Assume $\mathcal{A}$ is a $C^*$-algebra with unit $1$ and $\mathcal{B}\subset\mathcal{A}$ is a $C^*$-subalgebra (i.e. a closed $*$-subalgebra) such that $1\in\mathcal{B}$. It is said that under these assumptions, for any $a\in\mathcal{B}$ the spectrum $\sigma_\mathcal{B}(a)$ of $a$ in $\mathcal{B}$ coincides with $\sigma_\mathcal{A}(a)$, that is: If $a-\lambda 1$ has an inverse $b\in\mathcal{A}$, then $b\in\mathcal{B}$.
Now, my questions are the following:
- Do you know a nice proof of the above statement? I have found a proof that goes like this: If $b=(a-\lambda)^{-1}$ exists in $\mathcal{A}$, it can be expressed as a convergent power series, i.e. it is the norm limit of partial sums each belonging to $\mathcal{B}$, hene also $b\in\mathcal{B}$. Although this argument looks really nice and I'm aware of Neumann series, I do not see why $b$ can be expressed as a power series. Do you?
- Under the above assumptions, is the more general statement $\mathcal{B}^\times=\mathcal{A}^\times\cap B$ true? (here, we denote by $\mathcal{A}^\times$ and $\mathcal{B}^\times$ the set of invertible elements in $\mathcal{A}$ and $\mathcal{B}$, respectively)