Editing Von Neumann algebra (section)

===Type II factors===
A factor is said to be of '''type II''' if there are no minimal projections but there are non-zero [[Von Neumann algebra#Comparison theory of projections|finite projections]]. This implies that every projection ''E'' can be "halved" in the sense that there are two projections ''F'' and ''G'' that are [[Von Neumann algebra#Comparison theory of projections|Murray–von Neumann equivalent]] and satisfy ''E'' = ''F'' + ''G''.  If the identity operator in a type II factor is finite, the factor is said to be of type II<sub>1</sub>; otherwise, it is said to be of type II<sub>∞</sub>.  The best understood factors of type II are the [[hyperfinite type II-1 factor|hyperfinite type II<sub>1</sub> factor]] and the [[hyperfinite type II-infinity factor|hyperfinite type II<sub>∞</sub> factor]], found by {{harvtxt|Murray|von Neumann|1936}}. These are the unique hyperfinite factors of types II<sub>1</sub> and  II<sub>∞</sub>; there are an uncountable number of other  factors of these types that are the subject of intensive study. {{harvtxt|Murray|von Neumann|1937}} proved the fundamental result that a factor of type II<sub>1</sub> has a unique finite tracial state,  and the set of traces of projections is [0,1].

A factor of type II<sub>∞</sub> has a semifinite trace, unique up to rescaling, and the set of traces of projections is [0,∞]. The set of real numbers λ such that there is an automorphism rescaling the trace by a factor of λ is called the '''fundamental group''' of the type II<sub>∞</sub> factor.

The tensor product of a factor of type II<sub>1</sub> and an infinite type I factor has type II<sub>∞</sub>, and conversely any factor of  type II<sub>∞</sub> can be constructed like this. The '''fundamental group''' of a type II<sub>1</sub>  factor is defined to be the fundamental group of its tensor product with the infinite (separable) factor of type I. For many years it was an open problem to find a type II factor whose fundamental group was not the group of [[positive reals]], but [[Alain Connes|Connes]] then showed that the von Neumann group algebra of a countable discrete group with [[Kazhdan's property (T)]] (the trivial representation is isolated in the dual space), such as SL(3,'''Z'''),  has a countable fundamental group. Subsequently, [[Sorin Popa]] showed that the fundamental group can be trivial for certain groups, including the [[semidirect product]] of '''Z'''<sup>2</sup> by SL(2,'''Z''').

An example of a type II<sub>1</sub> factor is the von Neumann group algebra of a countable infinite discrete group such that every non-trivial conjugacy class is infinite.
{{harvtxt|McDuff|1969}} found an uncountable family of such groups with non-isomorphic von Neumann group algebras, thus showing the existence of uncountably many different separable type II<sub>1</sub> factors.