Spectrum of difference of two projections

Question

Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ?

The exercise is to show that $\lVert p-q \rVert \leq 1$.

Any hints are appreciated.

Answer 1

You have, since $0\leq q\leq I$ and $0\leq p\leq I$, $$ -I\leq -q\leq p-q\leq I-q\leq I. $$ So, as you mentioned, it follows that $\sigma(p-q)\subset[-1,1]$.

Note also that the argument does not use that $p,q$ are projections, only that they are positive elements of the unit ball.