 A von Neumann algebra is a \(*\)-subalgebra of the set of bounded operators B(H) on a Hilbert space H that is closed in the weak operator topology. It is usually assumed that the von Neumann algebra contains the identity operator. A \(*\)-subalgebra contains the adjoint of every operator in the algebra, where the “\(*\)” denotes the adjoint. There are special types of von Neumann algebras that are called “factors”. A von Neumann algebra is a factor, if its center (which is the set of elements that commute with all elements of the algebra) is trivial, meaning that it only contains scalar multiples of the identity element. Moreover, von Neumann showed in his reduction-theory paper that all von Neumann algebras that are not factors can be decomposed as a direct sum (or integral) of factors. There are three mutually exclusive and exhaustive factor types: type-I, type-II, and type-III. Each type has been classified into (mutually exclusive and exhaustive) sub-types: types I\(_n\) \((n = 1,2,\ldots ,\infty),\) II\(_n\) \((n = 1,\infty),\) III\(_z\) \((0\le z\le 1).\) As mentioned above, type-I\(_n\) correspond to finite dimensional Hilbert spaces, while type-I\(_{\infty}\) corresponds to the infinite dimensional separable Hilbert space that provides the rigorous framework for wave and matrix mechanics. Von Neumann and Murray distinguished the subtypes for type-I and type-II, but were not able to do so for the type-III factors. Subtypes were not distinguished for these factors until the 1960s and 1970s – see Chapter 3 of Sunder (1987) or Chapter 5 of Connes (1994) for details.