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

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 $A$ 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. 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 $X$. 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 $u\in Z(M(A))$.