Quasitrace

A quasitrace on a C^*-algebra is a functional that is nonnegative on positive elements, satisfies , is linear on every commutative -subalgebra, and obeys

for self-adjoint . A 2-quasitrace also admits a quasitrace extension to agreeing with on the upper-left corner. For unital , normalization means ; the corresponding matrix extension takes value 2 on the matrix identity. A tracial state satisfies these conditions, but the definition does not assume additivity on arbitrary pairs of positive elements.