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6 improved the LaTeX

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?

5 corrected some typos

We know the fact that $K_0(-)$ and $K_1(-)$ are continuous under inductive sequence of $C^*$-algebras (in fact inductive system), i.e. $K_0(\lim_{\rightarrow} A_n)=\lim_{\rightarrow} K_0(A_n)$ similar for $K_1(-)$. In fact it is also true that $M_k(\lim_{\rightarrow} A_n)=\lim_{\rightarrow} M_k(A_n)$ for $k\in \mathbb N$.

Q1: $\widetilde{(\lim_{\rightarrow} A_n)}=\lim_{\rightarrow}\tilde{(A_n)}$? In fact this is a claim in someones' book, but without a proof. If we let $(X,\lambda_n)$ be the inductive limit of $\tilde{A_1}\rightarrow \tilde{A_2}^~\cdots$, then by universal property we get a unique morphism $\lambda: X\rightarrow \widetilde{\lim_{\rightarrow} A_n}$. 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 functors functor which is not continuous?

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We know the fact that $K_0(-)$ and $K_1(-)$ are continuous under inductive sequence of $C^*$-algebras (in fact inductive system), i.e. $K_0(\lim_{\rightarrow} A_n)=\lim_{\rightarrow} K_0(A_n)$ similar for $K_1(-)$. In fact it is also true that $M_k(\lim_{\rightarrow} A_n)=\lim_{\rightarrow} M_k(A_n)$ for $k\in \mathbb N$.

Q1: $\widetilde{(\lim_{\rightarrow} A_n)}=\lim_{\rightarrow}\tilde{(A_n)}$? In fact this is a claim in someones' book, but without a proof. If we let $(X,\lambda_n)$ be the inductive limit of $\tilde{A_1}\rightarrow \tilde{A_2}^~\cdots$, then by universal property we get a unique morphism $\lambda: X\rightarrow \widetilde{\lim_{\rightarrow} A_n}$. 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 $C^$-algebras, C^{\star}$-algebras, tensor product of$C^$-algebras, C^{\star}$-algebras, cross product of $C^*$-algebras C^{\star}\$-algebras and so on?

Q3: Do we know any functors which is not continuous?

3 added 4 characters in body; edited body
2 added 18 characters in body; deleted 5 characters in body; added 22 characters in body
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