The stabilized homotopy category of graded C* algebra - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:41:42Z http://mathoverflow.net/feeds/question/66218 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66218/the-stabilized-homotopy-category-of-graded-c-algebra The stabilized homotopy category of graded C* algebra m07kl 2011-05-27T20:19:16Z 2011-05-27T21:13:57Z <p>Hi everyone On page 147 of the note "Group C*-Algebras and K-theory" by N.Higson and E.Guentner there are something about the stabilized homotopy category of graded C* algebra, which is a category whose objects are the graded C* -algebras and morphisms from A to B are the homotopy classes of graded $\ast$-homomorphisms from A to $B\otimes K(H)$. But the exact definition of composition of morphisms and the identity morphisms are not mentioned. I think the definition is dual to the defition of amplified category of graded C*-algebras, is there someone knows some references about this?\newline Thinks</p> http://mathoverflow.net/questions/66218/the-stabilized-homotopy-category-of-graded-c-algebra/66230#66230 Answer by Paul Siegel for The stabilized homotopy category of graded C* algebra Paul Siegel 2011-05-27T21:13:57Z 2011-05-27T21:13:57Z <p>In what follows, all tensor products are graded.</p> <p>The comments about the existence of canonical (up to homotopy) $\ast$-homomorphisms $\mathbb{C} \to K(H)$ and $K(H) \otimes K(H) \to K(H)$ right before the definition of the category in question are key. If you have $\ast$-homomorphisms $A \to B \otimes K(H)$ and $B \to C \otimes K(H)$ then the composition is just $A \to B \otimes K(H) \to C \otimes K(H) \otimes K(H) \to C \otimes K(H)$. And the identity morphism is just the map $A \to A \otimes K(H)$ given by $a \mapsto a \otimes e$ where $e$ is the projection onto a one dimensional grading-degree zero subspace of $H$ (well-defined up to homotopy). </p> <p>I'm not totally sure where you can find further discussion of this category, but you might try the AMS Memoire "Equivariant E-Theory for C* algebras" by Higson, Guentner, and Trout.</p>