Editing Von Neumann algebra (section)

===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 in the past, but recently that notation has been superseded by the notation III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.