50-16:40 Adam Dor-On (Technion)

C*-envelopes of tensor algebras and their applications to dilations and Hao-Ng isomorphisms

*Abstract:*

This will be the third lecture in which we will study the paper "Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture" by Raphael Clouatre.

*Abstract:*

This will be the second of two talks in which we will study the recent preprint "Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture", by Raphael Clouatre. Link:

https://arxiv.org/pdf/1709.01649.pdf

*Abstract:*

This will be the first of two talks in which we will study the recent preprint "Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture", by Raphael Clouatre. Link:

https://arxiv.org/pdf/1709.01649.pdf

*Abstract:*

TBA

*Abstract:*

TBA

*Abstract:*

TBA

*Abstract:*

(This is is the second of two lectures on this subject)

We shall present the background of Arveson-Douglas conjecture on essential normality, and discuss two papers by Ron Douglas and Yi Wang on the subject:

1) "Geometric Arveson-Douglas Conjecture and Holomorphic Extension"

link: https://arxiv.org/pdf/1511.00782.pdf

2) "Geometric Arveson-Douglas Conjecture - Decomposition of Varieties"

*Abstract:*

We shall present the background of Arveson-Douglas conjecture on essential normality, and discuss two papers by Ron Douglas and Yi Wang on the subject:

1) "Geometric Arveson-Douglas Conjecture and Holomorphic Extension"

link: https://arxiv.org/pdf/1511.00782.pdf

2) "Geometric Arveson-Douglas Conjecture - Decomposition of Varieties"

*Abstract:*

The fourth lecture in the series.

*Abstract:*

NOTE: The series continues to January 29th and February 5th.

**Abstract: **

In algebraic topology, the Borsuk-Ulam theorem and its extensions place restrictions on maps between compact spaces. Essential to this story is the antipodal action on the sphere, which sends each point x to -x, so equivariant maps are commonly called "odd". The original Borsuk-Ulam theorem then says that there is no odd map from a sphere of high dimension to a sphere of low dimension. This may be extended to more general compact spaces with free actions of finite groups by considering connectivity of a domain X and dimension of a codomain Y.

I will present my work on extending this theorem and similar results to C*-algebras, as motivated by the results and conjectures of other researchers (Yamashita, Taghavi, Baum-Dabrowski-Hajac). Along the way, we will see how this point of view may be used to improve topological results, and how the noncommutative setting differs from the commutative setting.

*Abstract:*

Ben Passer will give the second lecture in his series of lectures on Noncommutative Borsuk Ulam theorems.

**Abstract: **

In algebraic topology, the Borsuk-Ulam theorem and its extensions place restrictions on maps between compact spaces. Essential to this story is the antipodal action on the sphere, which sends each point x to -x, so equivariant maps are commonly called "odd". The original Borsuk-Ulam theorem then says that there is no odd map from a sphere of high dimension to a sphere of low dimension. This may be extended to more general compact spaces with free actions of finite groups by considering connectivity of a domain X and dimension of a codomain Y.

I will present my work on extending this theorem and similar results to C*-algebras, as motivated by the results and conjectures of other researchers (Yamashita, Taghavi, Baum-Dabrowski-Hajac). Along the way, we will see how this point of view may be used to improve topological results, and how the noncommutative setting differs from the commutative setting.

*Abstract:*

On the 15, 22 and perhaps also 29 of January, Ben Passer will give a series of lectures on Noncommutative Borsuk-Ulam Theorems.

**Abstract: **

In algebraic topology, the Borsuk-Ulam theorem and its extensions place restrictions on maps between compact spaces. Essential to this story is the antipodal action on the sphere, which sends each point x to -x, so equivariant maps are commonly called "odd". The original Borsuk-Ulam theorem then says that there is no odd map from a sphere of high dimension to a sphere of low dimension. This may be extended to more general compact spaces with free actions of finite groups by considering connectivity of a domain X and dimension of a codomain Y.

I will present my work on extending this theorem and similar results to C*-algebras, as motivated by the results and conjectures of other researchers (Yamashita, Taghavi, Baum-Dabrowski-Hajac). Along the way, we will see how this point of view may be used to improve topological results, and how the noncommutative setting differs from the commutative setting.

