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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

Let 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

Abelian von Neumann algebras

In a commutative von Neumann algebra, two projections are equivalent if and only if they are equal.

Representation

Let τ