C*-submodule preserving module mappings on Hilbert C*-modules
Michael Frank
TL;DR
The paper addresses when bounded module operators on Hilbert C*-modules preserve all C*-submodules and characterizes the resulting operators. It proves that for a full Hilbert A-module X, any bijective bounded A-linear preserver of all norm-closed A-submodules must be a central multiplier scalar $T = d \cdot id_X$ with $d \in Z(M(A))$ invertible; dropping norm-closedness introduces injective but non-surjective examples when the center is infinite-dimensional. In Morita-equivalence/bimodule contexts, the work shows that certain submodules are invariant under all bounded operators, and that bijective operators preserving both A- and B-valued inner products on imprimitivity bimodules are precisely central unitary scalars. These results clarify the structure of preserver problems in the setting of Hilbert C*-modules and highlight the roles of multiplier centers and Morita theory.
Abstract
Let $A$ be a (non-unital, in general) C*-algebra with center $Z(M(A))$ of its multiplier algebra, and let $\{ X, \langle .,. \rangle \}$ 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 $T=d \cdot {\rm id}_X$ where $d \in Z(M(A))$ is invertible. As an example of a merely injective bounded module operator with that preserver property serves $T =d \cdot {\rm id}_X$ 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. \newline From a different point of view, for two given strongly Morita equivalent C*-algebras $A$ and $B$ and a Hilbert $B$-$A$ bimodule $\{ X, \langle .,. \rangle \}$ 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 $X$. \newline 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 $T= u \cdot {\rm id}_X$ for a unitary element $u\in Z(M(A))$.