# Summer School in Operator Algebras

### June 14-18, 2021

### Online, hosted by the Fields Institute and the University of Ottawa

# Contributed talks

Click on talk titles to reveal abstracts.Kevin Aguyar Brix (University of Glasgow) - Conjugacy of local homeomorphisms - slides, video

Local homeomorphisms on locally compact Hausdorff spaces define a large and interesting class of dynamical systems. Important examples include one-sided shifts of finite type or boundary-path spaces of directed graphs or their topological analogues. A C*-algebra is canonically associated to such systems via a groupoid construction, and it turns out that whether two such dynamical systems are conjugate is completely described by *-isomorphism of these C*-algebras that intertwine a certain family of weighted gauge actions.

This is based on joint work with Becky Armstrong, Toke Meier Carlsen, and Søren Eilers.

M. Ali Asadi-Vasfi (Institute of Mathematics of the Czech Academy of Sciences) - The radius of comparison of the tensor product of a C*-algebra with C(X) - slides, video

The radius of comparison of a C*-algebra, based on the Cuntz semigroup, and the dimension-rank ratio of an AH algebra are numerical invariants which were introduced by Toms in 2006 to study exotic examples of simple amenable C*-algebras that are not \(\mathcal{Z}\)-stable. For example, Elliott classifiable C*-algebras have zero radius of comparison.

Let X be a compact metric space. In the commutative setting, it is well known that the radius of comparison of C(X) is dominated by one half of the covering dimension of X. This gives us permission to thinking of the radius of comparison of not necessarily commutative C*-algebras. In this regard, comparison theory can be viewed as a non-commutative dimension theory. In this talk, we will give some preliminary results related to the radius of comparison of C*-algebras of the form C(X, A), where A is a unital C*-algebra.

Sayan Das (UC Riverside) - On the free group factor problem, and Popa's MV-property - slides, video

One of the most important outstanding problems in von Neumann algebras asks if the group von Neumann algebra of the free group on two generators, denoted by L(F_{2}), is isomorphic to the group von Neumann algebra of the free group on infinitely many generators, denoted by L(F_{∞}). Recently, S. Popa established a roadmap for showing the nonisomorphism of L(F_{2}) and L(F_{∞}). The first step of the proposed roadmap is to establish the so called mean value property (abbreviated MV-property) for L(F_{2}).

In this talk I shall describe the proof of the result that L(F_{2}) has the MV-property, thereby establishing the first step of Popa's roadmap. This talk is based on a recent joint work with Prof. Jesse Peterson.

Daniel Drimbe (Leuven) - New examples of W* and C*-superrigid groups - slides, video

A group G is called W*-superrigid (resp. C*-superrigid) if it is completely "remembered" by its von Neumann algebra (resp. its reduced C*-algebra). An important, but difficult problem in operator algebras is to find new classes of such superrigid groups. Although this problem goes back to the pioneers of the field, currently only a few classes of W*-superrigid and C*-superrigid groups are known in the literature.

The goal of this talk is to present new constructions of W* and C*-superrigid groups arising from various natural constructions in group theory including direct products, amalgamated free products, HNN extensions, wreath products and coinduced groups. The proofs of these results are based on Popa's deformation/rigidity theory combined with a natural interplay with C*-algebraic techniques such as the unique trace property and the absence of nontrivial projections. This is based on a very recent joint work with Ionut Chifan and Alec Diaz-Arias.

Marzieh Forough (Institute of Mathematics of the Czech Academy of Sciences) - Asymptotic lifting for completely positive equivariant maps - slides, video

In this talk, I will begin with giving a brief introduction to the lifting problem for completely positive maps between C*-algebras. The importance of the existence of completely positive lifts can be traced back to Arveson’s work on extensions of C*-algebras. By proving the existence of the lifts for maps from C(X), he gave simpler proof of the fact that Ext(X) is a group. More generally, the Choi-Effros lifting theorem implies that Ext(A) is a group for any separable nuclear C*-algebra A. Motivated by the wide range applications of this celebrated lifting theorem and the recent increased interests in the structure of C*-dynamical systems, it is natural to look for an equivariant version of the lifting result for completely positive maps. I will discuss the following equivariant result: Let G be a locally compact second countable group, A and B be G-algebras and let I be a G-invariant ideal of B. Then every completely positive contractive map from A into B/I admits asymptotically equivariant lifts.

