We are pleased to announce that the 6th annual workshop on Operator Algebras and their Applications will be held at the School of Mathematics of Institute for Research in Fundamental Sciences (IPM) on January 6-10, 2019. These workshops intend to bring home new advances in modern analysis and give graduate students and young researchers an opportunity to meet well-known researchers in the field of operator algebras. This year, the workshop will focus on recent developments on Quantum Groups. It will consist lectures, invited talks and a limited number of contributed talks. As in the previous years, there will be a 1-day pre-workshop mini courses delivered by local speakers which is aiming for students or non-experts.
We look forward to welcoming you in Tehran and meeting you at IPM.

#### Organizers

• Massoud Amini (IPM and Tarbiat Modares University, Iran)
• Mehrdad Kalantar (University of Houston, USA)
• Fatemeh Khosravi (IPM, Iran)

• #### Important Dates

Registration and accommodation request: December 7, 2018 (Azar 16, 1397)
To pay registration and accommodation fees: December 14, 2018 (Azar 23, 1397)

#### Previous workshops

• January 2014: Quantum Groups and Harmonic Analysis
• January 2015: Actions and Crossed Products
• January 2016: $\text{C}^*$-Dynamics
• December 2016: Classification of $\text{C}^*$-algebras
• January 2017: Approximation Properties,
• February 2018: Coarse Geometry

# Program

## Invited Speakers

Abstract: I will overview how the quantum group techniques can be applied to the subfactors.
First we observe that the classification of subfactors can be interpreted as a classification of actions of tensor categories, which can be seen as a slight generalization of actions of quantum groups and I will present some results on this direction. Then I explain approximation properties of tensor categories which is important in such classification.

Reference:
[1] Y. Arano: Unitary spherical representations of Drinfeld doubles, J. Reine Angew. Math. 742 (2018), 157–186.
[2] Y. Arano: Comparison of unitary duals of Drinfeld doubles and complex semisimple Lie groups, Comm. Math. Phys. 351 (2017), no. 3, 1137–1147.
[3] S. K. Ghosh, C. Jones: Annular representation theory for rigid $\text{C}^*$-tensor categories, J. Funct. Anal. 270 (2016), no. 4, 1537–1584.
[4] S. Neshveyev, M. Yamashita:Drinfeld center and representation theory for monoidal categories, Comm. Math. Phys. 345 (2016), no. 1, 385–434.
[5] S. Popa, S. Vaes: Representation theory for subfactors, $\lambda$-lattices and $\text{C}^*$-tensor categories, Comm. Math. Phys. 340 (2015), no. 3, 1239–1280.
Abstract: A quantum Markov semi-group is a continuous time semi-group of normal ucp maps on a von Neumann algebra. Quantum Markov semi-groups are natural analogues of classical Markov processes in quantum probability. In this talk I plan to explain two applications of them: (1) quantum Markov semi-groups provide new tools to study structural results for von Neumann algebras (deformation-rigidity theory), (2) quantum Markov semi-groups can be used as a tool in non-commutative harmonic analysis and we give applications to perturbations of commutators. Though that the results of (1) and (2) may seem quite different at first sight, some of the underlying tools (coming from classical/commutative harmonic analysis) are the same. This will be explained in the talk.

References:
[1] M. Caspers - Gradient forms and strong solidity of free quantum groups - arXiv: 1802.01968
[2] M. Caspers, M. Junge, F. Sukochev, D. Zanin : BMO-estimates for non-commutative vector valued Lipschitz functions.
Abstract: We are interested in the Hochschild cohomology of the Hopf $*$-algebra of a discrete or compact quantum group. A discrete quantum group has the Haagerup property iff there exists a $*$-representation with a proper 1-cocycle, and it has property (T) iff all 1-cocycles of $*$-representations are trivial. The first and second cohomology groups also play a crucial role in the construction and classification of generating functionals and L'evy processes on compact quantum groups. In my talk I will determine residually finite dimensional quotient of the universal unitary quantum group $U^+_Q$ of Wang and Van Daele. Then I will present new (partial) results about Hochschild cohomology groups of (finite-dimensional) $*$-representations of this quantum group, and discuss some of there implications. This talk is based on joint work with Biswarup Das, Anna Kula, and Adam Skalski.
Abstract: Coideals in the context of quantum group are related with the concept of quantum subgroup, idempotent state and group-like projection. I will first discuss left coideal subalgebras of Hopf algebra, (co)integrals and group-like projections in them, and Frobenius property, semisimplicity and unimodularity of them. If time permits I will exemplify these concepts by discussing left coideal subalgebras of Taft Hopf algebras. In the second part of my lecture, I will discuss von Neumann coideals in a locally compact quantum group in correspondence with idempotent states and idempotent contractive functionals, group-like projections and their shifts, and ternary rings of operators. In particular a generalization of Illie-Spronk theorem will be presented, where the latter relates idempotent states (idempotent contractive functionals) on the universal group C*-algebra of G and characteristic functions of open subgroups of G (characteristic functions of cosets of open subgroups resp.).

