# SCIENTIFIC PROGRAMS AND ACTIVITIES

March  1, 2021

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

February 5-7, 2015
Workshop on
Recent Developments in Quantum Groups, Operator Algebras and Applications
at the University of Ottawa

 Organizing Committee Benoit Collins - University of Ottawa Thierry Giordano – University of Ottawa Sutanu Roy - University of Ottawa and Carleton University Scientific Advisory Committee George A. Elliott - University of Toronto Kenneth R. Davidson - University of Waterloo Stanislaw L. Woronowicz - University of Warsaw
 Registration on site Fee: \$50 for faculty, \$30 for postdocs, \$20 for students. Fee waived for invited speakers Schedule Abstracts Participant List Getting to UOttawa Speaker Abstracts Richard Blute, University of Ottawa Hopf Algebras and The Logic of Tensor Categories In categorical proof theory, the primary goal is build a category whose objects are formulasfrom a given logic and whose arrows are proofs in that logic. This was well understoodfor traditional logics for some time. But in 1987 Jean-Yves Girard introduced linear logic.Linear logic is a common refinement of both classical and intuitionistic logic, combining the best properties of each, but emphasizing in particular the subtleties of merging and duplicating resources.The basic structure of linear logic yields tensor (monoidal) categories of which representationsof Hopf algebras provide canonical examples. Such categories are monoidal and closed and hence provide models of a novel logical system, the linear logic of Jean-Yves Girard. Linear logic is a common refinement of both classical and intuitionistic logic, combining the best properties of each. Jason Crann, Carleton University A unified framework for complementarity in quantum information In open quantum system dynamics there is an important duality between the system and its environment which is usually represented through the notion of complementary channels. One particular manifestation of this duality is the complementarity of private and correctable subsystems in finite-dimensions. In this talk, we will generalize the notion of private subsystems to the level of von Neumann algebras and prove a generalized complementarity theorem with correctable subalgebras in arbitrary dimensions. This is joint work with David Kribs, Rupert Levene and Ivan Todorov. Douglas Farenick, University of Regina Operator systems from discrete groups In this lecture I will discuss some results obtained in collaboration with A. Kavruk, V. Paulsen, and I.G. Todorov on certain operator subsystems of C*-algebras of discrete groups. Of special interest and importance are operator systems on free groups with finitely many generators, and I will discuss this case in particular, especially with regards to their tensor products and their use in quantum information theory. David Handelman, University of Ottawa Good and not-so-good traces on (AF) C*-algebras (Joint with Sergey Bezuglyi) Akin introduced the notion of a {\it good\/}measure on a Cantor set, in order to classify measures up to homeomorphism. The notion translates directly to dimension groups, in particular, to simple dimension groups. These arise as the ordered groups associated to minimal homeomorphisms on Cantor sets. The translation to this setting permits relatively simple characterizations of goodness, yielding lots of examples of good and not good measures, which do not seem to be tractible in the original setting. When we apply the definitions in the not necessarily simple (i.e., minimal) situation, some interesting phenomena occur. If the dimension group (AF C*-algebra) arises as a fixed point subalgebra of a xerox product type action of an$n$-torus, the extremal traces are known (the faithful ones can be identified, via the weighted moment map, with the interior of an octant in$R^n$), and the characterization among them of the good (or almost good) is in terms of a Zariski topology property (over the integers). Akin also looked at a weaker version of goodness, called {\it refinability\/}; this can be studied via compact convex sets of traces (not necessarily faces in the trace space). This leads to a notion of goodness for such sets of traces on dimension groups, and when applied in the setting of the previous paragraph, leads to an algebraic-geometric characterization. Matthew Kennedy, Carleton University C*-simplicity and the unique trace property for discrete groups In joint work with M. Kalantar, we established necessary and sufficient conditions for the simplicity of the reduced C*-algebra of a discrete group. More recently, in joint work with E. Breuillard, M. Kalantar and N. Ozawa, we proved that any tracial state on the reduced C*-algebra of a discrete group is supported on the amenable radical. Hence every C*-simple group has a unique tracial state. I will discuss these results, along with recent work (in progress) to generalize these results to quantum groups. James A. Mingo, Queen's University Freeness and the Transpose In most examples of asymptotic freeness one starts with an assumption of classical independence of the entries of the random variables. I have shown recently that rearranging the entries of a matrix in a systematic way is enough tp produce asymptotic freeness. I will illustrate this with some standard ensembles of matrices. This is joint work with Mihai Popa. Matthias Neufang, Carleton University Amenability and injectivity of locally compact quantum groups As is well known, the equivalence between amenability of a locally compact group$G$and injectivity of its von