C*-submodule preserving module mappings on Hilbert C*-modules
Michael Frank

TL;DR
This paper characterizes bijective bounded module morphisms on Hilbert C*-modules that preserve all submodules, showing they are essentially scalar multiples of the identity by invertible elements in the center of the multiplier algebra.
Contribution
It provides a complete description of module mappings preserving all submodules, extending to non-closed submodules and linking to Morita equivalence and imprimitivity bimodules.
Findings
Bijective bounded module morphisms preserving submodules are scalar multiples of the identity.
Such morphisms are characterized by elements in the center of the multiplier algebra.
Preservation of inner product values occurs iff the operator is a unitary scalar multiple.
Abstract
Let be a (non-unital, in general) C*-algebra with center of its multiplier algebra, and let be a full Hilbert -module. Then any bijective bounded module morphism , for which every norm-closed -submodule of is invariant, is of the form where is invertible. As an example of a merely injective bounded module operator with that preserver property serves where has a positive spectrum, but not bounded away from zero. The same assertions are true if the restriction on the C*-submodules to be norm-closed is dropped. From a different point of view, for two given strongly Morita equivalent C*-algebras and and a Hilbert - bimodule with faithful compact right action of , for any two two-sided norm-closed ideals $I…
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