*Abstract:*

We will study the preprint by Fritz, Netzer and Thom with the title as above, available on the arxiv:

https://arxiv.org/abs/1609.07908

*Abstract:*

We continue studying the paper of Alexeev, Netzer and Thom discussed in the first two lectures.

*Abstract:*

We will continue to study the preprint by the name of the title by Alekseev, Netzer and Thom, see : https://arxiv.org/abs/1602.01618.

Following this, we will study the paper "Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization" by Fritz, Netzer and Thom, see: https://arxiv.org/abs/1609.07908v1

*Abstract:*

In the first talk we will discuss various Positivstellensatze and quadratic modules, both in the commutative setting (Stengle, Schmudgen and Putinar) as well as the noncommutative setting (Helton). Then we will move on to describe the C*-algebra associated to a quadratic module, following recent work of Alekseev, Netzer and Thom.

*Abstract:*

Fourth and last lecture in the sequence of talks on Katsoulis and Ransey's paper http://arxiv.org/pdf/1512.08162.pdf.

*Abstract:*

This is the second lecture (out of four) where we study the recent preprint "Crossed products of operator algebras" by Katsoulis and Ramsey.

*Abstract:*

We will study the paper "Crossed products of operator algebras" by Katsoulis and Ramsey, where the theory of crossed products of (not-necessarily selfadjoint) operator algebras by a group is developed.

*Abstract:*

The fundamental nonselfadjoint operator-algebra associated with a countable directed graph is its tensor-algebra. Ten years ago, Katsoulis and Kribs showed that its C*-envelope --- the noncommutative counterpart of the Shilov boundary --- is the Cuntz-Krieger algebra of the graph.

My aim in this talk is to describe the noncommutative counterpart of points in the Choquet boundary of the tensor-algebra and to provide a full characterization of them. This leads both to a new proof of Katsoulis-Kribs theorem mentioned above and to a characterization --- in terms of the graph itself --- of the tensor-algebra hyperrigidity inside the Cuntz-Krieger algebra.

The talk is based on joint work with Adam Dor-On.

*Abstract:*

Convolution semigroups of semi-inner products on colagebras give rise to subproduct systems of Hilbert spaces. By a concrete construction, we show that the Arveson systems generated by these coalgebra subproduct systems are type I, that is, Fock spaces. By an application of this result,we reprove Michael Schürmann’s result that every quantum Lévy process posses a representation on the Fock space. This is joint work with Malte Gerhold and part of our seeking for finite-dimensional subproduct systems. The proof is actually inspired by our paper with Michael Schürmann and his former MSc student Sylvia Volkwardt, which in turn is inspired by our joint work with Volkmar Liebscher how to construct units in product systems

*Abstract:*

During the month of March Prof. Michael Skeide will be visiting our department, and he has kindly agreed to give a series of lectures on "von Neumann modules". (A von Neumann module is a module over a von Neumann algebra M that has a M-valued inner product defined on it. So it is like a Hilbert space, with a von Neumann algebra taking the place of the scalars).

Michael Skeide developed a large part of the theory of von Neumann modules to tackle problems in non-commutative dynamics, such as the study of one parameter semigroups of endomorphisms or completely-positive semigroups on operator algebras.

To understand what is going on in this seminar, one needs to know some basic C*-algebras and von Neumann algebras theory, as well as basic facts on C*-correspondences.

One should refresh one's memory on the following topics:

1) C*-algebras, see http://www.math.uni-sb.de/ag/speicher/lehre/hmwise1516/Cstern.pdf

2) von Neumann algebras, see the first four pages of http://www.math.uni-sb.de/ag/speicher/lehre/hmwise1516/vNeumann.pdf

3) Basics of Hilbert C*-modules: chapters 1 and 2 and pages 39-42 in the book "Hilbert C*-modules: a Toolkit for Operator Algebraists" by E.C. Lance.

*Abstract:*

The last lecture in our learning seminar this semester.

*Abstract:*

We continue our learning seminar.

*Abstract:*

We will study the paper "Coactions on Cuntz-Pimsner algebras" by KALISZEWSKI, QUIGG, and ROBERTSON.

*Abstract:*

We will study the paper "Functoriality of Cuntz Pimsner Algebras", by Kaliszewski, Quigg and Robertson.

