Orthogonality-preserving, C ∗-conformal and conformal module mappings on Hilbert C ∗-modules ✩

We investigate orthonormality-preserving, C ∗-conformal and conformalmodulemappings on full Hilbert C ∗-modules to obtain their general structure. Orthogonality-preserving bounded module maps T act as a multiplication by an element λ of the center of the multiplier algebra of the C ∗-algebra of coefficients combined with an isometric module operator as long as some polar decomposition conditions for the specific element λ are fulfilled inside that multiplier algebra. Generally, T always fulfills the equality T (x),T (y) = |λ|2 x,y for any elements x, y of the Hilbert C ∗-module. At the contrary, C ∗-conformal and conformal bounded module maps are shown to be only the positive real multiples of isometric module operators. © 2010 Elsevier Inc. All rights reserved.