Let $A$ be a C*-algebra and $A_0$ a dense *-subalgebra. Is there a condition which ensures that all projections in $A$ already lie in $A_0$ ?
In particular, I think of $A$ being an inductive limit of $(A_n,\varphi_n)_{n =1}^\infty$ and $A_0$ the union of $\varphi_n(A_n)$.
Which C*-algebra $A$ has the property that $A \otimes \mathbb K$ contains a projection which is not contained in $A \otimes M_n(\mathbb C)$ for any $n$ ?
One example is the compact operators with $\bigcup_{\mathbb N} M_n(\mathbb C)$ as dense $*$-subalgebra. A compact projection must have finite rank.
Another example (working for any $A$) is $A_0 = \mathrm{Ped}(A)$, the Pedersen ideal. It is the smallest dense (algebraic) ideal in $A$. It is also a dense $*$-subalgebra which is hereditary.