Abstract
LetH be a separable infinite-dimensional complex Hilbert space. We prove that if Ф: ℬ(H)→ℬ(H) is a*-preserving ring homomorphism whose range contains a rank-one operator and an operator with dense range, then Ф is an isometric linear or conjugate-linear algebra automorphism of ℬ(H). In particular, if the unilateral shift is contained in the range of a*-endomorphism Ф of ℬ(H), then Ф is bijective.
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