Let $p$ be a projection in a unital C*-algebra $A$. What is the kernel of the map $a\mapsto pap$?

Question

Let $p$ be a non-zero projection in a unital C*-algebra $A$, i.e. $p$ is a self-adjoint idempotent. Can we say something about the kernel of the linear map $$\varphi\colon A\to A,\qquad a\mapsto pap?$$ All I can say is that $\varphi(1)=p1p=p^{2}=p\neq0$, i.e. that $1\notin\ker(\varphi)$. I think that the kernel of this map is zero, i.e. that the map is injective. Is this correct? Thanks in advance!

Answer 1

Since $1-p\in \ker\varphi$, you have that

$\ker\varphi=\{0\}\ \iff\ p=1.$

This follows easily from the fact that $1-p\in \ker\varphi$.

I don't think you can expect to say much more. If you write the elements of $A$ as $2\times2$ matrices in terms of $p$, you have that $$ \ker\varphi=\left\{\begin{bmatrix} 0&x\\ y&z\end{bmatrix}:\ x,y,z\in A\right\}. $$