Bilinear maps on C${}^{\ast }$-algebras that have product property at a compact element

We study bounded bilinear maps on a C${}^{\ast }$-algebra $A$ having product property at $c\in A$. This leads us to the question of when a C${}^{\ast }$-algebra is determined by products at $c\mathrm{.}$ In the first part of our paper, we investigate this question for compact C${}^{\ast }$-algebras, and in the second part, we deal with von Neumann algebras having non-trivial atomic part. Our results are applicable to descriptions of homomorphism-like and derivation-like maps at a fixed point on such algebras.