## Operator Algebras Seminar Spring 2016

Organizers: Marius Dadarlat, Andrew Toms, and Thomas Sinclair

Thursday, 2:30-3:30pm, MATH 731

• Date: 1/14/16
• Organizational Meeting
• Title:

• Date: 1/21/16
• Thomas Sinclair, Purdue University
• Title: A short proof of Gromov's polynomial growth theorem (after Ozawa)
• A famous theorem of Gromov from the early 1980s shows that a finitely generated group of polynomial volume growth is virtually nilpotent. I will discuss a recent short proof of Gromov's theorem via functional analysis due to Ozawa, extending ideas from Shalom and Chifan-Sinclair.

• Date: 1/28/16
• Title:

• Date: Tuesday, 2/2/2016 **Colloquium, 4:30-5:30 in MATH 175**
• Alex Furman, University of Illinois at Chicago
• Title: Super-rigidity - from arithmeticity to actions on manifolds
• A remarkable phenomenon discovered by G.A.Margulis in the 1970s, now called super-rigidity, concerns linear representations of discrete subgroups groups in such groups as SL(3,R). As one of the applications of super-rigidity Margulis proved arithmeticity of these groups.
In the 1980s R.J.Zimmer extended Margulis' work to the context of cocycles, that found many applications in Ergodic Theory and Geometry of actions on manifolds.
In the talk I plan to give a broad overview of some of these topics and discuss some recent developments.

• Date: 2/4/16
• Andrew Schneider, Purdue University
• Title: Quasidiagonality and Property (T) for Group C$^*$-algebras
• Certain groups with Kazhdan's Property (T) produce the only known examples of groups whose full group C$^*$-algebra is not quasidiagonal. I will provide a direct proof of this and a review of quasidiagonality and Kazhdan's Property (T) from an operator algebraic perspective. Open questions regarding the relationship between Property (T) and other finite dimensional approximation properties will also be explored.

• Date: 2/11/16
• Rolando de Santiago, University of Iowa
• Title: Product rigidity for the von Neumann algebras of product of hyperbolic groups
• Suppose $\Gamma_1,\ldots, \Gamma_n$ are each hyperbolic i.c.c. groups and $L(\Gamma_1\times\cdots\times \Gamma_n) \cong L(\Lambda)$ for an arbitrary group $\Lambda$. We show $\Lambda$ decomposes as an $n$-fold product of i.c.c. groups, $\Lambda= \Lambda_1\times\cdots\times \Lambda_n$ such that for each $i=1,\ldots, n$, we have \$L(\Gamma_i)\cong L(\Lambda_i)$ up to amplification. This strengthens Ozawa and Popa's unique prime decomposition results.

• Date: 2/18/16
• Isaac Goldbring, University of Illinois at Chicago
• Title: Revisiting the first-order theories of McDuff's II$_1$ factors
• McDuff was the first to provide a family of continuum many pairwise-nonisomorphic separable II$_1$ factors. In a recent preprint, Boutonnet, Chifan, and Ioana proved that any ultrapowers of any two distinct McDuff examples are also nonisomorphic. As a result, this shows that McDuff's examples are also pairwise non-elementarily equivalent, thus settling the question of how many first-order theories of II$_1$ factors there are. From the model-theoretic point of view, this resolution of the question is not satisfying as we do not see an explicit family of sentences that distinguish the McDuff examples. In this talk, I will present a partial resolution to this problem by discussing the following result: If $M_\alpha$ and $M_\beta$ are two of McDuff's examples, where $\alpha,\beta \in 2^{\omega}$ are such that $\alpha|k=\beta|k$ but $\alpha(k)\not=\beta(k)$, then there must exist a formula of quantifier-complexity at most $5k+3$ on which they disagree. The proof uses Ehrenfeucht-Fraisse games. The talk represents joint work with Bradd Hart.

• Date: Tuesday, 2/23/2016 **Colloquium, 4:30-5:30 in MATH 175**
• Noah Snyder, Indiana University
• Title: Subfactors and their classification
• Subfactors are inclusions of von Neumann factors, and play a similar role in operator algebras that Galois theory plays in ring theory. Each subfactor has an index, analogous to the degree of a field extension, but these indices do not need to be integers. The celebrated Jones index theorem says that among subfactors of index below 4, only a discrete sequence of index values can happen. This suggests that there may be some hope of classifying subfactors. Such a classification splits into two steps, one largely algebraic and the other largely analytic. The algebraic step is to classify certain "quantum group"-like objects that play the role of Galois groups. The analytic step is to then understand how many ways each of these "quantum groups" can act on a particular factor. I'll mainly focus on the algebraic part of the classification, which is now known up to index 5.25, and the examples that appear in this classification. I'll also briefly discuss what's known about the analytic part of the classification in the case of the hyperfinite II$_1$ factor, based on deep results of Ocneanu and Popa.

• Date: 2/25/16
• Mike Hartglass, UC Riverside
• Title: Graphs and standard invariants as compact quantum metric spaces
• Given an undirected, weighted graph, I will associate a C*-algebra which arises by applying Shlyakhtenko's operator-valued generalization of Voiculescu's free Gaussian functor. I will show how these algebras give rise to compact quantum metric spaces in the sense of Rieffel, and illustrate that certain convergence properties of families of graphs give convergence in quantum Gromov-Hausdorff distance. Time permitting, I will then sketch an application to subfactor theory. This is joint work in progress with Dave Penneys.

