Quasinormalizers in crossed products of von Neumann algebras
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909135288664064 |
|---|---|
| author | Bannon, Jon Cameron, Jan Chifan, Ionut Mukherjee, Kunal Smith, Roger Wiggins, Alan |
| author_facet | Bannon, Jon Cameron, Jan Chifan, Ionut Mukherjee, Kunal Smith, Roger Wiggins, Alan |
| contents | We study the relationship between the dynamics of the action $α$ of a discrete group $G$ on a von Neumann algebra $M$, and structural properties of the associated crossed product inclusion $L(G) \subseteq M \rtimes_αG$, and its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion $L(G) \subseteq M \rtimes_αG$ such as the Haagerup Approximation Property, and is essential to capturing "almost periodic" behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorems for general, tracial $W^*$-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02735 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quasinormalizers in crossed products of von Neumann algebras Bannon, Jon Cameron, Jan Chifan, Ionut Mukherjee, Kunal Smith, Roger Wiggins, Alan Operator Algebras 46L10, 46L55, 37A55 We study the relationship between the dynamics of the action $α$ of a discrete group $G$ on a von Neumann algebra $M$, and structural properties of the associated crossed product inclusion $L(G) \subseteq M \rtimes_αG$, and its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion $L(G) \subseteq M \rtimes_αG$ such as the Haagerup Approximation Property, and is essential to capturing "almost periodic" behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorems for general, tracial $W^*$-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras. |
| title | Quasinormalizers in crossed products of von Neumann algebras |
| topic | Operator Algebras 46L10, 46L55, 37A55 |
| url | https://arxiv.org/abs/2308.02735 |