Rank-preserving multiplicative maps on B(X)

AbstractLet X (H) be a Banach space (Hilbert space) and let B(X) (B(H)) be the algebra of all bounded linear operators on X(H). In this paper, we get some characterizations of rank-preserving multiplicative maps on B(X). As applications, we show that every multiplicative local approximate automorphism of B(X) with the set of all rank-1 idempotents contained in its range is in fact an automorphism. We describe the structure of corank-preserving multiplicative maps on B(H). We also get a characterization of a ∗-isomorphism (or a conjugate ∗-isomorphism) on B(H) by showing that there exists a unitary or conjugate linear unitary operator U∈B(H,K) such that Φ(T)=UTU∗ for all T∈B(H) if and only if Φ is multiplicative with the range containing all rank-1 projections and, for any A, B∈B(H), A∗B=0⇔Φ(A)∗Φ(B)=0