## Operator Algebras Seminar Fall 2015

Organizers: Marius Dadarlat, Andrew Toms, and Thomas Sinclair

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

• Date: 8/27/15
• Thomas Sinclair, Purdue University
• Title: W$^\ast$-Rigidity for the von Neumann Algebras of Products of Hyperbolic Groups
• Abstract: We show that if $\Gamma = \Gamma_1\times\cdots\times \Gamma_n$ is a product of $n\geq 2$ non-elementary ICC hyperbolic groups then any discrete group $\Lambda$ which is W$^\ast$-equivalent to $\Gamma$ decomposes as a $k$-fold direct sum exactly when $k=n$. This gives a group-level strengthening of Ozawa and Popa's unique prime decomposition theorem by removing all assumptions on the group $\Lambda$. This result in combination with Margulis' normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II$_1$ factors. This is joint work with Ionut Chifan and Rolando de Santiago.

• Date: 9/3/15
• Thomas Sinclair, Purdue University
• Title: W$^*$-Rigidity for the von Neumann Algebras of Products of Hyperbolic Groups, II
• Abstract: We show that if $\Gamma = \Gamma_1 \times\cdots\times \Gamma_n$ is a product of $n\geq 2$ non-elementary ICC hyperbolic groups then any discrete group $\Lambda$ which is W$^*$-equivalent to $\Gamma$ decomposes as a $k$-fold direct sum exactly when $k=n$. This gives a group-level strengthening of Ozawa and Popa's unique prime decomposition theorem by removing all assumptions on the group $\Lambda$. This result in combination with Margulis' normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II$_1$ factors. This is joint work with Ionut Chifan and Rolando de Santiago.

• Date: 9/10/15
• Hung-Chang Liao, Pennsylvania State University
• Title: A Rokhlin Type Theorem for Automorphisms of Simple $\mathbf C^*$-algebras of Finite Nuclear Dimension
• Abstract: Following Connes' work on automorphisms of the hyperfinite II$_1$ factor and Kishimoto's work on automorphisms of AF and $A\mathbb T$ algebras, we prove a Rokhlin type theorem for automorphisms of unital simple separable stably finite $C^*$-algebras of finite nuclear dimension. More precisely, under suitable assumptions on the trace space and/or the induced action on it, we show that strongly outer $\mathbb Z$-actions have finite Rokhlin dimension. In particular the crossed product formed by such an action has finite nuclear dimension. This result is inspired by an unpublished work of Hiroki Matui and Yasuhiko Sato.

• Date: 9/12-9/13/15

• Date: Tuesday, 9/15/15 **Colloquium, 4:30-5:30 in MATH 175**
• Joachim Zacharias, University of Glasgow
• Title: Noncommutative Generalizations of Covering Dimension
• Abstract: By a well-known slogan, which we will explain in the talk, C$^*$-algebras can be regarded as noncommutative locally compact spaces. Classical covering dimension theory is a theory of dimension for locally compact or even general topological spaces based on combinatorial properties of arbitrarily small open covers. One can think of an open cover as an approximation of the space. It has thus been an important question to define a topological dimension concept for noncommutative algebras. Several different approaches to this problem have been suggested. We will explain one such approach which is more recent and based on approximation. The resulting dimension concepts have turned out to be of great importance in the structure theory of C$^*$-algebras. There is also a dimension theory for dynamical systems which has some interesting connections to the so-called coarse geometry. We will sketch some of these developments in an elementary fashion without requiring too special background knowledge.

• Date: 9/17/15
• No Seminar due to Colloquium

• Date: 9/24/15
• Title: Deformations of groups, C$^*$-algebras and K-homology
• Abstract: Let G be a countable torsion free nilpotent group and let I(G) be the kernel of the trivial representation $\iota:C^*(G)\to \mathbb{C}$. We show that the K-homology group of I(G) is isomorphic to the homotopy classes of asymptotic homomorphisms $\{\phi_t:I(G)\to M_\infty(\mathbb{C})\}_{t\in [1,\infty)}$. This is joint work with Ulrich Pennig.

• Date: 10/1/15
• Wilhelm Winter, University of Münster
• Title: QDQ vs. UCT
• Abstract: None.

