Editing Von Neumann algebra (section)

==Terminology==
Some of the terminology in von Neumann algebra theory can be confusing, and the terms often have different meanings outside the subject.

*A '''factor''' is a von Neumann algebra with trivial center, i.e. a center consisting only of scalar operators.
*A [[Finite von Neumann algebra|'''finite''' von Neumann algebra]] is one which is the [[direct integral#Direct integrals of von Neumann algebras|direct integral]] of finite factors (meaning the von Neumann algebra has a faithful normal tracial state <math>\tau: M \rarr \mathbb{C}</math><ref>[http://perso.ens-lyon.fr/gaboriau/evenements/IHP-trimester/IHP-CIRM/Notes=Cyril=finite-vonNeumann.pdf An Introduction To II1 Factors] ens-lyon.fr</ref>).  Similarly, '''properly infinite''' von Neumann algebras are the direct integral of properly infinite factors.
*A von Neumann algebra that acts on a separable Hilbert space is called '''separable'''.  Note that such algebras are rarely [[separable space|separable]] in the norm topology.
*The von Neumann algebra '''generated''' by a set of bounded operators on a Hilbert space is the smallest von Neumann algebra containing all those operators.
*The '''tensor product''' of two von Neumann algebras acting on two Hilbert spaces is defined to be the von Neumann algebra generated by their algebraic tensor product, considered as operators on the Hilbert space tensor product of the Hilbert spaces.

By [[forgetting (mathematics)|forgetting]] about the topology on a von Neumann algebra, we can consider it a (unital) [[star-algebra|*-algebra]], or just a ring. Von Neumann algebras are [[semihereditary ring|semihereditary]]: every finitely generated submodule of a [[projective module]] is itself projective. There have been several attempts to axiomatize the underlying rings of von Neumann algebras, including [[Baer *-ring]]s and [[AW*-algebra]]s. The [[*-algebra]] of [[affiliated operator]]s of a finite von Neumann algebra is a [[von Neumann regular ring]]. (The von Neumann algebra itself is in general not von Neumann regular.)