most recent 30 from http://mathoverflow.net 2010-08-01T09:35:53Z http://mathoverflow.net/feeds/question/5945 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5945/what-morphisms-morita-equivalences-induce-the-2-periodicity-isomorphisms-of-kk Michael 2009-11-18T10:27:14Z 2010-03-05T16:08:58Z

<p>In Kasparov's paper, the canonical isomorphisms <code>$KK_* \rightarrow KK_{*+2k}$</code> are defined rather implicitely (by tensoring and stabilization).</p> <p>Are there morphisms of $C^*$-algebras which induce them (e.g. I've heard that the morphism $\varphi: \mathbb{C} \rightarrow \mathbb{C}_2$ sending 1 to $1 + i e_1 e_2$ induces the iso <code>$KK_* \rightarrow KK_{*+2}$</code>), and how to see that?</p> <p>Similarly, a graded irreducible representation of <code>$\mathbb{C}_{2k}$</code> gives a Morita equivalence between $\mathbb{C}$ and <code>$\mathbb{C}_{2k}$</code>. Does it induce the periodicity isomorphism, and if yes, which of the two gradings should one use? (relationship to the above question: $\varphi$ is easily seen to be the $KK$-inverse of the standard graded irreducible representation $W$ of <code>$\mathbb{C}_2$</code> where the complex volume element implements the grading, since then $\varphi(1)$ projects onto the even part of $W$)?</p>