description of an ideal generated by the projections in a $C^*$ algebra

Question

If $A$ is a $C^*$ algebra,$P_i$ are projections in $A$,$I$ is the ideal generated by the projections.I think $I$ is the $C^*$ algebra generated by $P_iAP_i$.How to charectarize $I$,is there a precise description of $I$?

Answer 1

It is not the C$^*$-algebra generated by $P_iAP_i$. For instance take $A=M_2(\mathbb C)$, $P=E_{11}$. The ideal generated by $P$ is $A$, and not $PAP=\mathbb C P$.

The ideal generated by elements $x_1,\ldots,x_m$ in $A$ is the C$^*$-subalgebra generated by $$\{ ax_jb:\ a,b\in A,\ j=1,\ldots,m\}.$$ I don't think you can get anything specific here. Unless the projections are central; in the case the ideal would be $P_1A+\cdots+P_mA$.