This talk is based on joint work with Eusebio Gardella and Klaus Thomsen.

Priyanga Ganesan (TAMU) - Quantum graphs - slides, video

Quantum graphs are an operator space generalization of classical graphs. In this talk, I will present the different notions of quantum graphs that arise in operator systems theory, non-commutative topology and quantum information theory. I will then introduce a non-local game with quantum inputs and classical outputs, that generalizes the graph homomorphism game for classical graphs. This is based on joint work with Michael Brannan and Samuel Harris.

Sergio Giron Pacheco (Oxford) - Anomalous symmetries of classifiable C*-algebras - slides, video

In this talk I will start by giving an elementary introduction to the symmetries of the hyperfinite II_{1} factor R. I will discuss classification results of these by Connes, Jones, Ocneanu and Popa. I will then discuss how the existence of these symmetries carry over to a particularly nice class of C*-algebras, those classified by the Elliott programme. It is known that any countable discrete group G acts faithfully on any classifiable C*-algebra. However, even for types of quantum symmetries closely related to group actions (anomalous symmetries) the existence question is subtle and can have both positive and negative answers. This talk is based on joint work with Samuel Evington.

Lucas Hall (Arizona State University) - Skew products for topological quivers - slides, video

Given a cocycle on a topological quiver by a locally compact group, the author constructs a skew-product topological quiver, and discovers conditions under which a topological quiver can be identified as a skew-product. We investigate the relationship between the C*-algebra of the skew product and a certain native coaction on the C*-algebra of the original quiver, finding that the crossed product by the coaction is isomorphic to the skew product.

Zahra Hasanpour (IPM Institute for Research in Fundamental Sciences) - Frame existence problem in Hilbert C*-modules - slides, video

Kasparov’s stabilization theorem, which plays an important role in Kasparov’s KK-theory, asserts that every countably generated Hilbert C*-module over a C*-algebra A is a direct summand of the free A-module \(\bigoplus_{j \in \mathbb{N}} A\). Indeed, it shows that every countably generated Hilbert C*-module admits a standard frame. It is a natural question to ask whether this result can be generalized to an arbitrary Hilbert C*-module. In other words, whether every Hilbert C*-module admits a standard frame. In this talk, we use the vector bundle approach to Hilbert C*-modules to propose the structure of a Hilbert C*-module admitting no frames (Joint work with Mohammad Bagher Asadi and Michael Frank).

Ishan Ishan (Vanderbilt) - Von Neumann equivalence and group approximation properties - slides, video

The notion of measure equivalence of groups was introduced by Gromov as the measurable counterpart to the topological notion of quasi-isometry. Another well-studied notion is that of W*-equivalence which states that two groups Γ and Λ are W*-equivalent if they have isomorphic group von Neumann algebras, i.e., LΓ ≃ LΛ. We introduce a coarser equivalence, which we call von Neumann equivalence, and show that it encapsulates both measure equivalence and W*-equivalence. We will also discuss the stability of many group approximation properties under von Neumann equivalence, particularly, that of the new and a wide class of groups called properly proximal groups, introduced by Rémi Boutonnet, Adrian Ioana, and Jesse Peterson, and thereby obtaining first examples of non-inner-amenable, non-properly proximal groups. This is based on joint work with Jesse Peterson and Lauren Ruth.

