If $A$ is a unital direct limit of C*- algebras, why can we assume that the connecting maps are unital? [closed]

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I know that we may assume that each $phi _n$ is injective. Then how to show that we may assume $phi_n$ is unit preserving when $n geq N$ for some $N$ ?

c-star-algebras

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edited Jul 18 at 17:10

André S.

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asked Jul 18 at 11:43

user540663

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closed as off-topic by Alex Francisco, Adrian Keister, José Carlos Santos, Shailesh, Parcly Taxel Jul 19 at 1:10

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up vote
2
down vote

favorite

I know that we may assume that each $phi _n$ is injective. Then how to show that we may assume $phi_n$ is unit preserving when $n geq N$ for some $N$ ?

c-star-algebras

share|cite|improve this question

edited Jul 18 at 17:10

André S.

1,565313

asked Jul 18 at 11:43

user540663

826

closed as off-topic by Alex Francisco, Adrian Keister, José Carlos Santos, Shailesh, Parcly Taxel Jul 19 at 1:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex Francisco, Adrian Keister, José Carlos Santos, Shailesh, Parcly Taxel

If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

up vote
2
down vote

favorite

up vote
2
down vote

favorite

I know that we may assume that each $phi _n$ is injective. Then how to show that we may assume $phi_n$ is unit preserving when $n geq N$ for some $N$ ?

c-star-algebras

share|cite|improve this question

edited Jul 18 at 17:10

André S.

1,565313

asked Jul 18 at 11:43

user540663

826

I know that we may assume that each $phi _n$ is injective. Then how to show that we may assume $phi_n$ is unit preserving when $n geq N$ for some $N$ ?

c-star-algebras

share|cite|improve this question

edited Jul 18 at 17:10

André S.

1,565313

asked Jul 18 at 11:43

user540663

826

share|cite|improve this question

share|cite|improve this question

edited Jul 18 at 17:10

André S.

1,565313

edited Jul 18 at 17:10

André S.

1,565313

edited Jul 18 at 17:10

André S.

1,565313

1,565313

asked Jul 18 at 11:43

user540663

826

asked Jul 18 at 11:43

user540663

826

asked Jul 18 at 11:43

user540663

826

826

closed as off-topic by Alex Francisco, Adrian Keister, José Carlos Santos, Shailesh, Parcly Taxel Jul 19 at 1:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex Francisco, Adrian Keister, José Carlos Santos, Shailesh, Parcly Taxel

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by Alex Francisco, Adrian Keister, José Carlos Santos, Shailesh, Parcly Taxel Jul 19 at 1:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex Francisco, Adrian Keister, José Carlos Santos, Shailesh, Parcly Taxel

If this question can be reworded to fit the rules in the help center, please edit the question.

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1 Answer
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If all connecting maps are injective you can do the following:

• First show that almost all $A_n$ are unital. This can be done by approximating $1_A$ by positive contractions and applying functional calculus to get projections that are close to $1_A$.


• Then use that an invertible projection is already the unit.


• Show that the connecting maps preserve the units you just have found.

share|cite|improve this answer

answered Jul 18 at 13:53

André S.

1,565313

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
2
down vote

If all connecting maps are injective you can do the following:

• First show that almost all $A_n$ are unital. This can be done by approximating $1_A$ by positive contractions and applying functional calculus to get projections that are close to $1_A$.


• Then use that an invertible projection is already the unit.


• Show that the connecting maps preserve the units you just have found.

share|cite|improve this answer

answered Jul 18 at 13:53

André S.

1,565313

add a comment | 

up vote
2
down vote

If all connecting maps are injective you can do the following:

• First show that almost all $A_n$ are unital. This can be done by approximating $1_A$ by positive contractions and applying functional calculus to get projections that are close to $1_A$.


• Then use that an invertible projection is already the unit.


• Show that the connecting maps preserve the units you just have found.

share|cite|improve this answer

answered Jul 18 at 13:53

André S.

1,565313

add a comment | 

up vote
2
down vote

up vote
2
down vote

If all connecting maps are injective you can do the following:

• First show that almost all $A_n$ are unital. This can be done by approximating $1_A$ by positive contractions and applying functional calculus to get projections that are close to $1_A$.


• Then use that an invertible projection is already the unit.


• Show that the connecting maps preserve the units you just have found.

share|cite|improve this answer

answered Jul 18 at 13:53

André S.

1,565313

If all connecting maps are injective you can do the following:

• First show that almost all $A_n$ are unital. This can be done by approximating $1_A$ by positive contractions and applying functional calculus to get projections that are close to $1_A$.


• Then use that an invertible projection is already the unit.


• Show that the connecting maps preserve the units you just have found.

share|cite|improve this answer

answered Jul 18 at 13:53

André S.

1,565313

share|cite|improve this answer

share|cite|improve this answer

answered Jul 18 at 13:53

André S.

1,565313

answered Jul 18 at 13:53

André S.

1,565313

answered Jul 18 at 13:53

André S.

1,565313

1,565313

add a comment | 

add a comment |