Something I've been thinking about for a while that I'm not sure I understand is why $\mathcal{Z}$ stability, as opposed to say $\mathcal{O}\_\infty$-stability or even $\mathcal{K}$-stability is so important to representation theory. I know that the Jiang-Su algebra has a lot of interesting properties such as being strongly self-absorbing, or projectionless, simple, KK-equivalent to $\mathbb{C}$. I can certainly see that it is an interesting object, but I think I struggle to understand the relevance for classification.


I was wondering if someone a bit closer to classification might be able to explain a bit is why $\mathcal{Z}$-stability is the type of stability we are interested in for Elliot classification. One thing I remember hearing in the YMC\*A minicourse Chris Schafhauser gave this year is that it's the analogue to the hyperfinite $II\_1$ factor in von Neumann algebras. If anyone could expand on this I would be really interested.


Bit of a soft question I suppose, is more out of interest than anything. Probably there is an abstract in a paper/introduction which explains this and pointing me in the right direction would be good.



What is the significance of the Jiang Su algebra in classification of C$^*$ -algebras?
Diego's answer in the comments above is related to why we would expect any classifiable $C^\ast$-algebra to satisfy $A\cong A\otimes \mathcal Z$:
Since $\mathcal Z$ is separable, nuclear, unital, simple and UCT (the properties of $C^\ast$-algebras we wish to classify by $K$-theory and traces),
tensoring with any other $C^\ast$-algebras with these properties will have all the same properties.
And since $\mathcal Z$ has the same $K$-theory and traces as $\mathbb C$, one doesn't change $K$-theory and traces by tensoring with $\mathcal Z$.
Hence if we expect to have classification by $K$-theory and traces, we would expect $A \cong A \otimes \mathcal Z$ for all $C^\ast$-algebras classified by $K$-theory and traces.


Now, I really want to address a misconception which I think is quite common amongst people not working in the
classification or structure programme of nuclear C\*-algebras:


*$\mathcal Z$-stability (essentially) has nothing to do with the Jiang-Su algebra $\mathcal Z$!*


For instance, the UHF algebras, the irrational rotation algebras, and the Cuntz algebras are all $\mathcal Z$-stable, but they have nothing to do with $\mathcal Z$.


This is similar to how McDuff factors\* don't really have anything to do with $\mathcal R$,
or how $\mathcal O\_\infty$-stable $C^\ast$-algebras have nothing to do with $\mathcal O\_\infty$\*\*


One should consider the very natural and frequently occuring regularity property "$\mathcal Z$-stability" as being equivalent
(by very deep, surprising, and non-constructive theorems!) to "$A \cong A\otimes \mathcal Z$".
Similar to the McDuff property, $\mathcal Z$-stability can be characterised by the (norm-)central sequence algebra $\frac{\prod\_{\mathbb N} A}{\bigoplus\_{\mathbb N} A} \cap A'$ (for separable unital $A$)
being suitably non-trivial, e.g. by requiring that it contains a unital copy of $Z(2,3)$;
the $C^\ast$-algebra of continuous functions $f\colon [0,1] \to M\_2(\mathbb C) \otimes M\_3(\mathbb C)$ with $f(0) \in M\_2(\mathbb C) \otimes 1$ and $f(1) \in 1 \otimes M\_3(\mathbb C)$.


Unfortunately, it would be much too technical to explain exactly how this property is used in the classification thereom.
A key technique is that these "almost central homotopies" coming from $Z(2,3)$ can for instance be used to show that $\mathcal Z$-stable $C^\ast$-algebras have cancellation of full projections by $K\_0$, that $K\_1(A) = U(A)/U\_0(A)$
(this was proved in an unpublished paper by Jiang in the late 90's; the paper can be found on arXiv), and to prove that the Cuntz semigroup is almost unperforated (Rørdam).


$\mathcal Z$-stability turns out to be a very mild property that can be verified in an abundance of examples, such as through the Toms-Winter conjecture in the cases where this is known to hold, such as separable, simple, non-type I $C^\ast$-algebras with finite nuclear dimension.


In fact, it is very hard to construct separable simple nuclear non-type I $C^\ast$-algebras which are not $\mathcal Z$-stable (cf. Villadsen, Rørdam, Toms).
So if you have any natural construction that gives you a separable, nuclear, simple $C^\ast$-algebra, you are almost certain to obtain a $\mathcal Z$-stable $C^\ast$-algebra
unless you tried very very hard to construct one without this property.




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(\*) McDuff factor: separable $II\_1$-factors such that $M \cong M\overline{\otimes} \mathcal R$. Equivalently, their ($W^\ast$-)central cequence algebras are non-abelian.


(\*\*) For separable, nuclear, simple $C^\ast$-algebras being $\mathcal O\_\infty$-stable is equivalent to being purely infinite by a theorem of Kirchberg.
In particular, the Cuntz algebras $\mathcal O\_n$ all satisfy $\mathcal O\_n \cong \mathcal O\_n \otimes \mathcal O\_\infty$ (this is quite spectacular; you can't express this isomorphism explicitly!).
Being $\mathcal O\_\infty$-stable is equivalent to the central sequence algebra $\frac{\prod\_{\mathbb N} A}{\bigoplus\_{\mathbb N} A} \cap A'$ (for separable unital $A$)
containing two isometries $s\_1,s\_2$ such that $s\_1^\ast s\_2 = 0$.
And this is the property which is used for classifying purely infinite $C^\ast$-algebras by "moving things around" in a paradoxical way.