References:
[1] Alexandru Chirvasitu, Paweł Kasprzak, and Piotr Szulim: Integrals in coideal subalgebras and group-like projections. ArXiv e-prints, August 2018
[2] Ramin Faal and Paweł Kasprzak: Group-like projections for locally compact quantum groups. Journal of Operator Theory, 80(1):153–166, 2018
[3]Paweł Kasprzak: Generalized (co)integrals on coideal subalgebras. ArXiv e-prints, October 2018
[4] Paweł Kasprzak: Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups. International Journal of Mathematics, to be published, 2018
[5] Paweł Kasprzak and Fatemeh Khosravi: Coideals, quantum subgroups and idempotent states. Quarterly Journal of Mathematics, 68(2):583–615, 2017
Abstract: I will report on recent joint work with Vadim Alekseev, concerning group rings with a unique $\text{C}^*$‐completion. It is easy to see that any locally finite group satisfies this property, and utilizing the so‐called Atiyah conjecture we provide some partial evidence for the converse statement.

References:
[1] Vadim Alekseev and David Kyed: Uniqueness questions for $\text{C}^*$-norms on group rings.
[2] Rostislav Grigorchuk, Magdalena Musat and Mikael Rørdam: Just-infinite $\text{C}^*$-algebras.
Abstract: We will study (discrete) group actions on compact quantum groups and will derive conditions for the action to be ergodic, weak mixing, mixing, compact, etc. We will construct several examples of such actions. We will also see how the crossed product formed by such actions is also a quantum group. If time permits, some questions on the topological structure of automorphism groups of compact quantum groups will also be discussed.

References:
[1] Automorphism of compact quantum groups, K. Mukherjee and I. Patri, PLMS 116(2), 330-377, 2018.
[2] Topological automorphism groups of compact quantum groups, A. Chirvasitu and I. Patri, Math. Z., 290 (1-2), 577-598, 2018
[3] Tensor products and crossed products of compact quantum groups, S. Wang, PLMS 71(3), 695-720, 1995.
Abstract: A discrete group is called $\text{C}^*$-superrigid if it can be recovered from its reduced group $\text{C}^*$-algebra. In this series of lectures we will motivate this notation by putting it in the context of Higman's unit conjecture and Connes' conjecture on $\text{C}^*$-superrigidity. Having described one basic strategy to show $\text{C}^*$-superrigidity and a recent esult with Caleb Eckhardt on $\text{C}^*$-superrigidity of 2-step nilpotent groups, we proceed to explain the latter's proof in detail.
Topics: $\text{C}^*$-superrigidity, primitive ideal space, twisted group $\text{C}^*$-algebras, $\text{C}^*$-bundles, K-theory

