Hyperfinite type II factor
In mathematics, there are up to isomorphism exactly two separably acting hyperfinite type II factors; one infinite and one finite.Murray and von Neumann proved that up to isomorphism there is a unique von Neumann algebra that is a factor of type II1 and also hyperfinite; it is called the hyperfinite type II1 factor.The hyperfinite II1 factor R is the unique smallest infinite dimensional factor in the following sense: it is contained in any other infinite dimensional factor, and any infinite dimensional factor contained in R is isomorphic to R. The outer automorphism group of R is an infinite simple group with countable many conjugacy classes, indexed by pairs consisting of a positive integer p and a complex pth root of 1.The projections of the hyperfinite II1 factor form a continuous geometry.It consists of those infinite square matrices with entries in the hyperfinite type II1 factor that define bounded operators.