A theorem by Kirchberg and Phillips states that two unital separable nuclear simple purely infinite $C^*$-algebras (so called Kirchberg algebras) satisfying the Universal Coefficient Theorem are isomorphic if and only if their topological $K$-theory groups are isomorphic.

Taking here direct sums of abelian groups at the $K$-theory side, what does it mean at the Kirchberg algebras side?

  • $\begingroup$ You want to know which operation on Kirchberg algebras corresponds to direct sum of $K$-groups (of course direct sum of simple algebras takes you out of simple algebras). But for your question to make sense, you need to specify the range of the $K$-theory invariants (i.e. which are the pairs of abelian groups that you can get as $(K_0,K_1)$ of a Kirchberg algebra). If direct sum of $K$-groups takes you out of the range, your question may be meaningless... $\endgroup$ – Alain Valette Mar 4 '13 at 17:28
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    $\begingroup$ @Alain: By Proposition 4.3.4 in the book "Classification of Nuclear, Simple $C^*$-algebras" by Rordam, any pair $(G_0,G_1)$ of countable abelian groups can be realized as $(K_0(A), K_1(A))$ of a unital Kirchberg algebra $A$, where $A$ can even be chosen to be in the UCT class. $\endgroup$ – Ulrich Pennig Mar 4 '13 at 18:25
  • $\begingroup$ @Ulrich: yes this fact I had in mind. One can even choose to be A a graph C∗-algebra, if one likes. – $\endgroup$ – Hans Mar 4 '13 at 19:59
  • $\begingroup$ @Ulrich: OK, then I eventually made sense of the OP. But can you describe, for instance, the operation on Kirchberg algebras that flips $K_0$ and $K_1$, i.e. which has the same effect on K-theory as tensoring by $C_0(\mathbb{R})$? $\endgroup$ – Alain Valette Mar 4 '13 at 20:19
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    $\begingroup$ @Alain: My guess would be that you can take any Kirchberg algebra with K0(A)=0 and K1(A)=ℤ, form the tensor product with that and use the Künneth theorem to see that the result has switched K-groups. $\endgroup$ – Ulrich Pennig Mar 4 '13 at 22:17

The most concrete way of constructing Kirchberg algebras with given $K$-theory that I know of, is due to Rordam and Elliott-Rordam, and is given in a series of papers "Classification of Certain Infinite Simple C*-Algebras" (Rordam) and "Classification of Certain Infinite Simple C*-Algebras, II" (Elliott-Rordam). There, Kirchberg algebras are constructed as crossed products of either AF-algebras or AT-algebras (depending on whether there is torsion on $K_1$ or not) by endomorphisms, similar to how the Cuntz algebras are obtained from UHF-algebras. I suppose one could go through their construction and see how they get the AF/AT-algebras and the endomorphism from the given $K$-theoretical data (which in general will also include the class of the unit), and then try to see if there is any way of relating the algebras one gets for two different choices, and the algebra one gets for the sum of the $K$-groups. One complication is that all the AF and AT-algebras involved are simple (so taking the sum of the AF/AT-algebras won't work either).

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