Certain resolvent functions generate $C_0(\mathbb{R})$?

Question

I have two questions about the following passage taken from Higson-Roe's Analytic K-Homology:

1) What does "$T$ graded-commutes with $(i\pm D)^{-1}$" mean? In particular, what grading is being placed on the bounded operators on $H$?

2) In what sense do "$(i\pm x)^{-1}$ generate the $C^*$-algebra $C_0(\mathbb{R})$"?

Thanks!

Answer 1

For $T$ and $D$ as described, $$ TD=-DT \\ T(\lambda I+D) = (\lambda I-D)T \\ (\lambda I-D)^{-1}T = T(\lambda I+D)^{-1} \\ ((\lambda I-D)^{-1}-(\overline{\lambda}I-D)^{-1})T=T((\lambda I+D)^{-1}-(\overline{\lambda}I+D)^{-1}). $$ If $\varphi$ is a bounded Borel function on $\mathbb{R}$, then the above leads to $$ \varphi(D)T=T\varphi(-D). $$ This is because \begin{align} \mbox{s-}\lim_{v\downarrow 0} \frac{1}{2\pi i}\int_{a}^{b}\varphi(u)\{&((u-iv)I-D)^{-1} \\ -&((u+iv)I-D)^{-1}\}du \\ = &(\chi \varphi)(D) \end{align} where $\chi$ is $1$ on $(a,b)$, is $1/2$ at $a$, $b$, and is $0$ otherwise, which follows from properties of the Poisson integral.