Trying to understand a proof from math overflow regarding direct limit
Direct limit of $mathbb {C^*}$ behaves well with quotients.
The following solution is from mathoverflow:
Direct limits do behave well with respect to quotients. Suppose $A$ is the direct limit of a sequence $(A_n)$ with connecting $*$-homomorphisms $phi_n: A_n to A_{n+1}$, and let $I$ be a closed ideal of $A$. Then $I$ pulls back to an ideal $I_n$ of $A_n$ for each $n$, and the connecting maps $phi_n$ are compatible with the quotients, i.e., they lift to connecting maps $tilde{phi}_n: A_n/I_n to A_{n+1}/I_{n+1}$. Moreover, $A/I$ is then the direct limit of the sequence $(A_n/I_n)$.
This is easy because the maps $tilde{phi}_n$ have no kernel and hence are isometric, and the whole sequence isometrically embeds in $A/I$.
I understand the whole argument whatever is written but I don’t follow how does this proves that $A/I$ is direct limit? Please explain what are we using here?
functional-analysis operator-algebras limits-colimits
edited Dec 6 at 13:08
Arpit Kansal
6,83911135
asked Dec 6 at 10:45
Math Lover
934315
add a comment |
Direct limit of $mathbb {C^*}$ behaves well with quotients.
The following solution is from mathoverflow:
Direct limits do behave well with respect to quotients. Suppose $A$ is the direct limit of a sequence $(A_n)$ with connecting $*$-homomorphisms $phi_n: A_n to A_{n+1}$, and let $I$ be a closed ideal of $A$. Then $I$ pulls back to an ideal $I_n$ of $A_n$ for each $n$, and the connecting maps $phi_n$ are compatible with the quotients, i.e., they lift to connecting maps $tilde{phi}_n: A_n/I_n to A_{n+1}/I_{n+1}$. Moreover, $A/I$ is then the direct limit of the sequence $(A_n/I_n)$.
This is easy because the maps $tilde{phi}_n$ have no kernel and hence are isometric, and the whole sequence isometrically embeds in $A/I$.
I understand the whole argument whatever is written but I don’t follow how does this proves that $A/I$ is direct limit? Please explain what are we using here?
functional-analysis operator-algebras limits-colimits
edited Dec 6 at 13:08
Arpit Kansal
6,83911135
asked Dec 6 at 10:45
Math Lover
934315
add a comment |
Direct limit of $mathbb {C^*}$ behaves well with quotients.
The following solution is from mathoverflow:
Direct limits do behave well with respect to quotients. Suppose $A$ is the direct limit of a sequence $(A_n)$ with connecting $*$-homomorphisms $phi_n: A_n to A_{n+1}$, and let $I$ be a closed ideal of $A$. Then $I$ pulls back to an ideal $I_n$ of $A_n$ for each $n$, and the connecting maps $phi_n$ are compatible with the quotients, i.e., they lift to connecting maps $tilde{phi}_n: A_n/I_n to A_{n+1}/I_{n+1}$. Moreover, $A/I$ is then the direct limit of the sequence $(A_n/I_n)$.
This is easy because the maps $tilde{phi}_n$ have no kernel and hence are isometric, and the whole sequence isometrically embeds in $A/I$.
I understand the whole argument whatever is written but I don’t follow how does this proves that $A/I$ is direct limit? Please explain what are we using here?
functional-analysis operator-algebras limits-colimits
edited Dec 6 at 13:08
Arpit Kansal
6,83911135
asked Dec 6 at 10:45
Math Lover
934315
Direct limit of $mathbb {C^*}$ behaves well with quotients.
The following solution is from mathoverflow:
Direct limits do behave well with respect to quotients. Suppose $A$ is the direct limit of a sequence $(A_n)$ with connecting $*$-homomorphisms $phi_n: A_n to A_{n+1}$, and let $I$ be a closed ideal of $A$. Then $I$ pulls back to an ideal $I_n$ of $A_n$ for each $n$, and the connecting maps $phi_n$ are compatible with the quotients, i.e., they lift to connecting maps $tilde{phi}_n: A_n/I_n to A_{n+1}/I_{n+1}$. Moreover, $A/I$ is then the direct limit of the sequence $(A_n/I_n)$.
This is easy because the maps $tilde{phi}_n$ have no kernel and hence are isometric, and the whole sequence isometrically embeds in $A/I$.
I understand the whole argument whatever is written but I don’t follow how does this proves that $A/I$ is direct limit? Please explain what are we using here?
functional-analysis operator-algebras limits-colimits
functional-analysis operator-algebras limits-colimits
edited Dec 6 at 13:08
Arpit Kansal
6,83911135
asked Dec 6 at 10:45
Math Lover
934315
edited Dec 6 at 13:08
Arpit Kansal
6,83911135
asked Dec 6 at 10:45
Math Lover
934315
edited Dec 6 at 13:08
Arpit Kansal
6,83911135
edited Dec 6 at 13:08
Arpit Kansal
6,83911135
edited Dec 6 at 13:08
Arpit Kansal
6,83911135
6,83911135
asked Dec 6 at 10:45
Math Lover
934315
asked Dec 6 at 10:45
Math Lover
934315
asked Dec 6 at 10:45
Math Lover
934315
934315
add a comment |
add a comment |
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