the $C^*$ algebra generated by $P_nAP_n$ is isomorphic to $c_0$ direct sum of the algebras $P_nAP_n$ Suppose $A$ is a $C^*$ algebra,if $P_n$ are pairwise orthogonal central projections in $A$. Then the $C^*$ algebra generated by $P_nAP_n$ is isomorphic to $c_0$ direct sum of the algebras $P_nAP_n$.
My question is: how to show the $C^*$ algebra generated by $P_nAP_n$ is  isomorphic to $c_0$ direct sum of the algebras $P_nAP_n$? why the generated algebra cannot be isomorphic to $l^\infty$ direct sum of the algebras $P_nAP_n$.
 A: For an obvious reason why the $\ell^\infty$ sum does not work, start with $A=c_0$, and take $P_n=e_n$. The $\ell^\infty$ sum is $\ell^\infty(\mathbb N)$, which is not separable while $A$ is. 
Let us call $B$ your subalgebra. Since the projections $\{P_n\}$ are pairwise orthogonal, any product of the form $(P_nTP_n)(P_mSP_m)$ is zero. 
The C$^*$-algebra generated by a set consists of all the closure of all the noncommutative polynomials on the generators and their adjoints. The aforementioned behaviour of the products guarantees that $B$ is the closure of the elements of the form 
$$
\sum_{j=1}^m P_{n_j}T_jP_{n_j},
$$
for all different choices $n_1,\ldots,n_m$. Again the orthogonality implies that 
$$
\left\|\sum_{j=1}^m P_{n_j}T_jP_{n_j}\right\|=\max\left\{\|P_{n_j}T_jP_{n_j}\|:\ j\right\}.
$$
It follows that the map 
$$
\Phi:\sum_{j=1}^m P_{n_j}T_jP_{n_j}\longmapsto \bigoplus_{j=1}^m P_{n_j}T_jP_{n_j}
$$
is well-defined, isometric, $*$-homomorphism. As such it extends uniquely to the closure, giving us a $*$-isomorphism $\phi:B\to \bigoplus_{c_0}P_nAP_n.$.
