In the first part of paper, we use states on $C^*$-algebras in order to establish some equivalent statements equality triangle inequality, as well parallelogram identity for elements a pre-Hilbert $C^*$-module. We also characterize case inequality adjointable operators Hilbert Then give certain necessary and sufficient conditions Pythagoras two vectors $C^*$-module under assumption that their i...

Let M be a space of weakly*-measurable functions F : ? B(H) on measure (?,?,?), for which the function F*F is Gel'fand integrable and integral d? nuclear operator Hilbert H. We show that H*-module contains an orthonormal basis.

Let E and F be two Hilbert C∗-modules over C∗-algebras A and B, respectively. Let T be a surjective linear isometry from E onto F and φ a map from A into B. We will prove in this paper that if the C∗-algebras A and B are commutative, then T preserves the inner products and T is a module map, i.e., there exists a ∗-isomorphism φ between the C∗-algebras such that 〈Tx, Ty〉 = φ(〈x, y〉), and T (xa) ...

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F (x) of degree n in A[x] such that A ⊗k k[x](x)/(F (x)) is a free A–module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x](x). Intr...