In mathematics, a finite von Neumann algebra is a von Neumann algebra whose identity element is a finite projection i.e. the identity is not Murray-von Neumann equivalent to a proper subprojection in the von Neumann algebra. A defining feature of these von Neumann algebras is the existence of a unique center-valued trace.
Definition
editLet N ⊆ B(H) be a von Neumann algebra with center Z. We say that N is finite if for any two Murray-von Neumann equivalent projections p, q in N such that q ≤ p, we have that p = q.
Examples
editAbelian von Neumann algebras
editIn a commutative von Neumann algebra, two projections are equivalent if and only if they are equal.
Finite-dimensional von Neumann algebras
editII_1 factors
editCenter-valued Trace
editRepresentation
editLet τ