nLab von Neumann algebra factor

Definition

For $A$ a von Neumann algebra write $A'$ for its commutant in the ambient algebra $B(\mathcal{H})$ of bounded operators.

Definition

A von Neumann algebra $A$ is called a factor if its center is trivial, equivalently if $A$ and its commutant $A'$ generate the full algebra of bounded operators $B(\mathcal{H})$:

$$Z(A) \coloneqq A \cap A' = \mathbb{C}1 \,.$$

Properties

Every von Neumann algebra may be written as a direct integral over factors. (von Neumann 1849)

Classification

Factors are classified in terms of the K-theory of their categories of finite W*-modules?. A W*-module? over a factor $A$ is finite if it is not isomorphic to its proper submodule.

Type I

Type I factors are characterized by the condition that the K-theory of finite modules is isomorphic to $\mathbf{Z}$, the group of integers. The only factors of this type are of the from $B(H)$, bounded operators on a Hilbert space $H$.

Type II

Type II factors are characterized by the condition that the K-theory of finite modules is isomorphic to $\mathbf{R}$, the group of real numbers.

Type II factors are subdivided into two classes: type II$_1$ factors are characterized by the condition that $A$ is a finite $A$-module, whereas for a type II$_\infty$ factor $A$ is not a finite $A$-module.

Type III

Type III factors are characterized by the condition that the K-theory of finite modules is trivial, i.e., only the zero module is finite.

Type III factors are further subdivided into three classes, according to the structure of the center of their modular algebra?, which is a commutative von Neumann algebra graded by purely imaginary numbers, whose graded components are noncommutative L^p-spaces?.

By the von Neumann duality? for commutative von Neumann algebras, the spectrum of this center is a measurable space equipped with a σ-ideal of negligible sets and the grading yields an action of the additive group of real numbers $\mathbf{R}$. This action is known as the noncommutative flow of weights.

If the center is trivial (so that the spectrum is a point), then the factor has type III$_1$. If the action of $\mathbf{R}$ is not periodic, then the factor has type III$_0$. If the action is periodic with period $\lambda$, a positive real number, then the factor has type III$_{\exp(-\lambda)}$.

Related concepts

• C-star algebra

• uniformly hyperfinite algebra

References

General

The original articles:

The classification of factors into types I, II, III and the construction of examples not of type I:

• Francis J. Murray, John von Neumann, On Rings of Operators, Annals of Mathematics, Second Series, 37 1 (1936) 116-229 [doi:10.2307/1968693, jstor:1968693]

Discussion of traces on these factors:

• Francis J. Murray, John von Neumann, On rings of operators. II, Trans. Amer. Math. Soc. 41 (1937) 208-248 [doi:10.2307/1989620, jstor:1989620]

On isomorphism of factors and proof of a single isomorphism class of approximately finite type $II_1$ factors:

• Francis J. Murray, John von Neumann, On rings of operators. IV, Annals of Mathematics, Second Series, 44 4 (1943) 716-808 [doi:10.2307/1969107, jstor:1969107]

On decomposing von Neumann algebras as a direct integral of factors:

• John von Neumann, On rings of operators, reduction theory, Annals of Mathematics Second Series, Vol. 50, No. 2 (1949) [jstor:1969463]

The classification of type III factors:

• Alain Connes, Structure Theory for Type III Factors, Proceedings of the International Congress of Mathematics, Vancouver 1975, vol 2 (1975) [pdf]

Recollection of the history which made von Neumann turn to discussion of these “factors”, motivated from considerations in the foundations of quantum mechanics and quantum logic:

• Miklos Rédei, Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead), Studies in History and Philosophy of Modern Physics 27 4 (1996) 493-510 [doi:10.1016/S1355-2198(96)00017-2]

Lecture notes:

• V.S. Sunder, von Neumann algebras, $II_1$-factors, and their subfactors [pdf]

• Hideki Kosaki, Type III factors and index theory (1993) [pdf]

Exposition with an eye towards discussion of entanglement entropy:

• Jonathan Sorce, Notes on the type classification of von Neumann algebras [arXiv:2302.01958]

Subfactors

The mathematics of inclusions of subfactors is giving deep structural insights. See also at planar algebra.

• Vaughan Jones,

  Index for subfactors, Invent. Math. 72, I (I983);

  A polynomial invariant for links via von Neumann algebras, Bull. AMS 12, 103 (1985);

  Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126, 335 (1987)

• Vaughan Jones, Scott Morrison, Noah Snyder, The classification of subfactors of index at most 5 (arXiv:1304.6141)

• Vaughan F. R. Jones, David Penneys, Infinite index subfactors and the GICAR categories, arxiv/1410.0856

Symmetries of depth two inclusions of subfactors may be described via associative bialgebroids,

• Lars Kadison, Kornél Szlachányi, Bialgebroid actions on depth two extensions and duality, Adv. Math. 179:1 (2003) 75-121 doi

Relation to quantum field theory

On von Neumann algebra factors as algebras of quantum observables in quantum physics and quantum field theory (which was their original motivation, cf. Rédei 1996):

• Jakob Yngvason, The role of type III factors in quantum field theory [arXiv:math-ph/0411058]

• Jonathan Sorce, Notes on the type classification of von Neumann algebras [arXiv:2302.01958]

• Roberto Longo, Edward Witten, A note on continuous entropy [arXiv:2202.03357, spire:2029393]

and particularly in quantum field theory on curved spacetime:

• Edward Witten, Notes On Some Entanglement Properties Of Quantum Field Theory, in APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory, Reviews of Modern Physics (2018) [arXiv:1803.04993, doi:10.1103/revmodphys.90.045003]

• Edward Witten, Why Does Quantum Field Theory In Curved Spacetime Make Sense? And What Happens To The Algebra of Observables In The Thermodynamic Limit?, in Dialogues Between Physics and Mathematics, Springer (2022) [arXiv:2112.11614, doi:10.1007/978-3-031-17523-7_11]

• Edward Witten, Algebras, Regions, and Observers [arXiv:2303.02837]

such as on de Sitter spacetime:

• Venkatesa Chandrasekaran, Roberto Longo, Geoff Penington, Edward Witten, An Algebra of Observables for de Sitter Space, Journal of High Energy Physics 2023 82 (2023) [arXiv:2206.10780, doi:10.1007/JHEP02(2023)082]

and potential application of von Neumann algebra factors to quantum gravity:

• Edward Witten, Gravity and the Crossed Product, Journal of High Energy Physics 2022 8 (2022) [arXiv:2112.12828, doi:10.1007/JHEP10(2022)008]

• Edward Witten, A Note On The Canonical Formalism for Gravity [arXiv:2212.08270, inspire:2615434]