symmetric monoidal (∞,1)-category of spectra
For $A$ a von Neumann algebra write $A'$ for its commutant in the ambient algebra $B(\mathcal{H})$ of bounded operators.
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 $B(\mathcal{H})$.
Every von Neumann algebra may be written as a direct integral? over factors. (von Neumann 49)
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The original sources are
Murray, John von Neumann, …
John von Neumann, On rings of operators, reduction theory, Annals of Mathematics Second Series, Vol. 50, No. 2 (1949) (jstor)
Lecture notes include
V.S. Sunder, von Neumann algebras, $II_1$-factors, and their subfactors (pdf)
Hideki Kosaki, Type III factors and index theory (1993) (pdf)
The mathematics of inclusions of subfactors is giving deep structural insights. See also at planar algebra.
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)