Let $B_0(H)$ be the compact operators on the Hilbert space $H$ and let $B \subseteq B_0(H)$ a $C^*$-subalgebra that acts non-degenerately on $H$. Let $\{p_i: i \in I\}$ be a maximal family of pairwise orthogonal minimal projections. The existence of this family is ensured by Zorn's lemma. Is it true that $$H = \bigoplus_i p_i (H)$$
Attempt:
Write $K$ for the direct sum. If $K$ is a proper subspace, we may fix a non-zero $\xi \in K^\perp$. The idea is now to use this vector to construct a minimal projection that will contradict maximality, but I'm not sure how to use the non-degeneracy to do this.
Let $ p = \sum_{i\in I}p_i, $ where the sum is known to converge strongly. Then, for every $a$ in $B$, notice that $$ pa = \sum_{i\in I}p_ia, $$ where the convergence is now in norm since right-multiplication by compact operators turns strong convergence (of bounded nets) into norm convergence. Therefore $pa\in B$.
Now, assuming by contraction that $p\neq 1$, and since $B$ is non-degenenerate, there must be some element $a$ in $B$ such that $(1-p)a\neq 0$, and hence also that $$ c:= (1-p)aa^*(1-p) \neq 0. $$ Since $c$ is a self-adjoint compact operator in $B$, any one of its spectral projections, say $q$, also lie in $B$. It is also clear that $q\perp p_i$ for every $i$, and since $q$ is necessarily finite dimensional, we may choose a minimal projection among the projections of $B$ dominated by $q$. The existence of such a projection then contradicts the maximality of the originaly chosen family.