Dissertation
Rigidity results for group von Neumann algebras with diffuse center
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2025
DOI: 10.25820/etd.007856
Abstract
This work compiles structural results for von Neumann algebras associated with groups having infinite center, established through a series of collaborations with I. Chifan, H. Tan, and D. Osin. These results provide explicit examples of groups where the infinite center can be recovered from the associated von Neumann algebra.
The primary result of our joint work with Chifan and Tan \cite{cfqt23} can be summarized as follows: let $G=Z(G)\times W$, where the center $Z(G)$ of $G$ is infinite and $W$ is an ICC wreath-like product group \cite{cios22,amcos23} with property (T) and trivial abelianization. If $H$ is a group such that $\mathcal{L}(G)$ is $\ast$-isomorphic to $\mathcal L(H)$, then $H$ necessarily decomposes as a product of its infinite center and $W$. Furthermore, we explicitly describe the $*$-isomorphism between $\mathcal L(G)$ and $\mathcal L(H)$. This result leads to new applications in the classification of group C$^*$-algebras, including examples of non-amenable groups that are distinguishable from their reduced C$^*$-algebras but not from their von Neumann algebras.
In the subsequent part of this work \cite{cfqot25}, together with Chifan, Osin, and Tan, we propose a natural extension of Connes' Rigidity Conjecture (1982) to the setting of property (T) groups with infinite center. Using methods at the rich intersection between von Neumann algebras and geometric group theory, we identify several instances (specifically, nonsplit central extensions of the form $G=Z(G)\rtimes_cW$) where this conjecture holds. In particular, we provide the first examples of W$^*$-superrigid property (T) groups with infinite center. In the course of proving our main results, we also generalize the main W$^*$-superrigidity result from \cite{cios22} to twisted group factors.
Details
- Title: Subtitle
- Rigidity results for group von Neumann algebras with diffuse center
- Creators
- Adriana Fernández I Quero
- Contributors
- Ionut Chifan (Advisor)Benjamin Cooper (Committee Member)Raúl Curto (Committee Member)Daniel Drimbe (Committee Member)Palle Jorgensen (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2025
- DOI
- 10.25820/etd.007856
- Publisher
- University of Iowa
- Number of pages
- vi, 114 pages
- Copyright
- Copyright 2025 Adriana Fernández I Quero
- Language
- English
- Date submitted
- 04/11/2025
- Description bibliographic
- Includes bibliographical references (pages 109-114).
- Public Abstract (ETD)
- Von Neumann algebras, introduced by John von Neumann in his work on quantum mechanics, provide a framework for studying complex mathematical structures like groups. A central question in this field is understanding how the properties of these algebras reflect the underlying groups that generate them. Specifically, when two von Neumann algebras are equivalent but arise from different groups–referred to as the source and target groups–what shared features can be identified, and to what extent can the groups themselves be reconstructed? This dissertation focuses on product groups where one component is commutative, and the other is highly rigid, meaning its structural properties are fully captured by the von Neumann algebra. We demonstrate that if the source is such a product group and its von Neumann algebra is equivalent to that of a target group, the target group can be decomposed into a product group with the same characteristics as the source. This represents a significant step forward in understanding the interplay between group structure and von Neumann algebras, marking the first study of such groups in this context. Additionally, we contribute to the classification of reduced group C∗-algebras by showing that the groups considered are fully reconstructible from these algebras. These results highlight novel forms of rigidity, providing the first known examples of non-amenable groups with this behavior. In the second part of this dissertation, we extend these results to identify new families of groups with commutative components whose structural properties are entirely preserved at the level of their von Neumann algebras.
- Academic Unit
- Mathematics
- Record Identifier
- 9984831231202771
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