Multiplicative mappings at unit operator on B(H) 1

Let A be a subalgebra of B(H). We say that a linear mapping [straight phi] from A into itself is a multiplicative mapping at Z(Z ∈ A) if [straight phi](ST) = [straight phi](S)[straight phi](T) for any S, T ∈ A with ST = Z. Let H be an infinite dimensional complex Hilbert space, and let [straight phi] be a surjective linear map on B(H). In this paper, we prove that if [straight phi] is a multiplicative mapping at I and continuous in the weak operator topology, then [straight phi] is an automorphism. We also prove that if [straight phi] is a weak continuous multiplicative mapping at any invertible operator with [straight phi](I) = I then [straight phi] is an automorphism. [PUBLICATION ABSTRACT]