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Motivation:

For a topological space $X$ one can consider under certain circumstances the cohomology ring of suitable cohomology theories, for example:

1) The cohomology ring $H^*(X;R)=\oplus_{i\ge 0} H^i(X;R)$ in singular cohomology together with coefficients in a, lets say commutative ring $R$, with the cup-product $\cup$. For example, the complex projective space in dimension $n$, $X=\mathbb{C}P^n$, and $R=\mathbb{Z}$ you can use Poincaré-duality (for closed manifolds) to prove $H^*(\mathbb{C}P^n;\mathbb{Z})\cong \mathbb{Z}[\xi ]/(\xi^{n+1})$, where $\xi\in H^2(\mathbb{C}P^n;\mathbb{Z})$ is a generator.

2) $X$ compact Hausdorff, the fiberwise tensor product of vector bundles makes the topological $K$-theory $K(X)$ a unital ring. For example, as a ring ( note that K^1 vanishes) we have: $K^0(\mathbb{C}P^n)\mathbb{Z}[\eta ]/(\eta^{n+1})$, where $\eta=l-\tilde{1}$ is the Hopf bundle and $\tilde{1}$ is the trivial vector bundle $\mathbb{C}P\times\mathbb{C}\to \mathbb{C}P$ of dimension 1.

The setting for C*-algebras and my questions:

It's well known that the KK-theory of separable $C^*$-algebras $A$ together with the Kasparov-product allows to consider $KK_*(A,A)$ as a $\mathbb{Z}/2\mathbb{Z}$-graded unital ring. Furthermore, it's known that $KK$-theory contains the topological $K$-theory as a special case and that the Kasparov-product contains in special cases the K-theoretic cup-product, cap-product and so on. However, $KK$ is bivariant which can be seen as a cohomology theory in the first variable and as a homology theory in the second variable (i.e. it's another situation as in the motivation above). Thus, even though these questions are maybe somehow naive, but out of interest:

  1. I would like to know if there are known expressions and computations of the ring structure $KK_*(A,A)$ (analogously as for the topological cases) for certain $C^*$-algebras.

  2. Additionally, considering $KK_*(A,A)$ as a ring, how can one use this for classification of $C^*$-algebras; are there known results where the ring structure is an important consideration? (Not only to check if $C^*$-algebras are maybe non-homotopic or something like that.). Can anyone provide a reference?

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    $\begingroup$ I asked an even less ambitious question several years ago, and the answers there might help with your question. mathoverflow.net/questions/51203/… $\endgroup$
    – Paul Siegel
    Commented Mar 7, 2017 at 19:44
  • $\begingroup$ @PaulSiegel thank you. Well, now that you mention it, I have never seen calculations of $KK(A,B)$ in non-trivial cases as well (I wasn't aware of it). I have seen some calculations using the universal coefficient theorem, but mostly in order to calculate the $K$-theory of $C^*$-algebras. (Thus, I guess that the answer of question 1. is "no" for non-trivial cases ) $\endgroup$
    – Sabrina Gemsa
    Commented Mar 7, 2017 at 23:42
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    $\begingroup$ I think the UCT-answer in Paul's link is the canonical one here. Note that the Ext-part vanishes in many interesting cases, such as for instance when $K_*(A)$ is finitely generated, so all you are left with is the usual Hom-set of endomorphisms on $K_*(A)$. $\endgroup$
    – Gabor Szabo
    Commented Mar 8, 2017 at 6:56
  • $\begingroup$ @GaborSzabo Ext doesn't vanish for finitely generated abelian groups, for instance $\mathrm{Ext}(\mathbb Z/2,\mathbb Z/2)\cong\mathbb Z/2$. $\endgroup$
    – Rasmus
    Commented Apr 16, 2017 at 21:21
  • $\begingroup$ Oops -- I think I was confused about something there. Sorry about that. $\endgroup$
    – Gabor Szabo
    Commented Apr 18, 2017 at 11:45

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