Isometries of real Hilbert C*-modules

Let T : V -> W be a surjective real linear isometry between full real Hilbert C*-modules over real C*-algebras A and B, respectively. We show that the following conditions are equivalent: (a) T is a 2-isometry; (b) T is a complete isometry; (c) T preserves ternary products; (d) T preserves inner products; (e) T is a module map. When A and B are commutative, we give a full description of the structure of T. (C) 2016 Elsevier Inc. All rights reserved.