# von Neumann algebra The classification of von Neumann algebras provides a way to understand the classification of entanglement patterns in quantum physics. A $*$-subalgebra, $\mathcal{A}$, is a **von Neumann algebra** if it is equal to its own double commutant, where a $*$-subalgebra is one that is closed under scalar multiplication, operator multiplication, operator addition, and adjoints, and contains the identity operator. This definition is powerful because the double commutant is easier to work with and think about than closure under the weak limit, which is an alternative definition of the von Neumann algebra. By the bi-commutant theorem, however, these two definitions are equivalent. $\mathcal{A}$ is a *factor* if the center is trivial, i.e. contain only (multiples of) the identity operator. A factor can be classified by properties of its projection operators. See below for the classification. Type I algebra is the operator algebra of ordinary QM, but the algebra of observables of a local region in QFT is type III, which does not have pure states, density matrices, or von Neumann entropies. A type II algebra does not have pure states but has density matrices and von Neumann entropies. ## Refs - introductory material - [[2018#Witten (Mar, Review)]] - [[2021#Witten (Dec, a, Review)]] - [[2023#Sorce (Feb, Notes)]] - recent developments - [[2021#Leutheusser, Liu (Oct)]] - [[2022#Chandrasekaran, Penington, Witten]] - renormalisation of Newton's constant - [[2023#Gesteau]] - [[0050 JT gravity|JT gravity]] - [[2023#Penington, Witten]] - [[2023#Kolchmeyer]] - relation to the decay of two-point function - [[2023#Furuya, Lashkari, Moosa, Ouseph]] - for general subregions - [[2023#Jensen, Sorce, Speranza]] - [[2023#Ahmad, Jefferson]] - [[2023#Banks]] - general background (not necessarily KMS) - [[2023#Kudler-Flam, Leutheusser, Satishchandran]] - background independent - [[2023#Witten (Aug)]] - trace inequality in holography - [[2023#Colafranceschi, Marolf, Wang]] - path integral understanding of RT without holography - [[2023#Colafranceschi, Dong, Marolf, Wang]] - covariant entropy, [[0405 Quantum null energy condition|QNEC]] and [[0418 Bekenstein bound|Bekenstein bound]] - [[2023#Kudler-Flam, Leutheusser, Rahman, Satishchandran, Speranza]] - crossed products and relation to [[0044 Extended phase space|extended phase space]] - [[2023#Klinger, Leigh (Jun)]] - [[2023#Klinger, Leigh (Dec)]] - boundary studies - [[2024#Basteiro, Di Giulio, Erdmenger, Xian]] - entropy difference and relative state counting - [[2024#Akers, Sorce]] - Page curve - [[2024#Gomez]] - observer dependence and quantum reference frames (QRF) - [[2024#De Vuyst, Eccles, Hoehn, Kirklin]] - semifiniteness - [[2024#Ahmad, Klinger, Lin]] - super-JT - [[2024#Penington, Witten]] - time-band algebras, coarse-graining - [[2024#Jensen, Raju, Speranza]] ## Classification - I$_n$ ($n\in \mathbb{Z}_+$): contains projectors of ranks $0,1,\dots,n$ - I$_\infty$: contains projectors of ranks $0,1,\dots,\infty$ - II$_1$: renormalised rank $\in[0,1]$ - II$_\infty$: renormalised rank $\in[0,\infty)$ - III: ranks either $0$ or $\infty$; trace does not exist - III$_0$, III$_\lambda$ ($\lambda\in(0,1)$), III$_\infty$ ## Properties - trace exists for Type I and II but not III