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 $\{X,⟨.,.⟩\}$ 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 $T=d\cdot {\mathrm{id}}_X$ where $d\in Z(M(A))$ is invertible. As an example of a merely injective bounded module operator with that preserver property serves $T=d\cdot {\mathrm{id}}_X$ where $|d|\in Z(M(A))$ 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 $\{X,⟨.,.⟩\}$ with faithful compact right action of $B$, for any two two-sided norm-closed ideals $I\in A$, $J\in B$, any full compatible norm-closed Hilbert $J$-$I$ subbimodule 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 $T=u\cdot {\mathrm{id}}_X$ for a unitary element $u\in Z(M(A))$.

8 pages, final version