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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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##### Article: Additive mappings decreasing rank one
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ABSTRACT: Let B(X) be the algebra of bounded operators on a real or complex Banach space X, and F(X) a subalgebra of finite rank operators. A complete description of additive mappings Φ:F(X)→F(X), which map rank one operators to operators of rank at most one, is given.
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