• Date: 10/3/15 - 10/4/15

• Date: 10/8/15
• Title:
• Abstract:

• Date: 10/10/15 - 10/11/15

• Date: 10/15/15

• Date: 10/22/15
• Martin Christensen, University of Copenhagen
• Title: Regularity and comparability properties of Villadsen algebras
• Abstract: In a recent article, Kirchberg and Rørdam asked the following question: Suppose that A is a unital and separable $C^*$-algebra. Does it follow that A absorbs the Jiang-Su algebra tensorially if and only if the central sequence algebra of A has no characters? Relying on work of several authors, most notably Kirchberg-Rørdam, Toms-Winter and Villadsen we show that this question has an affirmative answer when A is a unital and simple Villadsen algebras of either the first or second type. Time permitting we will also outline the construction of a Villadsen algebra of the second type which is unital, simple, separable, nuclear and has a unique tracial state and yet fails to satisfy the (strong) Corona Factorization Property.

• Date: 10/29/15
• Title:
• Abstract:

• Date: 11/5/15
• Brent Nelson, UC Berkeley
• Title: An application of free transport to mixed $q$-Gaussian algebras (joint with Qiang Zeng)
• Abstract: Given a symmetric array $Q=(q_{ij})\in M_{N\times N}([-1,1])$, Bo\.zejko and Speicher studied the commutation relation $l_i^*l_j - q_{ij}l_j l_i^*=\delta_{i=j}$ and showed it is satisfied by left creation operators $l_1,\ldots, l_N$ on a particular Fock space. Then the mixed $q$-Gaussian algebra $\Gamma_Q$, which is the von Neumann algebra generated by the operators $X_i=l_i+l_i^*$, can be thought of as a deformation of the free group factor $L(\mathbb{F}_N)$, which corresponds to the zero array. It is known that these deformations share various properties with the free group factors, and recently Guionnet and Shlyakhtenko showed that $\Gamma_Q\cong L(\mathbb{F}_N)$ when $Q=(q)$ is a constant array with $|q|$ less than a constant determined by $N$. They used a novel method called \emph{free transport} and relied on work of Dabrowski, who established that particular elements known as \emph{conjugate variables} could be represented as power series in $X_1,\ldots, X_N$ for sufficiently small $|q|$. In this talk, I will show that these techniques can be applied to a general mixed $q$-Gaussian algebra and that $\Gamma_Q\cong L(\mathbb{F}_N)$ provided $\max |q_{ij}|$ is less than a constant determined by $N$.

• Date: 11/12/15
• Dan Li, Purdue University
• Title: Topological insulators and NCG
• Abstract: Topological insulators can be characterized by a $\mathbb{Z}_2$-valued topological invariant, which is called the topological $\mathbb{Z}_2$ invariant. This mod 2 topological invariant can be understood as an index theorem with a $\mathbb{Z}_2$ symmetry. I will give some background about the index theory of the topological $\mathbb{Z}_2$ invariant, then talk about its generalizations in noncommutative geometry (NCG).

• Date: 11/19/15
• Daniel Hoff, UC San Diego
• Title: Von Neumann's Problem and Extensions of Non-Amenable Equivalence Relations
• Abstract: In 2007, Gaboriau and Lyons showed that any nonamenable group $\Gamma$ has a free ergodic pmp action $\Gamma \curvearrowright X$ whose orbit equivalence relation $\mathcal{R}(\Gamma \curvearrowright X)$ contains $\mathcal{R}(\mathbb{F}_2 \curvearrowright X)$ for some free ergodic pmp action of $\mathbb{F}_2$. This talk will focus on joint work with Lewis Bowen and Adrian Ioana in which we extend this result, showing that given any ergodic nonamenable pmp equivalence relation $\mathcal{R}$, the Bernoulli extension $\tilde{\mathcal{R}}$ over a nonatomic base space must contain $\mathcal{R}(\mathbb{F}_2 \curvearrowright \tilde{X})$ for some free ergodic pmp action of $\mathbb{F}_2$. We then show that any such $\mathcal{R}$ admits uncountably many extensions $\{\tilde{\mathcal{R}}_\alpha\}_{\alpha \in A}$ which are pairwise not stably von Neumann equivalent. From this we deduce that any nonamenable unimodular lcsc group $G$ has uncountably many free ergodic pmp actions which are pairwise not von Neumann equivalent (in particular, pairwise not orbit equivalent).

• Date: 11/26/15
• No Meeting, Thanksgiving Break.

• Date: 12/3/15
• Chenxu Wen, Vanderbilt University
• Title: Popa's AOP and amenable extensions inside II$_1$ factors
• Abstract: Popa introduced the notion of asymptotic orthogonality property (AOP) and used it to show the maximal amenability of the generator masa inside a free group factor. Since then it has become the most successful approach in studying maximal amenable subalgebras. In this talk I will explain usually how a stronger version of AOP leads to examples of unique maximal amenable extension inside II$_1$ factors.

• Date: 12/10/15
• Title:
• Abstract:

• End of Fall Semester.