David Jekel (UC San Diego) - A non-commutative optimal coupling problem - slides, video

We describe a free probabilistic analog of the Monge-Kantorovich optimal coupling problem that has connections with quantum information theory. Consider two tracial von Neumann algebras (A,τ_{A}) and (B,τ_{B}) generated by X = (X_{1},...,X_{m}) \(\in A_{s.a.}^m\) and Y = (Y_{1},...,Y_{m}) \(\in B_{s.a.}^m\) respectively. Following Biane and Voiculescu, a coupling of X and Y consists of a tracial von Neumann algebra (C,τ_{C}) and trace-preserving unital *-homomorphisms i: A → C and j: B → C. The cost of the coupling is ||i(X) - j(Y)||\(_{L^2(C,\tau_C)^m}\), and a coupling is said to be optimal if it achieves the minimal possible cost. By connecting optimal couplings with certain completely positive maps and using a result of Musat and Rordam, we show that for every n and d there exist m-tuples X and Y in A = B = M_{n}(\(\mathbb{C}\)) such that any algebra (C,τ_{C}) used for an optimal coupling of X and Y must have dimension at least d. In fact, based on the negative answer to the Connes-embedding problem (Ji, Natarajan, Vidick, Wright, Yuen) and a result of Haagerup and Musat, we conclude there are m-tuples of self-adjoint matrices which require a non-Connes-embeddable von Neumann algebra to optimally couple. We propose as a problem to young researchers in quantum information theory to study explicit examples of matrix tuples and their optimal couplings.

Krishnendu Khan (University of Iowa) - The fundamental group of certain property (T) factors - video

The calculation of fundamental group of type II_{1} factor is, in general, an extremely hard and central problem in the field of von Neumann algebras. In this direction, a conjecture due to A. Connes states that the fundamental group of the group von Neumann algebra associated to any icc property (T) group is trivial. Up to now there was no single example of property (T) factor satisfying the conjecture. In this talk, I shall provide the first examples of property (T) group factors (arising from group theoretic constructions) with trivial fundamental group. This talk is based on a joint work with Ionut Chifan, Sayan Das and Cyril Houdayer.

Kang Li (Leuven) - Rigidity of Roe algebras - slides, video

(Uniform) Roe algebras are C*-algebras associated to metric spaces, which reflect coarse properties of the underlying metric spaces. It is well-known that if X and Y are coarsely equivalent metric spaces with bounded geometry, then their (uniform) Roe algebras are (stably) *-isomorphic. The rigidity problem refers to the converse implication. The first result in this direction was provided by Ján Špakula and Rufus Willett, who showed that the rigidity problem has a positive answer if the underlying metric spaces have Yu's property A. I will in this talk review all previously existing results in literature, and then report on the latest development in the rigidity problem. This is joint work with Ján Špakula and Jiawen Zhang.

Shirin Moein (Mount Allison University) - On the stochastic operators on L^{1} - slides, video

In this lecture, we consider positive, integral-preserving linear operators acting on L^{1} space, known as stochastic operators or Markov operators. Stochastic operators are intimately connected to majorization theory. We show that, on finite-dimensional spaces, any stochastic operator can be approximated by a sequence of stochastic integral operators. We unify some of the literature on matrix majorization and multivariate majorization and give new results in the setting of vector valued functions on L^{1}.

Sanaz Pooya (IMPAN) - Higher Kazhdan projections, L²-Betti numbers, and Baum-Connes conjectures - slides, video

The Baum-Connes conjecture, if it holds for a certain group, provides topological tools to compute the K-theory of its reduced group C*-algebra. This conjecture has been confirmed for large classes of groups, such as amenable groups, but also for some Kazhdan’s property (T) groups. Property (T) and its strengthening are driving forces in the search for potential counterexamples to the conjecture. Having property (T) for a group is characterised by the existence of a certain projection in the universal group C*-algebra of the group, known as the Kazhdan projection. It is this projection and its analogues in other completions of the group ring, which obstruct certain method of proof for the Baum-Connes conjecture. In this talk, we will introduce a generalisation of Kazhdan projections. Employing these projections we establish a link between surjectivity of the (coarse) Baum-Connes assembly map and calculations of L²-Betti numbers of the group. This is based on joint work with Kang Li and Piotr Nowak.

