Operator Algebras Research Seminar

Title:  Non-Isomorphism Results for q-Araki-Woods Factors

Abstract:  Hiai’s construction of q-Araki-Woods factors ('02) unifies Free Araki-Woods factors (Shlyakhtenko, '97) and q-Gaussian algebras (Bożejko and Speicher, '91). In this talk, I will explain some non-isomorphism results of q-Araki-Woods factors, in the case of finite and infinite dimensional representations, that distinguish them from the free case. This is based on joint work with Changying Ding.

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Title:  SAD Neural Networks

Abstract:  When training artificial neural networks to fit a given polynomial function using gradient descent, one frequently observes that the parameters of the network diverge to infinity although the function approximation achieves arbitrary precision. After giving an overview to neural networks and their theoretical analysis, we will explain this phenomenon mathematically using the geometry of o-minimal structures in the sense of model theory.

Title:  Elementary equivalence and disintegration of tracial von Neumann algebras

Abstract:  We prove an analog of the disintegration theorem for tracial von Neumann algebras in the setting of elementary equivalence rather than isomorphism, showing that elementary equivalence of two direct integrals implies fiberwise elementary equivalence under mild, and necessary, hypotheses. This verifies a conjecture of Farah and Ghasemi. Our argument uses a continuous analog of ultraproducts where an ultrafilter on a discrete index set is replaced by a character on a commutative von Neumann algebra, which is closely related to Keisler randomizations of metric structures. We extend several essential results on ultraproducts, such as Łoś’s theorem and countable saturation, to this more general setting. This talk is based on joint work with David Jekel.

Title:  Structure and non-isomorphisms of q-Araki-Woods factors

Abstract:  Hiai’s construction of q-Araki-Woods factors generalizes both Shlyakhtenko’s free Araki-Woods factors and Bozejko-Speicher’s q-Gaussian algebras. In this talk, I will present that these q-Araki-Woods factors are strongly solid when almost periodic. Under a certain spectrum condition of the associated representation, I will show that q-Araki-Woods with infinite variables are not isomorphic to free Araki-Woods factors. This is a joint work with Hui Tan.

Title:  Von Neumann equivalence and group approximation properties

Abstract:  One of the themes in measured group theory is studying the stability of group approximations properties such as amenability, Haagerup property, property (T) etc. under measure equivalence which is an equivalence relation on the the class of groups. Not a long time ago the notion of von Neumann equivalence, which generalizes measure equivalence, was introduced by Jesse Peterson, Lauren Ruth, and myself. In this talk I will discuss the stability of some group approximation properties under von Neumann equivalence.

Title:  Projective representations of almost unimodular groups

Abstract:  Given a locally compact group G with a 2-cocycle ω: G × G → T , Colin Sutherland showed that any left Haar measure uniquely determines a faithful normal semifinite weight on the associated twisted group von Neumann algebra. This weight, which we call the twisted Plancherel weight, is tracial if and only if G is unimodular, and for countable discrete groups it is the usual tracial state. In the setting of non-unimodular groups, the modular automorphism group of the necessarily non-tracial twisted Plancherel weight is explicitly determined by the so-called modular function of G. The twisted group von Neumann algebra is generated by the left regular ω-projective representation of G. In 1958, George Mackey showed that ω-projective representations of G are connected to the representations of the central extension of G by T, when G is second countable. In this talk, we will introduce the class of "almost unimodular groups" for which the twisted Plancherel weight is almost periodic, in the sense of Connes from 1972.  We will also give some examples of such groups admitting a 2-cocycle such that the group von Neumann algebras are purely infinite and not factors, but the twisted group von Neumann algebras are semifinite factors.

Title:  Ultrapowers of free group C*-algebras

Abstract:  I will discuss recent work with Sri Kunnawalkam Elayavalli on the structure of the reduced group C*-algebras of free groups.  In particular, I will discuss the problem of when such C*-algebra have the same first order theory, or equivalently, when they have isomorphic ultrapowers.

Title:  The W* and C*-algebras of Similarity Structure Groups

Abstract:  Countable Similarity Structure (CSS) groups are a class of generalized Thompson groups. I will introduce CSS* groups, a subclass, that we prove to be non-acylindrically hyperbolic, that includes the Higman-Thompson groups Vd,r , the countable Röver-Nekrashevych groups Vd(G), and the topological full groups of subshifts of finite type of Matui. I will discuss how all CSS* groups give rise to prime group von Neumann algebras, which greatly expands the class of groups satisfying a previous deformation/rigidity result. I will then discuss how CSS* groups are either C*-simple with a simple commutator subgroup, or lack both properties. This extends C*-simplicity results of Le Boudec and Matte Bon and recovers the simple commutator subgroup results of Bleak, Elliott, and Hyde. This is joint work with Eli Bashwinger.

Title:  Almost unimodular groups

Abstract:  Just like discrete groups, locally compact groups provide a rich source of von Neumann algebras through their left regular representations. However, by moving beyond the discrete setting one loses the canonical tracial state on these algebras and must instead work with an infinite weight called the Plancherel weight. This weight can even have non-trivial modular theory (i.e. be non-tracial), but this is fortunately controlled by the so-called modular function of the group: a continuous homomorphism of the group into the positive reals that serves as the Radon–Nikodym derivative between left and right Haar measures. One example of this control is that the Plancherel weight is tracial if and only if the group is unimodular in the sense that the modular function is identically one. In this talk, I will discuss the class of locally compact groups one obtains when traciality of the Plancherel weight is loosened to almost periodicity. The resulting class of course contains all unimodular groups (e.g. all discrete or compact groups), but also all totally disconnected groups. Moreover, the von Neumann algebras associated to the groups in this class are easier to study because the almost periodicity of the Plancherel weight ensures there is a large tracial subalgebra that determines much of the structure. This talk is based on joint work with Aldo Garcia Guinto and will not assume any prior knowledge of locally compact groups beyond the existence of a left Haar measure. 

Title:  Planar algebras associated to cocommuting squares of II_1 subfactors

Abstract:  Planar algebras are a pictorial invariant of subfactors that describe 'generalized symmetry' of subfactors. They also serve as an invariant that connects subfactors to knot theory, category theory, statistical mechanics, representation theory, combinatorics, quantum groups, and other areas. Consequently, the classification of the planar algebras has been of interest across various fields of mathematics. We introduce planar algebras associated to quadrilateral inclusions of subfactors that have group-like properties and show their connection to the set partitions and the representations of the symmetric groups. This is based on a joint work with Dietmar Bisch. No background in subfactors or category theory is assumed.

Title:  Shlyakhtenko's operator-valued semicircular construction and applications 

Title:  On conjugate systems with respect to completely positive maps

Abstract:  In 2010, Dabrowski showed that a von Neumann algebra generated by self-adjoint operators is a factor when they admit a conjugate system. We extend this to the operator-valued case by defining an operator valued partial derivative and conjugate systems with respect to completely positive maps.

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