The spectrally bounded linear maps on operator algebras
Volume 150 / 2002
Abstract
We show that every spectrally bounded linear map ${\mit \Phi }$ 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.
