It is well-known that every C*-algebra is determined by its linear and multiplicative structure: Two C*-algebras are *-isomorphic if and only if they admit a multiplicative, linear bijection.

We study if instead of the whole multiplicative structure it suffices to record when two elements have zero product. While it is not clear if every C*-algebra is determined this way, we obtain many positive results. In particular, two unital, simple C*-algebras are *-isomorphic if and only if they admit a linear bijection that preserves zero products.

This is joint work with Eusebio Gardella.