The stabilized homotopy category of graded C* algebra

Question

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

Answer 1

In what follows, all tensor products are graded.

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).

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.