Editing Von Neumann algebra (section)

== Factors ==
A von Neumann algebra ''N'' whose [[center (algebra)|center]] consists only of multiples of the identity operator is called a '''factor'''.  As {{harvtxt|von Neumann|1949}} showed, every von Neumann algebra on a separable Hilbert space is isomorphic to a [[direct integral]] of factors.  This decomposition is essentially unique.  Thus, the problem of classifying isomorphism classes of von Neumann algebras on separable Hilbert spaces can be reduced to that of classifying isomorphism classes of factors.

{{harvtxt|Murray|von Neumann|1936}} showed that every factor has one of 3 types as described below.  The type classification can be extended to von Neumann algebras that are not factors, and a von Neumann algebra is of type X if it can be decomposed as a direct integral of type X factors;  for example, every commutative von Neumann algebra has type I<sub>1</sub>. Every von Neumann algebra can be written uniquely as a sum of von Neumann algebras of types I, II, and III.

There are several other ways to divide factors into classes that are sometimes used:

* A factor is called '''discrete''' (or occasionally '''tame''')  if it has type I, and '''continuous''' (or occasionally '''wild''') if it has type II or III.
* A factor is called '''semifinite''' if it has type I or II, and '''purely infinite''' if it has type  III.
* A factor is called '''finite''' if the projection 1 is finite and '''properly infinite''' otherwise. Factors of types I and II may be either finite or properly infinite, but factors of type III are always properly infinite.

===Type I factors===
A factor is said to be of '''type I''' if there is a minimal projection ''E ≠ 0'', i.e. a projection ''E'' such that there is no other projection ''F'' with 0 < ''F'' < ''E''.  Any factor of type I is isomorphic to the von Neumann algebra of ''all'' bounded operators on some Hilbert space;  since there is one Hilbert space for every [[cardinal number]], isomorphism classes of factors of type I correspond exactly to the cardinal numbers.  Since many authors consider von Neumann algebras only on separable Hilbert spaces, it is customary to call the bounded operators on a Hilbert space of finite dimension ''n'' a factor of type I<sub>''n''</sub>, and the bounded operators on a separable infinite-dimensional Hilbert space, a factor of type I<sub>∞</sub>.

===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.

===Type III factors===
Lastly, '''type III''' factors are factors that do not contain any nonzero finite projections at all.  In their first paper {{harvtxt|Murray|von Neumann|1936}} were unable to decide whether or not they existed; the first examples were later found by {{harvtxt|von Neumann|1940}}. Since the identity operator is always infinite in those factors, they were sometimes called type III<sub>∞</sub> in the past, but recently that notation has been superseded by the notation III<sub>λ</sub>, where λ is a real number in the interval [0,1]. More precisely, if the Connes spectrum (of its modular group) is 1 then the factor is of type III<sub>0</sub>, if the Connes spectrum is all integral powers of λ for 0&nbsp;<&nbsp;λ&nbsp;<&nbsp;1, then the type is III<sub>λ</sub>, and if the Connes spectrum is all positive reals then the type is III<sub>1</sub>. (The Connes spectrum is a closed subgroup of the positive reals, so these are the only possibilities.) The only trace on type III factors takes value ∞ on all non-zero positive elements, and any two non-zero projections are equivalent. At one time type III factors were considered to be intractable objects, but [[Tomita–Takesaki theory]] has led to a good structure theory. In particular, any type III factor can be written in a canonical way as the [[crossed product]] of a type II<sub>∞</sub> factor and the real numbers.