Projector in C*Algebra

Question

If $p \in P(A)$ a projector in unital $C^*$ Algebra $A$ then is it true that $(I + p) \in P(A)$? Of course the self-adjointness property is satisfied but what about idempotence?

Answer 1

$(I+p)^2=I+2p+p^2=I+3p$, hence $I+p$ is idempotent $ \iff I+3p=I+p \iff p=0.$

One remark: we have $(I-p)^2=I-2p+p^2=I-p$, thus $I-p$ is idempotent.

Answer 2

The spectrum of a proper projection is $\{0,1\}$. So the spectrum of $I+p$ is $\{1,2\}$ and it cannot be a projection. So, as Fred said, the only case where $I+p$ is a projection is when $p=0$.