I have C*-algebra $A$ and $\text{Prim}(A)$ (set of primitive ideals with hull-kernel topology). Then I also have abelian projection $p$. How does maximal ideal space of commutative C*-algebra $pAp$ look?
How it looks should follow from:
Lemma. If $A$ is a ring and $B$ is subring of the form $pAp$, where $p$ is projection, then there is one to one correspondence between primitive ideals of $B$ and those primitive ideals of $A$ not containing $B$. The mapping is done by $P \to P \cap B$ and it is homeomorphism.
However I have no idea what $P \cap pAp$ looks like and also it somewhat troubles me that only rings (not C*-algebras) are assumed in the lemmas.
Could someone help?