We know the fact that $K\_0(-)$ and $K\_1(-)$ are continuous under inductive sequence of $C^\*$-algebras (in fact inductive system), i.e. $K\_0(\varinjlim A\_n)=\varinjlim K\_0(A\_n)$ similar for $K\_1(-)$. In fact it is also true that $M\_k(\lim\_{\rightarrow} A\_n)=\varinjlim M\_k(A\_n)$ for $k\in \mathbb N$.


Q1: Does $\widetilde{(\varinjlim A\_n)}$ coincide with $\varinjlim\tilde{(A\_n)}$? In fact this is a claim in someone's book, but without a proof. If we let $(X,\lambda\_n)$ be the inductive limit of $\tilde{A\_1}\rightarrow \tilde{A\_2}\rightarrow~\cdots$, then by universal property we get a unique morphism $\lambda: X\rightarrow \widetilde{\varinjlim A\_n}$. 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?



Continuity of functors under inductive sequence of $C^*$-algebras.
The answer to Q1 is a standard fact, which can be found in the standard text books like Bruce Blackadar's book on $C^{\star}$-algebras. The answer to Q2 is "yes" and, again, looking in a text book helps.


For Q3, I think that it is worth noting that the Algebraic K-theory of $C^{\star}$-algebras is very interesting but not continuous, in fact the natural map


$$K^{alg}\_i({\mathbb C}) = \lim\_n \ K^{alg}\_i(M\_n{\mathbb C}) \to K^{alg}\_i(\lim\_n \ M\_n{\mathbb C}) = K^{alg}\_i(\mathcal K)$$


is only an isomorphism if $i=0$ (in fact, it is the zero-map if $i \neq 0$). The comparison map is much better behaved for commutative algebras if $i \leq 0$, but still not known to be continuous.


