# P2924: Openness of convolution on l1 of the integers

- ID: `P2924`
- Reference: `l1z-convolution-open-map`
- Page: https://theoremdb.org/statements/P2924
- Record maturity: Reviewed problem with recorded work

## Problem

Is the convolution map \(\ell_1(\mathbb Z)\times\ell_1(\mathbb Z)\to\ell_1(\mathbb Z)\), \((a,b)\mapsto a*b\), an open map?

### Problem setup

- **Remark.** For a,b in ell_1(Z), convolution is (a*b)_n=sum_{k in Z} a_k b_{n-k}, which again belongs to ell_1(Z).
- **Definition.** A map is open if it sends every open subset of the product Banach space to an open subset of ell_1(Z); equivalently, it is locally open at every pair (a,b).

### What counts as a solution

- Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to a*b that cannot be factored by pairs converging to (a,b).
- A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: The thread records local openness when one factor is finitely supported and failure of a uniform openness estimate. It does not establish or refute openness at every pair. Exact unresolved remainder: Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to a*b that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method. [3](#reference-3) [2](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: The thread records local openness when one factor is finitely supported and failure of a uniform openness estimate. It does not establish or refute openness at every pair. Exact unresolved remainder: Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to a*b that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: The thread records local openness when one factor is finitely supported and failure of a uniform openness estimate. It does not establish or refute openness at every pair.

Exact unresolved remainder: Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to a*b that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method.

### Background and intake notes

Fourier zero sets, finitely supported approximation lemmas, and quantitative local radii can be reused. The source already separates pointwise openness from the stronger uniform-open-map property.

- Original intake status: UNKNOWN as of 2026-07-27. The thread records local openness when one factor is finitely supported and failure of a uniform openness estimate. It does not establish or refute openness at every pair.
- The MathOverflow question and every visible answer and comment were checked on 2026-07-27. The partial argument covers finitely supported factors, a dense class, while the all-pairs question remains unanswered.
- Work on open bilinear mappings, including DOI 10.1215/17358787-3599741, was checked for a theorem applying directly to convolution on ell_1(Z). No such specialization was found.
- Norm-controlled inversion results for convolution algebras were checked because invertible factors yield local solution mechanisms. They do not cover arbitrary pairs whose Fourier transforms have common zeros.
- A local corpus search for open convolution map, ell_1(Z), and Banach algebra multiplication found no duplicate.

- Recorded example: If one factor is invertible in the convolution algebra, multiplication by that factor is a Banach-space isomorphism and gives local openness at the pair.

### Open directions

- **Route 1** (reported): Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to a*b that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `l1z-convolution-open-map`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>MathOverflow question 504613, “Openness of convolution on l1 of the integers,” checked 2026-08-01. Question 504613 and every visible answer and comment were checked on 2026-07-27. https://mathoverflow.net/questions/504613/is-convolution-in-ell-1-mathbb-z-an-open-map
   - Also cited at Full question, answers, and visible comments concerning Openness of convolution on l1 of the integers; checked 2026-08-01.
   - Also cited at Editorial research route recorded 2026-08-01.
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For Openness of convolution on l1 of the integers, the reviewed source scope is Full question, answers, and visible comments concerning Openness of convolution on l1 of the integers; checked 2026-08-01.. The packet makes no inference beyond that cited scope.
   - Source named by the research packet.
2. <a id="reference-2"></a>Marek Balcerzak, Ehrhard Behrends, and Filip Strobin, “On certain uniformly open multilinear mappings,” Banach Journal of Mathematical Analysis 10(3) (2016), 482-494. DOI 10.1215/17358787-3599741. abstract and uniform-openness results for pointwise multiplication and nonzero multilinear functionals https://doi.org/10.1215/17358787-3599741
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Openness of convolution on l1 of the integers, this source supplies techniques and examples for different multilinear maps without settling l1(Z) convolution.
3. <a id="reference-3"></a>Karlheinz Gröchenig and Andreas Klotz, “Norm-controlled inversion in smooth Banach algebras, I,” Journal of the London Mathematical Society 88(1) (2013), 49-64. DOI 10.1112/jlms/jdt004. main norm-controlled inversion inequalities for smooth Banach subalgebras https://doi.org/10.1112/jlms/jdt004
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Openness of convolution on l1 of the integers, this source controls inverse norms in convolution-type Banach algebras without proving openness of l1(Z) convolution.