Let $A$ be a $\mathbb{Z}_2$-graded $C^*$-Algebra. Then we can take the direct limit $$ colim_{A_\sigma} KK_*(\mathbb{C}, A_\sigma) $$ over all $\sigma$-unital graded $C^*$-sub-algebras $A_\sigma \subseteq A$. This gives me the $K$-theory of graded $C^*$-algebras. Now, this is supposed to be tha same as taking the direct limit over all separable, graded $C^*$-subalgebras. For that to be true for any $\sigma$-unital, graded $C^*$ subalgebra $A_\sigma$ should be contained in a separable, graded $C^*$-sub-algebra $A_s$. But I don't quite see why.
Thank you