I have a question concerning the spectrum of a projection in an abstract $C^*-$algebra. Let $A$ be $C^*-$algebra, $p\in A$ a projection (i.e. a selfadjoint idempotent). Can we say what is the spectrum of $p$?
Thank you for for attention and the help.
On
Martini's answer is the canonical one. But in this case one can check things explicitly:
We can assume that $A$ is unital, as we can always work on the unitization (we actually need it to define the spectrum).
If $p=I$ of $p=0$, it is straightforward to check that the spectra are respectively $\{1\}$ and $\{0\}$.
For non-trivial $p$: If $\lambda=0$, $p$ is not invertible, because $p(I-p)=0$. Similarly, $I-p$ is not invertible.
For $\lambda\not\in\{0,1\}$, one can check directly that $$ (p-\lambda I)^{-1}=\frac1{\lambda(1-\lambda)}\,p-\frac1\lambda\,I. $$ So $\sigma(p)=\{0,1\}$.
We can say that $\sigma(p) \subseteq \{0,1\}$. $p - p^2 = 0$, and hence, by the spectral theorem $$ \{0\} = \sigma(0) = \{\lambda - \lambda^2 \mid \lambda \in \sigma(p) \}$$