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# Additive preserving rank one maps on Hilbert \$C^\ast\$-modules

06/2007;
Source: arXiv

ABSTRACT In this paper, we characterize a class of additive maps on Hilbert \$C^\ast\$-modules which maps a "rank one" adjointable operators to another rank one operators.

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• ##### Article:Rank-preserving multiplicative maps on B(X)
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ABSTRACT: Let X (H) be a Banach space (Hilbert space) and let () be the algebra of all bounded linear operators on X(H). In this paper, we get some characterizations of rank-preserving multiplicative maps on . As applications, we show that every multiplicative local approximate automorphism of 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 . We also get a characterization of a ∗-isomorphism (or a conjugate ∗-isomorphism) on by showing that there exists a unitary or conjugate linear unitary operator such that Φ(T)=UTU∗ for all if and only if Φ is multiplicative with the range containing all rank-1 projections and, for any A, , A∗B=0⇔Φ(A)∗Φ(B)=0.
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Journal of Mathematical Analysis and Applications.
• Modules over operator algebras, Amer. I Kaplansky . 1953. J. Math 75 839-853.

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### Keywords

additive maps

adjointable operators

maps