Von Neumann factor ($Z(M)=\mathbb{C}I$)

A Von Neumann algebra whose center consists exactly of scalar multiples of the identity. A group von Neumann algebra is a factor exactly when every nonidentity conjugacy class is infinite. A central operator has a square-summable coefficient vector constant on those classes; an infinite class must therefore have coefficient zero. Conversely, the sum of left translations over a finite nonidentity class is a nonscalar central operator.