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

Michael Frank

arxiv: 2507.11206 · v4 · submitted 2025-07-15 · 🧮 math.OA · math.FA

C-submodule preserving module mappings on Hilbert C-modules

Michael Frank This is my paper

Pith reviewed 2026-05-19 04:52 UTC · model grok-4.3

classification 🧮 math.OA math.FA

keywords Hilbert C*-modulesmodule morphismssubmodule preservationcentral multipliersMorita equivalenceimprimitivity bimodulesC*-algebras

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The pith

Any bijective bounded module map on a full Hilbert C*-module that preserves all norm-closed submodules is multiplication by an invertible central multiplier.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that on a full Hilbert A-module X over a C*-algebra A, every bijective bounded A-module morphism that leaves every norm-closed A-submodule invariant must act as multiplication by an invertible element from the center of the multiplier algebra M(A). This gives a concrete description of the operators that respect the entire lattice of submodules. The same conclusion holds when the closedness requirement on submodules is removed. In the setting of strongly Morita equivalent algebras, the paper further shows that many full subbimodules are automatically invariant under all bounded module operators, while for imprimitivity bimodules the inner-product values are preserved precisely when the operator is a unitary central multiplication.

Core claim

Let A be a (possibly non-unital) 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 · id_X where d ∈ Z(M(A)) is invertible. The same holds without the norm-closed restriction. For strongly Morita equivalent C*-algebras A and B and a Hilbert B-A bimodule X with faithful compact right action of B, any full compatible norm-closed Hilbert J-I subbimodule is invariant under any left bounded B-module operator and any right bounded A-module operator. For any B-A imprimitivity bimodule, bijective bounded module operators T preserve C

What carries the argument

The invariance of every norm-closed A-submodule under a bijective bounded A-module morphism T, which forces T to equal multiplication by an invertible central multiplier.

If this is right

T must equal multiplication by an invertible central multiplier.
The same characterization holds when submodules are not required to be norm-closed.
In the Morita-equivalence setting, the collection of full compatible subbimodules cannot exclude any bounded module operator.
On an imprimitivity bimodule, bijective bounded operators preserve the C*-valued inner products exactly when they are unitary central multiplications.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit example of a merely injective map given by a non-invertible central d with positive spectrum shows that bijectivity is essential for obtaining an invertible multiplier.
The automatic invariance of many subbimodules under all bounded operators suggests that submodule lattices alone are too coarse to distinguish operators in the Morita setting.
The result supplies a concrete test for centrality: check whether a candidate bijective map preserves every closed submodule.

Load-bearing premise

The Hilbert A-module X is full and the map T is a bijective bounded module morphism.

What would settle it

Exhibit a full Hilbert A-module X together with a bijective bounded A-module morphism T that leaves every norm-closed A-submodule invariant yet is not equal to multiplication by any invertible element of Z(M(A)).

read the original 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 $u\in Z(M(A))$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a clean characterization of bijective bounded module maps preserving closed submodules as central multiplications, plus some automatic-invariance facts for Morita bimodules, but the claim that the same holds without closedness looks shaky.

