# Operator Algebras/Operator Theory[ Edit ]

## Moderator: Orr Shalit

*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.