Rank-preserving multiplicative maps on image Original Research Article

Let X (H) be a Banach space (Hilbert space) and let image ( image ) be the algebra of all bounded linear operators on X(H). In this paper, we get some characterizations of rank-preserving multiplicative maps on image . As applications, we show that every multiplicative local approximate automorphism of image 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 image . We also get a characterization of a *-isomorphism (or a conjugate *-isomorphism) on image by showing that there exists a unitary or conjugate linear unitary operator image such that Φ(T)=UTU * for all image if and only if Φ is multiplicative with the range containing all rank-1 projections and, for any A, image , A*B=0left right double arrowΦ(A)*Φ(B)=0.