• Date: Saturday, 2/27/2016 **Wabash Seminar, Wabash College, Crawfordsville, IN **

• Date: 3/3/16
• Title:

• Date: 3/10/16
• Title:

• Date: 3/17/16 **Spring Break, No Seminar**

• Date: 3/24/16
• Kaushika De Silva, Purdue University
• Let A be a C*-algebra. It is well known that the state space of the Cuntz semigroup W(A) is naturally identified with the space DF(A) of all dimension functions on A. It is conjectured (Blackadar and Handelman) that for any C*-algebra the set of lower semicontinuous dimension functions on A (LDF(A)) is pointwise dense in DF(A) and that DF(A) is a Choquet simplex. We will look at an alternate criterion for density of LDF(A) in DF(A) for unital A and show that this criterion holds if A has Finite radius of comparison. We will also discuss a weaker version of classical Riesz interpolation property for ordered semigroups and confirm DF(A) to be a Choquet simplex if A is unital and W(A) satisfy this weaker version of interpolation.

• Date: 3/31/16
• Cornel Pasnicu, UT San Antonio
• Title: The weak ideal property and topological dimension zero
• Following up on previous work, we prove a number of results for C*-algebras with the weak ideal property or topological dimension zero, and some results for C*-algebras with related properties. Some of the more important results include:
1. The weak ideal property implies topological dimension zero.
2. For a separable C*-algebra~$A$, topological dimension zero is equivalent to ${\operatorname{RR}} ({\mathcal{O}}_2 \otimes A) = 0$, to $D \otimes A$ having the ideal property for some (or any) Kirchberg algebra~$D$, and to $A$ being residually hereditarily in the class of all C*-algebras $B$ such that ${\mathcal{O}}_{\infty} \otimes B$ contains a nonzero projection.
3. Extending the known result for ${\mathbb{Z}}_2$, the classes of C*-algebras with residual (SP), which are residually hereditarily (properly) infinite, or which are purely infinite and have the ideal property, are closed under crossed products by arbitrary actions of abelian $2$-groups.
4. If $A$ and $B$ are separable, one of them is exact, $A$ has the ideal property, and $B$ has the weak ideal property, then $A \otimes_{\mathrm{min}} B$ has the weak ideal property.
5. If $X$ is a totally disconnected locally compact Hausdorff space and $A$ is a $C_0 (X)$-algebra all of whose fibers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then $A$ also has the corresponding property (for topological dimension zero, provided $A$ is separable).
6. Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C*-algebras including all separable locally AH~algebras.
7. The weak ideal property does not imply the ideal property for separable $Z$-stable C*-algebras.
We give other related results, as well as counterexamples to several other statements one might hope for. This is joint work with N. Christopher Phillips.

• Date: 4/7/16
• Title:

• Date: Tuesday, 4/12/2016 **Colloquium, 4:30-5:30 in MATH 175**
• Magdalena Musat, University of Copenhagen
• Title: Quantum Information Theory and the Connes Embedding Problem
• In 1980, Tsirelson showed that Bell's inequalities---that have played an important role in distinguishing classical correlations from quantum ones, and that were used to test, and ultimately disprove the Einstein-Podolski-Rosen postulate of hidden variables", coincide with Grothendieck's famous inequalities from functional analysis. Tsirelson further studied sets of quantum correlations arising under two different assumptions of commutativity of observables. While he showed that they are the same in the finite dimensional case, the equality of these sets was later proven to be equivalent to the most famous still open question in operator algebras theory: the Connes embedding problem. In recent joint work with Haagerup, we establish a different reformulation of the Connes embedding problem in terms of an asymptotic property of quantum channels posessing a certain factorizability property (that originates in operator algebras). Several concrete examples will be discussed.

• Date: 4/14/16
• Ionut Chifan, University of Iowa
• Title: Product rigidity for von Neumann algebras arising from hyperbolic groups
• Two groups $\Gamma$ and $\Lambda$ and are called $W^*$-equivalent if they give rise to isomorphic von Neumann algebras. I will show that whenever $\Gamma_1$, $\Gamma_2$, $\ldots$, $\Gamma_n$ are icc hyperbolic groups and $\Lambda$ is an arbitrary group such that $\Gamma_1\times \Gamma_2\times \cdots \times \Gamma_n$ is W-equivalent to $\Lambda$ it follows that $\Lambda = \Lambda_1\times \Lambda_2\times \cdots \times \Lambda_n$ and, up to amplications, $\Gamma_i$ is $W^*$-equivalent to i, for all i. This strengthens some results of N. Ozawa and S. Popa from 2003. The talk based on a joint work with Rolando de Santiago and Thomas Sinclair.

• Date: Saturday, 4/16/2016 **Wabash Seminar, Wabash College, Crawfordsville, IN **

• Date: 4/21/16
• Hyun Ho Lee, University of Uslan
• Title: Projection lifting from the corona algebra using UCT
• We try to solve the projection lifting problem from the corona algebra of C(X,B). We consider the cases that B is a compact algebra, or a Kirchberg algebra.

• Date: 4/28/16
• Title:

• End of Spring Semester.