read the letter

The main result here is that on a full Hilbert A-module, any bijective bounded module map T that leaves every norm-closed A-submodule invariant must be multiplication by an invertible central element d from the multiplier algebra of A. The paper also notes an injective but non-surjective example when the spectrum of |d| touches zero, and it extends the picture to strongly Morita equivalent algebras with faithful compact actions, where full compatible subbimodules coming from ideals are automatically invariant under all bounded left or right module operators. Finally it records that for imprimitivity bimodules the inner-product values are preserved precisely when T is unitary central multiplication. These pieces organize some standard facts about preservers in a useful way and flag the automatic-invariance observation for the Morita case, which does not seem to be a direct restatement of earlier work. The abstract is consistent on its own terms and the claims follow from routine properties of C*-modules and Morita equivalence once the bijectivity and fullness hypotheses are in place. The central-multiplier conclusion for the closed case looks solid on the surface. The soft spot is the assertion that the same characterization holds if one drops the closedness requirement on the submodules. Central multipliers need not map arbitrary (non-closed) submodules into themselves, even when the module is full; density of the inner-product range does not automatically give invariance for non-closed sets, and bijectivity plus boundedness do not obviously close the gap. The stress-test note on this point seems to land. The Morita section is clearer: it correctly shows that no single such subbimodule can be used to rule out a bounded operator, since every one is preserved anyway. This is a specialized note aimed at people already working on preserver problems or module morphisms in operator algebras. A reader who follows the literature on Hilbert C*-modules will find the organization and the automatic-preservation remark worth seeing. The paper is coherent enough and formally grounded enough to deserve a serious referee, mainly to check the non-closed extension and the details of the bimodule claims. I would send it to peer review with a request to clarify or restrict that one sentence in the abstract.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that for a full Hilbert A-module X over a C*-algebra A, any bijective bounded A-module morphism T preserving every norm-closed A-submodule is necessarily of the form T = d · id_X with d invertible in Z(M(A)). It asserts that the same characterization holds without requiring the submodules to be norm-closed, provides an example of a merely injective such operator, and discusses related results on invariance of full compatible norm-closed Hilbert J-I subbimodules for strongly Morita equivalent algebras as well as preservation of C*-valued inner products by bijective operators on imprimitivity bimodules.

Significance. If the central claims hold, the results provide a concrete characterization of submodule-preserving module morphisms on Hilbert C*-modules and clarify invariance properties in the setting of Morita equivalence and imprimitivity bimodules. This could be of interest to researchers working on operator modules and preservers, though the paper does not appear to include machine-checked proofs or reproducible code.

major comments (1)

[Abstract] Abstract: The assertion that 'the same assertions are true if the restriction on the C*-submodules to be norm-closed is dropped' is load-bearing for the main result but lacks visible justification. For T = d · id_X with d invertible in Z(M(A)) to preserve arbitrary (not necessarily closed) right A-submodules M of the full module X, the inclusion dM ⊆ M must hold for all such M; centrality of d does not automatically guarantee this for non-closed submodules, and the bijectivity/boundedness hypotheses do not obviously repair the gap. The manuscript should supply a proof or explicit verification of this extension in the statement of the main theorem.

minor comments (2)

[Abstract] Abstract: The sentence 'for any two two-sided norm-closed ideals I ∈ A, J ∈ B' is slightly awkward in phrasing; consider rewording for clarity as 'for any two-sided norm-closed ideals I of A and J of B'.
[Abstract] Abstract: The final sentence on imprimitivity bimodules introduces the condition 'iff T = u · id_X for a unitary element u ∈ Z(M(A))' without prior definition of 'imprimitivity bimodule' in the abstract; a brief parenthetical or reference would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for identifying the need for explicit justification of the claim regarding non-norm-closed submodules. We address this point directly below and will incorporate the requested clarification in the revised version.

read point-by-point responses

Referee: [Abstract] Abstract: The assertion that 'the same assertions are true if the restriction on the C*-submodules to be norm-closed is dropped' is load-bearing for the main result but lacks visible justification. For T = d · id_X with d invertible in Z(M(A)) to preserve arbitrary (not necessarily closed) right A-submodules M of the full module X, the inclusion dM ⊆ M must hold for all such M; centrality of d does not automatically guarantee this for non-closed submodules, and the bijectivity/boundedness hypotheses do not obviously repair the gap. The manuscript should supply a proof or explicit verification of this extension in the statement of the main theorem.

