von Neumann Dimension

The von Neumann dimension over a finite von Neumann algebra with normalized trace assigns to a finitely generated projective module represented by a projection the number

The extended dimension of an arbitrary module is the supremum of the dimensions of its finitely generated projective submodules, allowing infinity. For a group von Neumann algebra this uses the canonical trace. Thus the free module has dimension . The extension to homology modules defines L²-Betti numbers, which need not be integers.