Additive jointly separating maps and ring homomorphisms

Fereshteh Sady; Masoumeh Najafi Tavani

arxiv: 1907.10286 · v1 · pith:BV6FXZAMnew · submitted 2019-07-24 · 🧮 math.FA

Additive jointly separating maps and ring homomorphisms

Fereshteh Sady , Masoumeh Najafi Tavani This is my paper

Pith reviewed 2026-05-24 16:54 UTC · model grok-4.3

classification 🧮 math.FA

keywords additive mapsjointly separating mapscozero setsring homomorphismsvector-valued functionsLipschitz algebrasBanach algebrascontinuous functions

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The pith

Pairs of additive maps between vector-valued function spaces that jointly preserve disjoint cozero sets admit a partial description.

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

The paper establishes a partial description of pairs of additive maps S and T from a subspace A of vector-valued continuous functions on a compact Hausdorff space X to continuous functions on Y that satisfy the jointly separating condition based on cozero sets. If two functions have disjoint cozero sets, their images under S and T also have disjoint cozero sets. This description applies to spaces including vector-valued Lipschitz functions, absolutely continuous functions, and continuously differentiable functions. The results are then used to characterize continuous ring homomorphisms between corresponding Banach algebras. A sympathetic reader would care because this provides a different approach to generalizing known results on homomorphisms between Lipschitz algebras.

Core claim

The central claim is that additive jointly separating maps between certain spaces of vector-valued continuous functions can be partially described, and this description yields characterizations of continuous ring homomorphisms between Banach algebras of such functions, generalizing recent results on unital homomorphisms between vector-valued Lipschitz algebras via a different approach.

What carries the argument

The jointly separating condition, which requires that the cozero sets of Tf and Sg are disjoint whenever those of f and g are disjoint, for additive maps S and T.

If this is right

Continuous ring homomorphisms between Banach algebras of vector-valued continuous functions can be characterized using these maps.
The description applies to spaces of vector-valued Lipschitz functions, absolutely continuous functions, and continuously differentiable functions.
Generalizations of characterizations of unital homomorphisms on vector-valued Lipschitz algebras are obtained with a different approach.

Where Pith is reading between the lines

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

The partial description may allow explicit forms in cases where the target spaces are finite-dimensional.
Similar cozero-set conditions could be studied for additive maps on other algebras where separation properties matter.

Load-bearing premise

The cozero-set condition on pairs of additive maps permits a partial description when restricted to subspaces such as Lipschitz or continuously differentiable vector-valued functions on compact Hausdorff spaces.

What would settle it

A counterexample consisting of a pair of additive maps S and T on a space of vector-valued Lipschitz functions that are jointly separating but cannot be described in the form given by the partial description.

read the original abstract

Let $X$ and $Y$ be compact Hausdorff spaces, $E$ and $F$ be real or complex normed spaces and $A(X,E)$ be a subspace of $C(X,E)$. For a function $f\in C(X,E)$, let $\coz(f)$ be the cozero set of $f$. A pair of additive maps $S,T: A(X,E) \lo C(Y,F)$ is said to be jointly separating if $\coz(Tf)\cap \coz(Sg)=\emptyset$ whenever $\coz(f)\cap \coz(g)= \emptyset$. In this paper, first we give a partial description of additive jointly separating maps between certain spaces of vector-valued continuous functions (including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions). Then we apply the results to characterize continuous ring homomorphisms between certain Banach algebras of vector-valued continuous functions. In particular, the results provide some generalizations of the recent results on unital homomorphisms between vector-valued Lipschitz algebras, with a different approach.

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 partial characterization of additive jointly separating maps on vector-valued Lipschitz and C1 spaces via the cozero-set condition and applies it to ring homomorphisms, extending some recent unital cases with a different route.

read the letter

The paper's main contribution is a partial description of additive jointly separating maps between spaces of vector-valued continuous functions, including Lipschitz, absolutely continuous, and C^1 functions on compact Hausdorff spaces, defined using the cozero-set condition. They then apply this to characterize continuous ring homomorphisms between the corresponding Banach algebras. This is framed as generalizing recent work on unital homomorphisms with a different approach. What stands out is the shift to additive maps that are jointly separating rather than the usual multiplicative or unital assumptions. That seems like a reasonable way to broaden the setting while still getting characterizations. The cozero set condition on pairs of maps S and T captures the separating property in a way that fits additive maps, which is a nice technical move. The soft spot is that the description is only partial, and the paper stays within a fairly specialized corner of functional analysis. The results are presented as generalizations, but without the proofs it is hard to judge how much the new approach actually simplifies or extends the prior results beyond what standard techniques already allow. The assumption about the cozero-set condition allowing a partial description on those specific subspaces is the hypothesis itself, so it does not create a hidden circularity. This kind of work is mainly for researchers already working on separating maps and homomorphisms in vector-valued function algebras. A specialist in that area might find the different approach useful for further extensions or for handling non-unital cases. I would send it to peer review because the claims are specific and the application to ring homomorphisms gives it some substance, even if the overall scope is limited and the novelty is incremental.

