Minimal projections in a C* algebra

Question

Let $e$ be a projection in a C* algebra $A$. Is $eAe= \mathbb{C}e$ equivalent to the nonexistence of any projection in between $e$ and $0$? I know it is true if $A$ is a Von Neumann algebra because then you can use the Borel functional calculus. Takesaki states that the definition of minimality of a projection is $eAe= \mathbb{C}e$ "because it means" that there are no projections in between $e$ and $0$. I can't tell if "because it means" means "implies" or "is equivalent to."

Answer 1

It is easy to see that $eAe=\mathbb{C}e$ implies that there are no projections below $e$.

But the converse is not true. Consider for instance $A=C([0,1]\cup[2,3])$. Then $e=1_{[0,1]}$ is a projection in $A$ that admits no proper subprojection, and $eAe=C[0,1]\subset A$ is not $\mathbb{C}e$.