C*-submodule preserving module mappings on Hilbert C*-modules
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917390504165376 |
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| author | Frank, Michael |
| author_facet | Frank, Michael |
| contents | 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 $u\in Z(M(A))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | C*-submodule preserving module mappings on Hilbert C*-modules Frank, Michael Operator Algebras Functional Analysis 46L08 (Primary) 46C05, 47B49 (Secondary) 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 $u\in Z(M(A))$. |
| title | C*-submodule preserving module mappings on Hilbert C*-modules |
| topic | Operator Algebras Functional Analysis 46L08 (Primary) 46C05, 47B49 (Secondary) |
| url | https://arxiv.org/abs/2507.11206 |