dimension of a von Neumann algebra

Question

Is there any dimension on a von Neumann algebra? Is there any relationship between finite von Neumann algebras and finite dimensional von Neumann algebras?

Answer 1

A von Neumann algebra is a vector space, so it has a dimension in that sense, but it is meaningless (actually, it is meaningless for any Banach space as the dimension is either finite or uncountably infinite).

Any finite-dimensional von Neumann algebra is finite. But there are (many) finite, infinite-dimensional von Neumann algebras: precisely those of type II$_1$.