Orthogonality-preserving, C*-conformal and conformal module mappings on Hilbert C*-modules
Michael Frank, Alexander S. Mishchenko, Alexander A. Pavlov
TL;DR
The paper addresses the structure of orthogonality-preserving, $C^*$-conformal, and conformal module mappings on Hilbert $C^*$-modules, extending the analysis beyond Hilbert spaces. It develops a framework using Paschke's canonical extension and bidual techniques to lift and analyze maps, showing that orthogonality-preserving maps have the form $T = \lambda V$ with $\lambda$ in the center of the multiplier algebra and $V$ an isometric A-linear embedding on the extension; polar-decomposition considerations in the center govern when this lifts to the original module. For $C^*$-conformal and conformal maps, it proves that such maps are precisely positive real multiples of isometric module operators, leveraging a reduction to the discrete part of $A^{**}$ and analysis of minimal central projections. These results highlight the role of the multiplier-center and polar decomposition in determining map structure on Hilbert $C^*$-modules and propose a general conjecture for the orthogonality-preserving case. The findings clarify how module mappings behave in the presence of nontrivial centers and provide a pathway for characterizing similar maps in broader noncommutative settings.
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 λof 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 λare fulfilled inside that multiplier algebra. Generally, T always fulfils 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 C*-linear mappings are shown to be only the positive real multiples of isometric module operators.