Abstract 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.

We prove that log n is an almost everywhere convergence Weyl multiplier for the orthonormal systems of non-overlapping Haar polynomials. Moreover, it is done for the general systems of martingale difference polynomials.

In this note, we will discuss what kind of operators between C*-algebras preserves Jordan triple products

We define type A, type B, type C as well as C*-semi-finite C*-algebras.

Let X and Y be locally compact Hausdorff spaces. We give a full description of disjointness preserving Fredholm linear operators T from C0(X) into C0(Y), and show that is continuous if either Y contains no isolated point or T has closed range. Our task is achieved by writing T as a weighted composition operator Tf = h f . Through the relative homeomorphism , the structure of the range space of T can be completely analyzed, and X and Y are homeomorphic after removing finite subsets.