*Abstract:*

Continuing our learning seminar...

*Abstract:*

This is the second lecture in our learning seminar discussing the paper :

"Crossed products of C*-correspondences by amenable group actions" by Hao and Ng

*Abstract:*

C*-algebras constructed out of C*-correspondences have been a central theme in operator algebras for almost twenty years at least. This semester, the seminar will be dedicated to (co)actions on C*-correspondences, (co)actions on the associated algebras and the relations between them.

We will begin with the papers

"Crossed products of C*-correspondences by amenable group actions" by Hao and Ng (JMAA, 2008, link: http://www.sciencedirect.com/science/article/pii/S0022247X08004563 )

followed perhaps by

"Coactions on Cuntz-Pimsner algebras", by Kaliszewski & Quigg & Robertson, (Math. Scand., to appear, link http://arxiv.org/abs/1204.5822)

As these constructions rely on many operator-algebraic notions, we will require a few preliminaries. Most of them will be given during the talks, but in a succinct way. We therefore list several topics, with references, that the audience will be expected to be familiar with - at least at the level of knowing what they mean. We emphasize that up to some preparations, this seminar will be accessible to non-operator algebraists.

* Spatial tensor products of C*-algebras: definition and basics. See the book "Hilbert C*-modules" by Lance, pp. 31-32, or most books on C*-algebras.

* Multiplier algebras: definition and basic theorems. Browse through Chapter 2 of Lance.* Hilbert modules and C*-correspondences: again, definition and basic examples. See "Tensor algebras over C*-correspondences: representations, dilations, and C*-envelopes" by Muhly and Solel (JFA, 1998), Definition 2.1 and the following examples. A more complete overview on Hilbert modules is Lance, Chapter 1.

* Crossed product C*-algebras: we will define them from scratch, but to avoid a shock, it's better to be familiar with this construction. See Chapter 2 of "Crossed Products of C*-Algebras" by D. Williams.

* Cuntz-Pimsner algebras: ditto; look at the definition and see some examples. See Definition 3.5 in "On C*-algebras associated with C*-correspondence" by Katsura (JFA, 2004). For examples, see Muhly-Solel.

*Abstract:*

We will discuss prerequisites for the Operator Algebras Learning Seminar.

ABSTRACT:

C*-algebras constructed out of C*-correspondences have been a central theme in operator algebras for almost twenty years at least. This semester, the seminar will be dedicated to (co)actions on C*-correspondences, (co)actions on the associated algebras and the relations between them.

We will begin with the papers

"Crossed products of C*-correspondences by amenable group actions" by Hao and Ng (JMAA, 2008, link: http://www.sciencedirect.com/science/article/pii/S0022247X08004563 )

followed perhaps by

"Coactions on Cuntz-Pimsner algebras", by Kaliszewski & Quigg & Robertson, (Math. Scand., to appear, link http://arxiv.org/abs/1204.5822)

As these constructions rely on many operator-algebraic notions, we will require a few preliminaries. Most of them will be given during the talks, but in a succinct way. We therefore list several topics, with references, that the audience will be expected to be familiar with - at least at the level of knowing what they mean. We emphasize that up to some preparations, this seminar will be accessible to non-operator algebraists.

* Spatial tensor products of C*-algebras: definition and basics. See the book "Hilbert C*-modules" by Lance, pp. 31-32, or most books on C*-algebras.

* Multiplier algebras: definition and basic theorems. Browse through Chapter 2 of Lance.* Hilbert modules and C*-correspondences: again, definition and basic examples. See "Tensor algebras over C*-correspondences: representations, dilations, and C*-envelopes" by Muhly and Solel (JFA, 1998), Definition 2.1 and the following examples. A more complete overview on Hilbert modules is Lance, Chapter 1.

* Crossed product C*-algebras: we will define them from scratch, but to avoid a shock, it's better to be familiar with this construction. See Chapter 2 of "Crossed Products of C*-Algebras" by D. Williams.

* Cuntz-Pimsner algebras: ditto; look at the definition and see some examples. See Definition 3.5 in "On C*-algebras associated with C*-correspondence" by Katsura (JFA, 2004). For examples, see Muhly-Solel.