Nuclearity of C*-algebras is a fundamental property in the classification program, often viewed as an internal finite-dimensional approximation property. In 2012, Matui and Sato introduced a refined finite-dimensional approximation property using pure states for simple C*-algebras, which leads to a breakthrough in the remaining implication of the Toms-Winter conjecture - an important open problem in classification. The property was later generalized to non-simple C*-algebras. In this talk, we show that this property is equivalent to nuclearity and mention its relation to the classification program.
