# Suggested tag: approximation-property

I'm relatively new to MO (but have been on MSE for a longer time), which means that I don't have enough reputation to create new tags. Which is probably a good thing.

Lately I've been puzzling with Banach space tensor products and the approximation property, and on multiple occasions found that my search would have been easier if there had been a tag for questions related to the approximation property. It is my impression that it is a whole field of its own, with a rich body of theory and (especially) counterexamples, so to me a tag approximation-property would make sense.

Since I understand that we don't want to go haphazardly creating new tags for every specialization (or niche) within mathematics, I thought I'd ask what you think. Has this been considered before? Do we like the idea? Do we hate it? Should we consider variations on the proposal (e.g. a tag metric-theory-of-tensor-products)?

Added later: as per François' suggestion, here is my suggested tag info.

For questions relating to various approximation properties in functional analysis and (abstract) harmonic analysis, including approximation properties of Banach spaces, locally convex spaces, operator algebras, and locally compact groups.

Furthermore, I tried to compile a list of questions to which the new tag would apply. Note that this list might be incomplete, again because I found it difficult to find all relevant questions on this topic. (Some would only mention the term "AP", so those would be hard to track down. This applies to one of the questions below – I only found it because it happened to be linked to another question on my list.)

Questions directly related to the approximation property (sorted by publication date).

Questions tangentially related to the approximation property (sorted by publication date).

Questions containing the term "approximation property" which nevertheless fall outside the scope of the proposed tag (mostly regarding different approximation properties, e.g. in algebra or logic).

• My two cents: at the very least a tag metric-theory-of-tensor-products seems superfluous. Questions regarding e.g. tensor products of Banach spaces can easily be found by combining the tags banach-spaces and tensor-products. – J. van Dobben de Bruyn Aug 28 '19 at 20:41
• Are there existing questions where the tag would apply? (Tags can't be created for their own sake, they need to be applied to a question.) – François G. Dorais Aug 28 '19 at 21:12
• @FrançoisG.Dorais by my count, at least somewhere between 30 and 50 existing questions would fall under this tag. – J. van Dobben de Bruyn Aug 28 '19 at 21:20
• Allright, help people out and create a list of questions that would benefit from this tag! For additional traction, you could also propose a tag info blurb. – François G. Dorais Aug 28 '19 at 21:53
• Done! I think my proposed tag info should be checked by an expert. For instance, I don't know too much about the (Haagerup) approximation property for locally compact groups, but I get the feeling that it is related to other approximation properties in functional analysis (maybe via the group algebra?), and therefore should be included in this tag (which is what I've done now). But I'm not sure if that makes sense. – J. van Dobben de Bruyn Aug 29 '19 at 20:44
• This is wonderful! I think this is now a model tag request. (Which we needed.) I am not at all an expert in this area but I hope experts will chime in sooner than later. – François G. Dorais Aug 29 '19 at 21:06
• I read on Wikipedia that the term approximation property is also used in ring theory. Should it be explicitly mentioned in tag-info (perhaps tag-wiki if not tag-excerpt) that this is not meant for ring theory? Would it perhaps be better to choose the tagname which would distinguish this from algebra, for example, (approximation-property-analysis)? – Martin Sleziak Sep 1 '19 at 4:56
• I confirm that the various approximation properties for locally compact groups (mainly the AP of Haagerup and Kraus and weak amenability, but also the Haagerup property) fit perfectly well in the description of the tag. The reason for their introduction by Haagerup and coauthors is that for discrete groups, they characterize corresponding approximations properties (à la Grothendieck) of the operators algebras of the group. – Mikael de la Salle Sep 3 '19 at 9:35
• I think that all this is excellent, but unfortunately this tag will attract a large number of off-tag questions (typing "approximation" will propose this tag, and whatever good the tag info is, many users, especially new ones, do not read it or ignore it). – YCor Sep 6 '19 at 16:06
• @YCor That is somewhat similar situation to the (divisors) tag, in this case, there is a suggestion to change the name of the tag. Do you have some name that would fit 35 characters and be distinctive enough to prevent the incorrect usage? The name (approximation-theory-analysis) suggested in a previous comment could probably still be misused. And alternative would be (approximation-property-funct-an) - or something similar within the character limit - however I don't really like such shortening. – Martin Sleziak Sep 7 '19 at 20:06
• That's a fair point, but bear in mind that typing "approximation" first shows approximation-theory, diophantine-approximation, and approximation-algorithms. Wouldn't they go with approximation theory instead? – J. van Dobben de Bruyn Sep 8 '19 at 18:23
• Otherwise, perhaps operator-approximation-property? – J. van Dobben de Bruyn Sep 8 '19 at 18:24
• It has come to my attention that one particular version of the approximation property is called "operator approximation property" (OAP). (This is not the classical approximation property.) So I guess the suggestion from the preceding comment might falsely suggest that the tag focuses exclusively on this OAP, as opposed to any kind of AP. Therefore I change my previous suggestion to operator-approximation-properties. – J. van Dobben de Bruyn Sep 10 '19 at 16:18
• (By the way, I'm still in favour of calling it simply approximation-property. I don't think it'll be so bad. But then, what do I know?) – J. van Dobben de Bruyn Sep 10 '19 at 16:28
• @J.vanDobbendeBruyn I will just point out that mods can rename a tag - so it's possible to rename it later if the tag becomes problematic. (Technically, it is merging into another tag rather than renaming. But the end result is a tag containing the same question with a new name. See also:How to rename a tag? and Can you change the name of a tag?) – Martin Sleziak Sep 11 '19 at 4:56