Linear Orthogonality Preservers of Hilbert C∗-modules over C∗-algebras with Real Rank Zero
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چکیده
Let A be a C∗-algebra. Let E and F be Hilbert A-modules with E being full. Suppose that θ : E → F is a linear map preserving orthogonality, i.e., 〈θ(x), θ(y)〉 = 0 whenever 〈x, y〉 = 0. We show in this article that if, in addition, A has real rank zero, and θ is an A-module map (not assumed to be bounded), then there exists a central positive multiplier u ∈M(A) such that 〈θ(x), θ(y)〉 = u〈x, y〉 (x, y ∈ E). In the case when A is a standard C∗-algebra, when A is a properly infinite unital C∗-algebra, or when A is a W ∗-algebra, we also get the same conclusion with the assumption of θ being an A-module map weakened to being a local map.
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