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We study bounded bilinear maps on a C$^*$-algebra $A$ having product propertyat $c\in A$. This leads us to the question of when a C$^*$-algebra isdetermined by products at $c.$ In the first part of our paper, we investigatethis question for compact C$^*$-algebras, and in the second part, we deal withvon Neumann algebras having non-trivial atomic part. Our results are applicableto descriptions of homomorphism-like and derivation-like maps at a fixed pointon such algebras.

Bilinear maps on C$^*$-algebras that have product property at a compact element