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. There are an uncountable number of other factors of type II1. Connes proved that the infinite one is also unique.

Constructions

Properties

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.

The infinite hyperfinite type II factor

While there are other factors of type II, there is a unique hyperfinite one, up to isomorphism. It consists of those infinite square matrices with entries in the hyperfinite type II1 factor that define bounded operators.

See also

References

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