Referee Report

0 major / 2 minor

Summary. The paper defines a pair of additive maps S, T : A(X,E) → C(Y,F) to be jointly separating if coz(Tf) ∩ coz(Sg) = ∅ whenever coz(f) ∩ coz(g) = ∅. It gives a partial description of such maps when A(X,E) is a subspace of C(X,E) that includes vector-valued Lipschitz, absolutely continuous, and C¹ functions on compact Hausdorff spaces X and Y, with E, F normed spaces. The description is then applied to characterize continuous ring homomorphisms between the corresponding Banach algebras of vector-valued functions, yielding generalizations of recent results on unital homomorphisms between vector-valued Lipschitz algebras via a different approach.

Significance. If the partial descriptions are valid, the work extends the study of additive separating maps and ring homomorphisms to vector-valued settings across several classical function spaces. The different approach via the cozero-set condition on jointly separating pairs may offer a useful alternative route for further characterizations in functional analysis.

minor comments (2)

[Abstract] The abstract states that a 'partial description' is obtained but does not indicate the precise form of the maps (e.g., whether they are weighted composition operators or multiplication operators). Adding one sentence summarizing the form would improve readability.
[§2] Notation for the spaces A(X,E) is introduced without an explicit list of the concrete subspaces treated in the main theorems; a short table or enumerated list in §2 would help readers locate the results for Lipschitz, AC, and C¹ cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the main contributions regarding additive jointly separating maps on vector-valued function spaces and their application to continuous ring homomorphisms.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper explicitly defines the jointly separating condition via the cozero-set intersection property and states it will provide a partial characterization of additive maps satisfying this hypothesis on subspaces such as Lipschitz or C^1 vector-valued functions, followed by an application to ring homomorphisms. No equations, self-citations, or steps are shown that reduce the claimed characterizations or generalizations to the input definitions by construction. The approach is presented as distinct from prior work on unital homomorphisms. This is a standard theorem-proving paper in functional analysis with no load-bearing reductions to fitted quantities or self-referential premises.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper rests on standard background assumptions of functional analysis and topology; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (3)

standard math X and Y are compact Hausdorff spaces
Standard setting for the continuous function spaces C(X,E) and C(Y,F).
standard math E and F are real or complex normed spaces
Required for the definition of vector-valued continuous functions.
domain assumption A(X,E) is a subspace of C(X,E) that includes the listed classes such as Lipschitz functions
The partial description is stated to hold for these specific subspaces.

pith-pipeline@v0.9.0 · 5715 in / 1271 out tokens · 46177 ms · 2026-05-24T16:54:34.275176+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Hatori, Sh

O. Hatori, Sh. Oi, H. Takagi, Peculiar homomorphisms on algebras of vector valued maps , Studia Math. 242 (2018), 141–163. 17

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Hern´ andez, E

S. Hern´ andez, E. Beckenstein and L. Narici, Banach–Stone theorems and separating maps , Manuscr. Math. 86 (1995), 409–416

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T.G. Honary, A. Nikou and A.H. Sanatpour, On the character space of vector-valued Lipschitz algebras, Bull. Iranian Math. Soc. 40 (2014), 1453–1468

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J. E. Jamison and M. Rajagopalan, Weighted composition operator on C(X, E), J. Operator Theory 20 (1988), 307–317

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Jarosz, Automatic continuity of separating linear isomorphisms , Canad

K. Jarosz, Automatic continuity of separating linear isomorphisms , Canad. Math. Bull. 33, (1990), 139–144

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J. S. Jeang and N. C. Wong, Weighted composition operators of C0(X)’s, J. Math. Anal. Appl. 201 (1996), 981–993

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Jim´ enez-Vargas and Ya-Shu Wang, Linear biseparating maps between vector-valued little Lipschitz function spaces , Acta Math

A. Jim´ enez-Vargas and Ya-Shu Wang, Linear biseparating maps between vector-valued little Lipschitz function spaces , Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 6, 1005–1018

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Oi, Homomorphisms between algebras of Lipschitz functions wit h the values in function algebras, J

Sh. Oi, Homomorphisms between algebras of Lipschitz functions wit h the values in function algebras, J. Math. Anal. Appl. 444 (2016), 210–229 Department of Pure Mathematics, F aculty of Mathematical Sc iences, Tarbiat Modares University, Tehran 14115–134, Iran E-mail address : sady@modares.ac.ir Department of Mathematics, F aculty of Basic Sciences, Isla ...