Neumann algebra$L(G)$does not hold beyond inner-amenable groups. We will show that the equivalence persists for all locally compact groups if$L(G)$is considered as a$T(L_2(G))$-module with respect to a natural action. In fact, we will present an appropriate version of this result for every locally compact quantum group$\mathbb{G}$. This will lead us to several characterizations of quantum group amenability in terms of injectivity in the category of$T(L_2(\mathbb{G}))$-modules. This is joint work with Jason Crann. Sutanu Roy, University of Ottawa & Carleton University Quasitriangular quantum group twisted tensor product of C*-algebras For a quasitriangular C*-quantum group, we enrich a monoidal structure on the category of its continuous coactions on C*-algebras. We define braided C*-quantum groups, where the comultiplication takes values in a twisted tensor product. We show that compact braided C*-quantum groups yield compact quantum groups by a semidirect product construction. This is a joint work with Ralf Meyer and Stanislaw Lech Woronowicz. Zhong-Jin Ruan, University of Illinois at Urbana-Champaign Exotic Quantum Group C*-algebras Let$\hat{\mathbb{G}}$be a compact quantum group. Suppose that$\hat{\mathbb{G}}$is not co-amenable, i.e. the natural quotient map$C_u(\hat{\mathbb{G}}) \mapsto C(\hat{\mathbb{G}})$is proper. It is interesting to know whether there exists any compact quantum group C*-algebra$A$such that the sequence of quotients $C_u(\hat{\mathbb{G}}) \mapsto A \mapsto C(\hat{\mathbb{G}})$ is proper. We call such$A$an exotic quantum group C*-alegbra. Motivated by the work of Brown-Guentner and Okayasu, we study a special class of exotic quantum group C*-algebras associated with the orthogonal free quantum groups and unitary free quantum groups. This is a recent joint work with Michael Brannan. Nico Spronk, University of Waterloo Weak amenability of Fourier algebras Let$G$be a locally compact group. Amenability and weak amenability for its group algebra$L^1(G)$have been known for a long time now. These properties have also been settled for the Fourier algebra$A(G)$, when its natural operator space structure is considered. Even the characterisation of classical amenability (without operator space structure) has been known for some time. However, the problem of knowing when$A(G)$is weakly amenable remains unsettled. I wish to discuss recent work, conducted with H. H. Lee, J. Ludwig and E. Samei, which settles the issue for all Lie groups. I will indicate how the problem for general locally compact groups is "almost completed". Charles Starling, University of Ottawa Boundary Quotients of Semigroup C*-algebras The construction due to Li of a C*-algebra associated to a left-cancellative semigroup$P$generalizes many interesting classes of C*-algebras. These algebras are akin to Toeplitz algebras, and in this analogy their boundary quotients play the role of the Cuntz algebras. Li's recent work on these algebras focuses on the case where$P$embeds in a group. The class of semigroups which embed into groups is a large and rich class, though it does not include a great many interesting examples -- for instance semigroups obtained from self-similar groups. In this talk we discuss the boundary quotients of the C*-algebras of such$P$by using a canonical embedding into an inverse semigroup, and find algebraic conditions on$P$which guarantee that the boundary quotient is simple and purely infinite. Roland Vergnioux, Université de Caen Basse-Normandie Stabilizer subgroups of universal compact quantum groups and the Connes embedding property In a recent joint work with Brannan and Collins we show that$O_n^+$is generated by two copies of$O_{n-1}^+$and we deduce that the associated finite von Neumann algebras embed in$R^\omega$. I shall explain the strategy of the proof and the connection to free entropy dimension. Grazia Viola, Lakehead University Classification of spatial$L^p$AF algebra The talk will be an introduction to$L^p$operator algebras and spatial$L^p$AF algebra. The main theorem we will discuss is a classification theorem for spatial$L^p$AF algebras. We show that two spatial$L^p$AF algebras are isomorphic if and only if their scaled ordered$K_0$groups are isomorphic. Moreover, we prove that any countable Riesz group can be realized as the scaled ordered$K_0$group of a spatial$L^p$AF algebra. Therefore, the classification given by G. Elliott for AF algebras also holds for spatial$L^p$AF algebras. Lastly, we will talk about compressibility and p-incompressibility for Banach algebras, and discuss the results we have for a spatial$L^p$AF algebra. This is joint work with Chris Phillips. Stanisław Lech Woronowicz, IMPAN-Warsaw University Categories of C*-algebras subject to an action of a quantum group Given$C^*$-algebras$X$and$Y$one may consider the$C^*$-algebra$X\otimes Y$. If$X$and$Y$are equipped with actions of a locally compact quantum group$G$, then there exists unique action of$G$on$X\otimes Y$such that the natural embeddings of$X$and$Y$into$X\otimes Y$intertwine the actions of$G$. In the categorical language: tensor product$\otimes$defines a monoidal structure on the category$C^*_G$of all$C^*$-algebras equipped with the action of$G$. This is no longer the case if$G$is a quantum group. Let$G$be a locally compact quantum group. It turns out that the category of$C^*_G$admits a monoidal structure satisfying certain natural conditions if and only if$G$is quasitriangular. The monoidal structures are in bijective correspondence with unitary$R$-matrices. In general the monoidal structure is not given by$\otimes\$.