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

TL;DR

This paper characterizes orthogonality-preserving, C*-conformal, and conformal module mappings on Hilbert C*-modules, revealing their structure as scalar multiples of isometries or involving central multipliers under certain conditions.

Contribution

It provides a comprehensive structural description of these module mappings, including conditions for their representation involving central multipliers and polar decompositions.

Findings

Orthogonality-preserving maps act as scalar multiples of isometries under certain conditions.

C*-conformal and conformal maps are positive real multiples of isometric operators.

The paper establishes the general form of these mappings on Hilbert C*-modules.

Abstract

We investigate orthonormality-preserving, C*-conformal and conformal module mappings on Hilbert C*-modules to obtain their general structure. Orthogonality-preserving bounded module maps T act as a multiplication by an element \lambda 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 \lambda are fulfilled inside that multiplier algebra. Generally, T always fulfils the equality $<T(x),T(y) > = | \lambda |^2 < x,y>$ for any elements x,y of the Hilbert C*-module. At the contrary, C*-conformal and conformal bounded C*-linear mappings are shown to be only the positive real multiples of isometric module operators.