We know the fact that $K_0(-)$ and $K_1(-)$ are continuous under inductive sequence of $C^*$-algebras (in fact inductive system), i.e. $K_0(\lim_{\rightarrow} A_n)=\lim_{\rightarrow} K_0(\varinjlim A_n)=\varinjlim K_0(A_n)$ similar for $K_1(-)$. In fact it is also true that $M_k(\lim_{\rightarrow} A_n)=\lim_{\rightarrow} A_n)=\varinjlim M_k(A_n)$ for $k\in \mathbb N$.
Q1: Does $\widetilde{(\lim_{\rightarrow} A_n)}=\lim_{\rightarrow}\tilde{(A_n)}$? \widetilde{(\varinjlim A_n)}$ coincide with $\varinjlim\tilde{(A_n)}$? In fact this is a claim in someones' someone's book, but without a proof. If we let $(X,\lambda_n)$ be the inductive limit of $\tilde{A_1}\rightarrow \tilde{A_2}^~\cdots$, tilde{A_2}\rightarrow~\cdots$, then by universal property we get a unique morphism $\lambda: X\rightarrow \widetilde{\lim_{\rightarrow} widetilde{\varinjlim A_n}$. How can we show $\lambda$ is injective? NB morphisms need not be unital, even though $C^*$-algebras are unital.
Q2: Can we find any other continuous functors? What about the universal group $C^{\star}$-algebras, tensor product of $C^{\star}$-algebras, cross product of $C^{\star}$-algebras and so on?
Q3: Do we know any functor which is not continuous?