References:
[1] Higman: The units of group-rings (PhD thesis).
[2] Moore, Rosenberg: Groups with T_1 primitive ideal spaces.
Echterhoff: On maximal prime ideals in certain group $\text{C}^*$-algebras and crossed product algebras.
[3] Packer, Raeburn: On the structure of twisted group $\text{C}^*$-algebras.
[4] Elliott: On the K-theory of the $\text{C}^*$-algebra generated by projective representation of a torsion-free discrete abelian group.
[5] Eckhardt, Raum: $\text{C}^*$-superrigidity of 2-step nilpotent groups.
Abstract: We consider a new class of potentially exotic group $\text{C}^*$-algebras $\text{C}^*(\text{PF}_{p}^*(G))$ for a locally compact group $G$, and its connection with the class of potentially exotic group $\text{C}^*$-algebras $\text{C}^*_{L^p}(G)$ introduced by Brown and Guentner [BG]. Surprisingly, these two classes of $\text{C}^*$-algebras are intimately related. By exploiting this connection, we show $\text{C}^*_{L^p}(G)=\text{C}^*(\text{PF}_{p}^*(G))$ for $p\in (2,\infty)$, and the $C^*$-algebras $C^*_{L^p}(G)$ are pairwise distinct for $p\in (2,\infty)$ when $G$ belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of $C^*_{L^p}(G)$ and $C^*(\text{PF}_{p}^*(G))$. This greatly generalizes earlier results of Okayasu [Okay] and Wiersma [W-Fourier] on the pairwise distinctness of $C^*_{L^p}(G)$ for $2 < p < \infty$ when $G$ is either a noncommutative free group or the group $\text{SL}(2,\mathbb{R})$, respectively.
As a byproduct of our techniques, we present two applications to the theory of unitary representations of a locally compact group $G$. Firstly, we give a short proof of the well-known Cowling-Haagerup-Howe Theorem which presents sufficient condition implying the weak containment of a cyclic unitary representation of $G$ in the left regular representation of $G$ [CHH]. Also we give a near solution to a 1978 conjecture of Cowling stated in [Cow]. This conjecture of Cowling states if $G$ is a Kunze-Stein group and $\pi$ is a unitary representation of $G$ with cyclic vector $\xi$ such that the map $G\ni s\mapsto < \pi(s)\xi,\xi >$ belongs to $L^p(G)$ for some $2< p <\infty$, then $A_\pi\subseteq L^p(G)$. We show $B_\pi\subseteq L^{p+\epsilon}(G)$ for every $\epsilon>0$ (recall $A_\pi\subseteq B_\pi$). Here $A_\pi$ and $B_\pi$ are the closed span of coefficients of $\pi$ in the norm and $w^*$-topology of the Fourier-Stieljes algebra $B(G)=C^*(G)^*$, respectively. In particular, we have $A_\pi\subseteq B_\pi$.
This is based on a joint work with M. Wiersma [SW].

Refrences:
[BG] N. P. Brown and E. P. Guentner: New $\text{C}^*$-completions of discrete groups and related spaces, Bull. Lond. Math. Soc. 45 (2013), no. 6, 1181-1193.
[Cow] M. Cowling: The Kunze-Stein phenomenon. Ann. Math. (2), 107:209-234, 1978.
[CHH] M. Cowling, U. Haagerup and R. Howe: Almost $L^2$ matrix coefficients, J. Reine Angew. Math. 387 (1988), 97-110.
[Okay] R. Okayasu: Free group $\text{C}^*$-algebras associated with $l_p$, Internat. J. Math. 25 (2014), no. 7, 1450065, 12 pp.
[SW] E. Samei and M. Wiersma: Exotic $\text{C}^*$-algebras of geometric groups, submitted (22 pages), arXiv:1809.07007.
[W-Fourier] M. Wiersma: $L^p$-Fourier and Fourier-Stieltjes algebras for locally compact groups, J. Funct. Anal. 269 (2015), no. 12, 3928-3951.
Abstract: In the first lecture I will recall the definition of Wang's orthogonal and unitary free quantum groups, as well as Banica's description of their categories of corepresentations. I will also survey the known results about the structure of the associated $\text{C}^*$ and von Neumann algebras. In the second lecture I will discuss the notion of quantum Cayley graph and present some results in the case of free quantum groups. The most interesting object is the edge-reversing operator which plays a central role in the proof (obtained in collaboration with Brannan) that orthogonal free quantum group factors are not isomorphic to free group factors.
In the third lecture I will discuss random walks on discrete quantum group, and more specifically the associated boundaries. In the case of free quantum groups, the Poisson and Martin boundaries can be identified with a quantum analogue of the Gromov boundary which I will also present.

