Linear maps on W⁎-algebras preserving elements annihilated by a continuous function

Denote by F either the real field R or the complex field C . Let f : F → F be a continuous function such that its zero set Z F ( f ) contains an isolated point and at least one other point. For a given λ ∈ F , the zero set Z F ( f ) is said to be λ -convex if λ Z F ( f ) + ( 1 − λ ) Z F ( f ) ⊆ Z F ( f ) . Assume that Z F ( f ) is not λ -convex for any λ ∈ F with | λ | > 1 . When F = R , we prove that a bounded unital R -linear map Φ : M s a → N s a between the self-adjoint parts of W ⁎ -algebras is a Jordan homomorphism whenever f ( Φ ( A ) ) = 0 for every A in M s a satisfying f ( A ) = 0 . When F = C , we prove that a bounded unital C -linear map Φ : M → N between W ⁎ -algebras is a Jordan ⁎-homomorphism whenever f ( Φ ( A ) ) = 0 for every normal element A in M satisfying f ( A ) = 0 , and Φ maps projections to normal elements. Additionally, a scalar λ ∈ F is called a multiplier of Z F ( f ) if λ Z F ( f ) ⊆ Z F ( f ) . Assume that 0 is an isolated point of Z F ( f ) and that Z F ( f ) has no multiplier λ ∈ F with | λ | > 1 . When F = R , we prove that if Φ : M s a → N s a is a bounded R -linear map (not necessarily unital) satisfying f ( Φ ( A ) ) = 0 for every A in M s a with f ( A ) = 0 , then Φ ( I M ) Φ is a Jordan homomorphism. When F = C , we prove that if Φ : M → N is a bounded C -linear map satisfying f ( Φ ( A ) ) = 0 for every normal element A in M with f ( A ) = 0 , then there exists a positive integer m such that Φ ( I M ) m − 1 Φ is a Jordan ⁎-homomorphism, provided that Z C ( f ) contains a non-zero isolated point and that Φ maps projections to normal elements.