What is an example of a $C^{*}$ subalgebra $A$ of $B(H)$ such that $A$ contains the identity $I_{H}$ and satisfies the following properties:
1) For every $T\in A$, The orthogonal projection $\pi_{T}$ on the closure of $Range(T)$ belongs to $A$.
but
2) $A$ is not a Von Neumann Algebra.
The question is motivated by the fact that every Von Neumann algebra satisfies (1).
Let $A \subseteq B(H)$ be any AW*-algebra that is not a von Neumann algebra. (Actually, we don't need a full AW*-algebra, see below.)
Let $t \in A$ be any operator. By definition of AW*-algebra, every right-annihilator is generated by a projection. In particular, the right-annihilator of the singleton set $\{t\}$ is generated by a projection $q \in A$. That is to say: 1. for any $a \in A$, if $at = 0$ then $qa=a$ and 2. $qt=0$. Thus $(1-q)t=t$.
Thus we have $1-q \geq \mathrm{Range}(t)$. Conversely, $(1-\mathrm{Range}(t))t = t-\mathrm{Range}(t)t=0$ and so $q(1-\mathrm{Range}(t))=1-\mathrm{Range}(t)$. Thus $1-\mathrm{Range}(t) \leq q$. We conclude $\mathrm{Range}(t)=1-q \in A$.