For a unital $C^*$-algebra $\mathcal{A}$ the spectral permanence gives \begin{equation} \sigma_{\mathcal{B}}(a)=\sigma_{\mathcal{A}}(a) \end{equation} for any unital $C^*$-subalgebra $\mathcal{B}$.
It is natural to look at the smallest such subalgebras, namely, the $C^*$-subalgebra generated by $1,a$ and $a^*$. Then the permanence says if $\lambda-a$ is invertible, then $\lambda-a$ is in the closed linear span of products of $1,a$ and $a^*$ (although order of multiplications matters here and it is not actually a polynomial).
I am wondering whether there is some canonical way to construct these 'polynomials'. That is, given $a\in\mathcal{A}$ invertible, how can one find explicitly the linear span of products of $1,a$ and $a^*$ that converges to $a^{-1}$?
Thanks!
My feeling (not formally justified) is that that there is no canonical choice. Take a look at the simplest example: let $$ a=\begin{bmatrix}2&0\\0&3\end{bmatrix}, $$ $\lambda=1$. So $a$ is selfadjoint, and of course $$ (a-\lambda)^{-1}=\begin{bmatrix}1&0\\0&1/2\end{bmatrix}=p(a-\lambda) $$ for an appropriate polynomial. Now, what is the canonical polynomial that takes $1,2$ to $1,1/2$? Let us assume we want the minimum degree possible (i.e. $2$); this is already arbitrary. You could ask that the polynomial be monic, and in this case $$ p(t)=t^2-\frac72\,t+\frac72; $$ or you could want $p(0)=0$, in which case $$ p(t)=-\frac34\,t^2+\frac74\,t. $$ I don't really see a reason that makes one more canonical than the other.