Authors: We agree that the abstract statement would benefit from an explicit supporting argument rather than relying on the closed-case reasoning alone. In the body of the manuscript the characterization for norm-closed submodules is established by showing that any bijective bounded module map preserving all closed submodules must commute with the right A-action in a manner forcing it to be central multiplication. For the non-closed case, the forward direction (any such preserver is of the indicated form) follows immediately from the closed case, since preservation of all submodules implies preservation of all closed submodules. For the converse, we will add a short lemma verifying that if d lies in Z(M(A)) and is invertible, then left multiplication by d maps every right A-submodule M into itself. The argument proceeds by noting that any right submodule M satisfies M = {x ∈ X | ⟨x, y⟩ ∈ I for suitable ideals generated by the fullness condition}, combined with centrality ensuring that d commutes past the inner-product values; bijectivity of the map then guarantees the image remains inside M. This verification will be inserted as a dedicated remark immediately following the main theorem, together with a brief check that the example of a merely injective operator continues to work without closedness. We therefore revise the manuscript accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation is self-contained

full rationale

The paper establishes a characterization of bijective bounded module morphisms T on a full Hilbert A-module X that preserve all norm-closed A-submodules, concluding T = d · id_X for invertible d in Z(M(A)). This follows from standard C*-module properties, Morita equivalence, and invariance arguments without any equation or step reducing the claimed form of T to a fitted parameter, self-defined quantity, or input by construction. The abstract's extension to non-closed submodules is asserted separately and does not redefine the result in terms of itself. No load-bearing self-citations, uniqueness theorems imported from prior author work, or smuggled ansatzes appear in the derivation chain; the result is presented as following from external benchmarks in operator algebra theory and is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard axiomatic framework of Hilbert C*-modules, multiplier algebras, and Morita equivalence; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of full Hilbert C*-modules, norm-closed submodules, and the center of the multiplier algebra
Invoked throughout the abstract as background for the module morphism statements.

pith-pipeline@v0.9.0 · 5896 in / 1302 out tokens · 43956 ms · 2026-05-19T04:52:25.261628+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

any bijective bounded module morphism T, for which every norm-closed A-submodule of X is invariant, is of the form T=d · id_X where d ∈ Z(M(A)) is invertible

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

P. Ara, M. Mathieu . The centre of the local multiplier algebra. In: Local Multipli- ers of C*-Algebras. Springer Monographs in Mathematics. Springer, London. pp. 71-95, https://doi.org/10.1007/978-1-4471-0045-4 4

work page doi:10.1007/978-1-4471-0045-4
[2]

Frank, A

M. Frank, A. S. Mishchenko, A. A. Pavlov . Orthogonality-preserving, C*-conformal and conformal module mappings on Hilbert C*-modules. J. Funct. Anal. 260(2011). 327–339

work page 2011
[3]

M. Frank . A multiplier approach to the Lance-Blecher theorem. Zeitschr. Anal. Anwend. 16(1997), 565–573

work page 1997
[4]

Hines, E

T. Hines, E. W alsberg. Nontrivially Noetherian C*-algebras. Math. Scand. 111(2012), 135– 146

work page 2012
[5]

Kaliszewki, J

S. Kaliszewki, J. Quigg, D. P. Williams . The Rieffel correspondence for equivalent Fell bundles. J. Aust. Math. Soc. 117(2024), 288–307

work page 2024
[6]

E. C. Lance . Unitary operators on Hilbert C*-modules. Bull. London Math. Soc. 26(1994), 363–366

work page 1994
[7]

Linear orthogonality preservers of Hilbert C*-modules

Chi-W ai Leung, Chi-Keung Ng, Ngai-Ching Wong . Linear orthogonality preservers of Hilbert C*-modules. J. Operator Theory 71(2014), no. 2, 571–584

work page 2014
[8]

Bounded module maps and pure completely positive maps

Huaxin Lin . Bounded module maps and pure completely positive maps. J. Oper. Theory 26(1991), 121–138

work page 1991
[9]

M. Mathieu . The second local multiplier algebra of a separable C*-algebra. In: Algebraic Methods in Functional Analysis. eds.: I. Todorov, L. Turowska. Operator Theory: Advances and Applications, vol. 233, Birkh¨ auser, Basel, 2013, pp. 93–102

work page 2013
[10]

G. J. Murphy . C∗-Algebras and Operator Theory. Academic Press, Boston, 2004. C*-SUBMODULE PRESER VING MODULE MAPPINGS 5

work page 2004
[11]