## Contibuted Talks

Abstract: Properties on locally compact group $G$, can be reflected through function spaces on $G$ like $ap(G)$, $wap(G)$ and $luc(G)$. As a matter of this fact, $G$ is compact if and only if $wap(G)=luc(G)$, or $G$ is amenable if and only if $luc(G)$ is amenable. In this talk we introduce function spaces $ap(L^1(\mathbb{G}))$, $wap(L^1(\mathbb{G}))$ and $luc(L^1(\mathbb{G}))$ on locally compact quantum group $\mathbb{G}$ and characterize them in the language of locally compact quantum groups. Classically, $ap(G)$, $wap(G)$ and $luc(G)$ are m-admissible $C^*-$algebras, thus we attempt to find conditions that force these function spaces to be $C^*-$algebras. In this direction, we characterize the biggest $C^*-$algebra inside $wap(L^1(\mathbb{G}))$ and find a mild condition with which $luc(L^1(\mathbb{G}))$ is a $C^*-$algebra. Next we show taht for a large class of locally compact quantum groups containing Kac algebras $wap(L^1(\mathbb{G}))$ is amenable. Finally, by investigating the relation between function spaces, we characterize compactness and discreteness of locally compact quantum groups.
Abstract: We will generalize the concept of compact quantum hypergroup to compact quantum hypersystem. We will consider the condition that guarantee the existance of Haar satates on compact quantum hypersystems. Finally we will prove that all idempotent states on locally compact quantum groups arise as Haar states of compact quantum subhypersystems in a canonical way. This is an appropriate version of Kawada-Itô theorem for locally compact quantum group. This talk is based on an ongoing project with M. Amini.
Abstract: We introduce the notion of Zimmer amenability for actions of discrete quantum groups on von Neumann algebras. We prove generalizations of several fundamental results of the theory in the noncommutative case. In particular, we give a characterization of Zimmer amenability of an action $\alpha:{\mathbb{G}}\curvearrowright N$ in terms of $\hat{\mathbb{G}}$-injectivity of the von Neumann algebra crossed product $N\ltimes_\alpha\mathbb{G}$. As an application we show that the actions of any discrete quantum group on its Poisson boundaries are always amenable.

## Mini Course

Abstract: We begin by some historical remark which motivates studying quantum groups. We justify why and how quantum groups are defined. Next, we introduce some preliminiaries of coalgebra category. In this stage we can introduce compact quantum groups and describe it's Haar state, comultiplication, multiplicative unitary and representation. Furthermore, some examples of compact quantum groups will be peresented.

References:
[1] U. Franz, A. Skalski and P. M. Sołtan Introduction to compact and discrete quantum groups,arXiv:1703.10766v1.
[2] A. Maes and A. Van Daele, Notes on compact quantum groups, Nieuw Arch. Wisk. (4) 16 (1998), 73112.
[3] V. Runde, Characterizations of compact and discrete quantum groups through second duals, J. Op. Theory 60 (2008), 415-428.
[4] T. Timmermann, An Invitation to Quantum Groups and Duality, European Mathematical Society Publishing House, 2008.
Abstract: Quantum groups have been studied within several areas of mathematics and mathematical physics. It turns out that there are different approaches to the theory, but we will mainly focus on the approaches taken by J. Kustermans and S. Vaes [1,2,3]. We will start with a short history and motivations behind the theory of locally compact quantum group. Then we will state the definition of locally compact quantum groups in a von Neumann algebraic setting. The multiplicative unitary, which plays a crucial role in the theory, will be introduced. We will explain how the Pontrjagin dual quantum groups is constructed. At the end, the reduced and universal $C^*$-algebraic approaches will be introduced and the relations between these different approaches will be considered.

References:
[1] J. Kustermans: Locally compact quantum groups in the universal setting. Internat. J. Math., 12 (2001), 289-338.
[2] J. Kustermans, and S. Vaes:Locally compact quantum groups. Ann. Scient.  Ec. Norm. Sup., 33 (2000), 837-934.
[3] J. Kustermans, and S. Vaes: Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand., 92 (2003), 68-92.
Abstract: This Lecture is devoted to the Pontryagin dual of compact quantum groups, called ''discrete quantum groups''. We study the comultiplication, counit and Haar measures related to this dual object.

References:
[1] U. Franz, A. Skalski and P.M. Soltan, Introduction to compact and discrete quantum groups, arXiv:1703.10766.
[2] P. Podles and S.L. Woronowicz, Quantum deformation of Lorentz group, Comm. Math. Phys. 130 (1991).

## Abstracts of the Talks

You can download the pdf file of the abstracts by clicking here.

## Lecture Notes

You can download the pdf file of the Workshop Lecture Notes by clicking here
and the Short course Lecture Notes here.

# Registration

To register for the workshop, please fill out the Registration Form.

#### Registration fee for the workshop:

1) Registration fee for Iranian participants:

2) Registration fees for international participants:
The registration fee is 250 Euro for faculties and postdocs and 150 Euro for students. The registration fee for international participants will be due in cash at the time of registration on the first day of the meeting. Please note that standard credit cards; e.g., Visa, Master or AmEXP, cannot be used in Iran.

Registration fee includes: Participation in the workshop, documentation package, lunches and coffee breaks during the meeting.

Residence fee at IPM guest house is 40 Euro per night.