The nuclearity of C*-algebras is a fundamental property in the classification program, often viewed as an internal finite-dimensional approximation property. We introduced a refined finite-dimensional approximation property using pure states, called the Matui-Sato approximation property (MSAP). Such property first appeared in the work of Matui and Sato in 2012, where they show that strict comparison implies Z-stability through "small to large" comparison property, called property (SI), for simple nuclear C*-algebra with nice enough trace space. The MSAP was later shown to hold for a large class of separable nuclear C*-algebras in 2015.

Beyond its role in the classification program, this property is intriguing and natural in its own right; for instance, in the case of commutative C*-algebras, it arises naturally through the application of partitions of unity. In this work, we show that this property is equivalent to nuclearity for separable C*-algebras. As a consequence, we obtain a more general property (SI) result for nuclear maps.