Consider continuous complex-valued functions on real line which vanish at infinity $$ C_0 (\mathbb{R}) := \{f\in C(\mathbb{R}):\lim_{|t|\to+\infty}f(t) = 0,f(\mathbb{R}) \subset \mathbb{C}\} $$ as a $C^*$-algebra under the pointwise conjugation. In HG04 one talks about suspension algebra and KK-theory/E-theory and the author states that $$ C_0 (\mathbb{R}) := \overline{C^* [u,v]},\quad\quad u(x) = e^{-x^2},\,v(x)= xe^{-x^2} \in C_0(\mathbb{R}) $$ as $C^*$-algebra.($\overline{(..)}$ is taking closure under the $C^*$-norm on $C_0(\mathbb{R})$ which is pointwise maximal norms)
I do not know how to derive this results. This maybe relates to some approxiamation theorems by Hermite polynomials(What I Guess)? If somebody could figure it out I'll be much grateful.
The ultimate aim of this question is a starting point of $KK$-theory, where $C^*$-algebra $C_0(\mathbb{R})$ stands in a crucial stage in amplification of $C^*$-algebra. In HG04 one define a coalgebra structure: $$ \Delta: C_0(\mathbb{R}) \to C_0(\mathbb{R})\widehat{\otimes}C_0(\mathbb{R}) $$ And I do not know how to computer the relations below: $$ \qquad \Delta(u) = u\widehat{\otimes} u, \Delta(v) = u\widehat{\otimes} v + v \widehat{\otimes} u $$ where $\widehat{\otimes}$ is the minimal tensor product of $C^*$-algebra.
And $\Delta(f)$ is defined as the limit of functional calculation of $f(\mathrm{Id}_{[-R,R]}\widehat{\otimes} 1 + 1\widehat{\otimes} \mathrm{Id}_{[-R,R]})$, where $1,\mathrm{Id}_{[-R,R]} \in \mathcal{S}_R = C_0 (\mathbb{R})/ \mathrm{ker}\mathrm{(-)|}_{[-R,R]}$.
This issue is about suspension algebra $S(A):= C_0 (\mathbb{R})\otimes A$. But I do not know how to compute the $\Delta$ map. Can some experts on KK-theory and analytic index theory give some explanations?
Appreciating any useful comments.