The spectrally bounded linear maps on operator algebras
Volume 150 / 2002
Studia Mathematica 150 (2002), 261-271
MSC: Primary 47B48, 47L10, 47A10.
DOI: 10.4064/sm150-3-4
Abstract
We show that every spectrally bounded linear map from a Banach algebra onto a standard operator algebra acting on a complex Banach space is square-zero preserving. This result is used to show that if {\mit \Phi }_{2} is spectrally bounded, then {\mit \Phi } is a homomorphism multiplied by a nonzero complex number. As another application to the Hilbert space case, a classification theorem is obtained which states that every spectrally bounded linear bijection {\mit \Phi } from {\cal B}(H) onto {\cal B}(K), where H and K are infinite-dimensional complex Hilbert spaces, is either an isomorphism or an anti-isomorphism multiplied by a nonzero complex number. If {\mit \Phi } is not injective, then {\mit \Phi } vanishes at all compact operators.