Anshu Nirbhay (IISER Bhopal) - General stable rank of C*-algebras - slides, video

The goal of this talk is to discuss the general stable rank of a C*-algebra. The notion of general stable rank for C*-algebras was introduced by Marc Rieffel in the pursuit of understanding the stability properties of C*-algebras. We will provide estimates of general stable rank for upper semicontinuous C(X)-algebras and crossed product C*-algebras by finite groups. We will also prove that if A has general stable rank one, then the crossed product C*-algebra by an action with the Rokhlin property also has general stable rank one.

Pawel Sarkowicz (University of Ottawa) - Unitary groups and traces - slides, video

The unitary group of a C*-algebra is quite a large invariant which contains plenty of information the about algebra, and there are many classes of operator algebras where the unitary group can determine the algebra up to isomorphism. For a C*-algebra with nice regularity properties, the K-theory can be computed from the homotopy groups of the unitary group, and the (continuous affine functions on the) trace simplex can be computed "up to K_{0}" as a quotient of the connected component of the identity. With this in mind, any continuous group homomorphism between unitary groups (of sufficiently nice C*-algebras) gives rise to a map between spaces of affine functions on the trace simplices "up to K_{0}". We discuss an attempt at lifting this map.

Sushil Singla (Shiv Nadar University) - Orthogonality and the Gateaux derivative of the C* norm - slides, video

Birkhoff-James orthogonality is a generalization of Hilbert space orthogonality to normed spaces. In a given normed space V, an element v is said to be Birkhoff-James orthogonal to a subspace W if ||v||≤||v-w|| for all w in W. We characterize Birkhoff-James orthogonality of an element a in A to a subspace B of A. We will provide two different proofs of the following characterization - a is Birkhoff-James orthogonal to B if and only if there exists a state ϕ on A such that ϕ(a*b) = 0 for all b in B and ϕ(a*a = ||a||^{2}.

The first proof uses a characterization of bounded linear functionals on A in terms of cyclic representations on A. In the second proof, we use tools of convex analysis. Consider the function g(λ)=||v+λw||, mapping F into R_{+}. Since ||.|| is a convex function, lim_{t→0+} \(\frac{g(t)-g(0)}{t}\) always exists, known as Gateaux derivative of ||.|| at v. We shall give an expression for the Gateaux derivative of the C* norm in terms of states on A. As a consequence, we will get the proof for the characterization of orthogonality of a to B.

Ian Thompson (University of Manitoba) - Maximal C*-covers and residual finite-dimensionality - slides, video

A (possibly non-self-adjoint) operator algebra possesses many different fluctuations up to complete isometric isomorphism. The collection of C*-covers can describe these fluctuations. Recently, the C*-covers of a fixed operator algebra have been identified as a complete lattice. Here, we equate this lattice structure with another lattice arising from the spectrum of the so-called maximal C*-cover. This allows us to provide applications on residual finite-dimensionality (RFD) and a generalization of Hadwin’s characterization of separable RFD C*-algebras.

Gerrit Vos (TU Delft) - Noncommutative BMO spaces in sigma-finite von Neumann algebras - slides, video

In classical analysis, BMO spaces have many uses, for example as endpoints of complex interpolation. One can define such BMO spaces also in the noncommutative setting, using non-commutative Lp-spaces. In this presentation we give a brief introduction to noncommutative Lp-spaces in the 'easier' case of finite von Neumann algebras and the 'slightly harder' case of sigma-finite von Neumann algebras. We consider how these cases affect the definition of the corresponding BMO space. We extend some properties of BMO spaces from the finite to the sigma-finite case, such as the existence of a predual and interpolation results. These results can be applied to obtain Lp-bounded Fourier-Schur multipliers on the quantum group SU_{q}(2), but due to time reasons we will not discuss this here. This is joint work with Martijn Caspers.