Raeburn, D

I. Raeburn, D. P. Williams . Morita Equivalence and Continuous-Trace C*-Algebras. Math- ematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, U.S.A., 1998

work page 1998
[12]

K. Sharifi . Invariant submodules of modular operators and Lomonosov type theorem for Hilbert C*-modules. arxiv:2506:01161, June 2025, 19 pp

work page 2025
[13]

B. Solel . Isometries of Hilbert C*-modules. Trans. Amer. Math. Soc. 353(2001), no. 11, 4637–4660. Hochschule f ¨ur Technik, Wirtschaft und Kultur (HTWK) Leipzig, F akult ¨at Infor- matik und Medien, PF 301166, D-04251 Leipzig, Germany. Email address: michael.frank@htwk-leipzig.de

work page 2001

[1] [1]

P. Ara, M. Mathieu . The centre of the local multiplier algebra. In: Local Multipli- ers of C*-Algebras. Springer Monographs in Mathematics. Springer, London. pp. 71-95, https://doi.org/10.1007/978-1-4471-0045-4 4

work page doi:10.1007/978-1-4471-0045-4

[2] [2]

Frank, A

M. Frank, A. S. Mishchenko, A. A. Pavlov . Orthogonality-preserving, C*-conformal and conformal module mappings on Hilbert C*-modules. J. Funct. Anal. 260(2011). 327–339

work page 2011

[3] [3]

M. Frank . A multiplier approach to the Lance-Blecher theorem. Zeitschr. Anal. Anwend. 16(1997), 565–573

work page 1997

[4] [4]

Hines, E

T. Hines, E. W alsberg. Nontrivially Noetherian C*-algebras. Math. Scand. 111(2012), 135– 146

work page 2012

[5] [5]

Kaliszewki, J

S. Kaliszewki, J. Quigg, D. P. Williams . The Rieffel correspondence for equivalent Fell bundles. J. Aust. Math. Soc. 117(2024), 288–307

work page 2024

[6] [6]

E. C. Lance . Unitary operators on Hilbert C*-modules. Bull. London Math. Soc. 26(1994), 363–366

work page 1994

[7] [7]

Linear orthogonality preservers of Hilbert C*-modules

Chi-W ai Leung, Chi-Keung Ng, Ngai-Ching Wong . Linear orthogonality preservers of Hilbert C*-modules. J. Operator Theory 71(2014), no. 2, 571–584

work page 2014

[8] [8]

Bounded module maps and pure completely positive maps

Huaxin Lin . Bounded module maps and pure completely positive maps. J. Oper. Theory 26(1991), 121–138

work page 1991

[9] [9]

M. Mathieu . The second local multiplier algebra of a separable C*-algebra. In: Algebraic Methods in Functional Analysis. eds.: I. Todorov, L. Turowska. Operator Theory: Advances and Applications, vol. 233, Birkh¨ auser, Basel, 2013, pp. 93–102

work page 2013

[10] [10]

G. J. Murphy . C∗-Algebras and Operator Theory. Academic Press, Boston, 2004. C*-SUBMODULE PRESER VING MODULE MAPPINGS 5

work page 2004

[11] [11]

Raeburn, D

I. Raeburn, D. P. Williams . Morita Equivalence and Continuous-Trace C*-Algebras. Math- ematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, U.S.A., 1998

work page 1998

[12] [12]

K. Sharifi . Invariant submodules of modular operators and Lomonosov type theorem for Hilbert C*-modules. arxiv:2506:01161, June 2025, 19 pp

work page 2025

[13] [13]

B. Solel . Isometries of Hilbert C*-modules. Trans. Amer. Math. Soc. 353(2001), no. 11, 4637–4660. Hochschule f ¨ur Technik, Wirtschaft und Kultur (HTWK) Leipzig, F akult ¨at Infor- matik und Medien, PF 301166, D-04251 Leipzig, Germany. Email address: michael.frank@htwk-leipzig.de

work page 2001