C*-Submodule Preserving Module Mappings on Hilbert C*-Modules

Abstract Let A be a (non-unital, in general) C*-algebra with center Z ( M ( A )) of its multiplier algebra and let $$\{ X, \langle .,. \rangle \}$$ { X , ⟨ . , . ⟩ } be a full Hilbert A -module. Then any bijective bounded module morphism T , for which every norm-closed A -submodule of X is invariant, is of the form $$T=d \cdot \textrm{id}_X$$ T = d · id X where $$d \in Z(M(A))$$ d ∈ Z ( M ( A ) ) is invertible. As an example of a merely injective bounded module operator with that preserver property serves $$T =d \cdot \textrm{id}_X$$ T = d · id X where $$|d| \in Z(M(A))$$ | d | ∈ Z ( M ( A ) ) 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 A and B and a Hilbert B - A bimodule $$\{ X, \langle .,. \rangle \}$$ { X , ⟨ . , . ⟩ } with faithful compact right action of B , for any two two-sided norm-closed ideals $$I \in A$$ I ∈ A , $$J \in B$$ J ∈ B , any full compatible norm-closed Hilbert J - I sub-bimodule of X is invariant for any left bounded B -module operator and any right bounded A -module operator. So these subsets of submodules of X cannot rule out any bounded module operator as a non-preserver of that subset collection, however any single element of this subset collection is preserved by any bounded module operator on X . For any B - A imprimitivity bimodule both the C*-valued inner product values are always preserved by bijective bounded module operators T on X iff $$T= u \cdot \textrm{id}_X$$ T = u ·