Linear Orthogonality Preservers of Hilbert C∗-modules
نویسنده
چکیده
We show in this paper that the module structure and the orthogonality structure of a Hilbert C∗-module determine its inner product structure. Let A be a C∗-algebra, and E and F be Hilbert A-modules. Assume Φ : E → F is an A-module map satisfying 〈Φ(x),Φ(y)〉A = 0 whenever 〈x, y〉A = 0. Then Φ is automatically bounded. In case Φ is bijective, E is isomorphic to F . More precisely, let JE be the closed two-sided ideal of A generated by {〈x, y〉A : x, y ∈ E}. We show that there exists a unique central positive multiplier u ∈ M(JE)+ such that 〈Φ(x),Φ(y)〉A = u〈x, y〉A (x, y ∈ E). As a consequence, the induced map Φ0 : E → Φ(E) is adjointable, and Eu1/2 is isomorphic to Φ(E) as Hilbert A-modules.
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