## Higson’s spectral picture of K-theory

On page 149 of the note "Group C*-Algebras and K-theory" by N.Higson and E.Guentner:

If $A$ is a graded $C^*$-algebra, define $K(A)=[S,A\widehat{\otimes}K(H)]$, where $S=C_0(R)$ and $[A,B]$ is the set of homotopy classes of graded *-homomorphisms between graded A and graded B. The addition on $K(A)$ is the direct sum of graded *-homomorphisms. But Higson/Guentner says that the inverse of a graded *-homomorphism $\psi \in K(A)$ obtained by composing $\psi$ with the grading automorphism on $S$ abd also reversing the grading on the Hilbert space $H$. I really don't understand why we need reverse the grading on $H$? But I notice that the rotation homotopy given on page 149 is not a path of Cayley transform for $A\widehat{\otimes}K(H)$ for exemple for $t=\pi/2$, are not equal to 1 modulo $A\widehat{\otimes} K(H \oplus H^{opp})$. I think the "stardard" homotopy (the one in Thm 4.2,9 [N.E. Wegge-olsen]) will work.

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