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).*

Examples of groups without the n-positive approximation property

Are the compact and Haagerup approximation properties equivalent?

What Approximation Property does the space of Schatten-p class operators have?

$C^*$-algebra generated by those operators that are bounded on every $\ell_p$

A question on metric characterization of approximation property

Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?

Does the Approximation Property (AP) pass to quotients by amenable subgroups?

For discrete groups, does the Haagerup property imply the AP of Haagerup-Kraus?

Is there a quotient of $c_0$ without the approximation property?

Does the Banach algebra of jets have the approximation property?

The approximation property for some spaces of holomorphic functions

Completely bounded maps approximately factoring through finite matrices

Approximation property counterexamples? (Also: relation to tensor products)

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

Approximation property of Fréchet if range is restricted to an embedded Hilbert space

A "slice-map" type problem for symmetric tensors in the square of a nuclear C*-algebra

Noncommutative analogs of classical Banach geometric properties

Outline of Generic Separable Banach Spaces don't have a Schauder Basis

Characterization of exact groups via the existence of amenable actions on unital C*-algebras, part 2

When does the dual to the space $K(X)$ of compact operators consist of nuclear functionals?

*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).*

- Real approximation for homogeneous spaces of linear algebraic groups
- Infinitely many solutions of a diophantine equation
- What could be some potentially useful mathematical databases?
- Class number of PGL_2
- Are there lightweight foundations for arbitrarily extendable objects?
- Analytic elements in non-archimedean geometry
- Is there a truth approximation on a cumulative hierarchy?
- A cohomology group which depends on the connection
- Preserving Jonsson cardinals
- The $\delta$-approximation property for ground models
- If the finitely additive measure of an open set is approximable by clopen sets, is it approximable from within?

`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`

. $\endgroup$approximation propertyis 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)? $\endgroup$`operator-approximation-property`

? $\endgroup$`operator-approximation-properties`

. $\endgroup$`approximation-property`

. I don't think it'll be so bad. But then, what do I know?) $\endgroup$