Bilinear maps on C⁎-algebras that have product property at a compact element

Abstract

We study bounded bilinear maps on a C⁎-algebra A having product property at c∈A. This leads us to the question of when a C⁎-algebra is determined by products at c. In the first part of our paper, we investigate this question for compact C⁎-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.