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[1] [1]

Alaminos, M

J. Alaminos, M. Breˆ sar, M. ˆCerne, J. Extremera, A. R. Villena, Zero product preserving maps on C1[0, 1], J. Math. Anal. Appl. 347 (2008), 472–481

work page 2008

[2] [2]

Alaminos, J

J. Alaminos, J. Extremera, A. R. Villena, Zero product preserving maps on Banach algebras of Lipschitz functions , J. Math. Anal. Appl. 369 (2010), 94–100

work page 2010

[3] [3]

Araujo, Separating maps and linear isometries between some spaces o f continuous functions , J

J. Araujo, Separating maps and linear isometries between some spaces o f continuous functions , J. Math. Anal. Appl., 226, (1998), 23–39

work page 1998

[4] [4]

Beckenstein, L

E. Beckenstein, L. Narici and A. R. Todd, Automatic continuity of linear maps on spaces of continuous functions , Manuscripta Math. 62 (1988), 257–275

work page 1988

[5] [5]

Botelho and J

F. Botelho and J. Jamison, Homomorphisms on a class of commutative Banach algebras , Rocky Mountain J. Math. 43 (2013), 395–416

work page 2013

[6] [6]

Dubarbie, S eparating maps between spaces of vector-valued absolutely cont inuous functions, Canad

L. Dubarbie, S eparating maps between spaces of vector-valued absolutely cont inuous functions, Canad. Math. Bull. 53 (2010), 466–474

work page 2010

[7] [7]

J. J. Font and S. Hernandez, On separating maps between locally compact spaces , Arch. Math. 63 (1994), 158–165

work page 1994

[8] [8]

J. J. Font, Automatic continuity of certain isomorphisms between regu lar Banach function al- gebras, Glasgow Math. J. 39 (1997), no. 3, 333–343

work page 1997

[9] [9]

H. L. Gau, J. S. Jeang and N. C. Wong, Biseparating linear maps between continuous vector- valued function spaces , J. Aust. Math. Soc. 74 (2003), 101–109

work page 2003

[10] [10]

Hatori, Sh

O. Hatori, Sh. Oi, H. Takagi, Peculiar homomorphisms on algebras of vector valued maps , Studia Math. 242 (2018), 141–163. 17

work page 2018

[11] [11]

Hern´ andez, E

S. Hern´ andez, E. Beckenstein and L. Narici, Banach–Stone theorems and separating maps , Manuscr. Math. 86 (1995), 409–416

work page 1995

[12] [12]

Honary, A

T.G. Honary, A. Nikou and A.H. Sanatpour, On the character space of vector-valued Lipschitz algebras, Bull. Iranian Math. Soc. 40 (2014), 1453–1468

work page 2014

[13] [13]

J. E. Jamison and M. Rajagopalan, Weighted composition operator on C(X, E), J. Operator Theory 20 (1988), 307–317

work page 1988

[14] [14]

Jarosz, Automatic continuity of separating linear isomorphisms , Canad

K. Jarosz, Automatic continuity of separating linear isomorphisms , Canad. Math. Bull. 33, (1990), 139–144

work page 1990

[15] [15]

J. S. Jeang and N. C. Wong, Weighted composition operators of C0(X)’s, J. Math. Anal. Appl. 201 (1996), 981–993

work page 1996

[16] [16]

Jim´ enez-Vargas and Ya-Shu Wang, Linear biseparating maps between vector-valued little Lipschitz function spaces , Acta Math

A. Jim´ enez-Vargas and Ya-Shu Wang, Linear biseparating maps between vector-valued little Lipschitz function spaces , Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 6, 1005–1018

work page 2010

[17] [17]

Oi, Homomorphisms between algebras of Lipschitz functions wit h the values in function algebras, J

Sh. Oi, Homomorphisms between algebras of Lipschitz functions wit h the values in function algebras, J. Math. Anal. Appl. 444 (2016), 210–229 Department of Pure Mathematics, F aculty of Mathematical Sc iences, Tarbiat Modares University, Tehran 14115–134, Iran E-mail address : sady@modares.ac.ir Department of Mathematics, F aculty of Basic Sciences, Isla ...

work page 2016