Projective (or inverse) limit of C*-algebras Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Inverse Limit in the category of $C^ast$-algebras or operator spacesAre projective limits always subobjects of a product and dualy, inductive limits quotient objects of a coproduct?The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor productС* algebras and projective limitsProjective limit of finite dimensional C* algebrasPurely infinite $C^*$-algebrastensorial product of C*-algebras and adjointnessInductive limits of $C^*$-algebrasProjections in C*-algebrasC*-algebras explanationContinuity in C$^*$-AlgebrasInclusion of multiplier algebras
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Projective (or inverse) limit of C*-algebras
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Inverse Limit in the category of $C^ast$-algebras or operator spacesAre projective limits always subobjects of a product and dualy, inductive limits quotient objects of a coproduct?The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor productС* algebras and projective limitsProjective limit of finite dimensional C* algebrasPurely infinite $C^*$-algebrastensorial product of C*-algebras and adjointnessInductive limits of $C^*$-algebrasProjections in C*-algebrasC*-algebras explanationContinuity in C$^*$-AlgebrasInclusion of multiplier algebras
$begingroup$
(I think that the term "inverse limit" is used when the index set is directed)
- To begin with, I'd like to know if projective limits of C*-algebras (in the category of C*-algebras) always exist, and if not, what are some sufficient conditions for their existence.
A rapid look at various articles in internet shows that people have been interested in projective limits of C*-algebras but within categories of topological algebras, e.g. Operator Algebra and its Applications: Vol. 1, David E. Evans, Masamichi Takesaki p.130-131
- In that particular excerpt, it says that the usual construction (which they recall equation ($*$)) does not yield the projective limit. I would be really interested to know why it fails, what should be modified to get the limit. (With my limited experience of C*-algebras, there often seems to be difficulties with the choice of a norm)
(I would also appreciate help for the tags if the question is also of interest in other fields)
category-theory operator-algebras c-star-algebras
asked Jun 20 '15 at 16:41
Noix07Noix07
1,241922
$endgroup$
add a comment |
$begingroup$
(I think that the term "inverse limit" is used when the index set is directed)
- To begin with, I'd like to know if projective limits of C*-algebras (in the category of C*-algebras) always exist, and if not, what are some sufficient conditions for their existence.
A rapid look at various articles in internet shows that people have been interested in projective limits of C*-algebras but within categories of topological algebras, e.g. Operator Algebra and its Applications: Vol. 1, David E. Evans, Masamichi Takesaki p.130-131
- In that particular excerpt, it says that the usual construction (which they recall equation ($*$)) does not yield the projective limit. I would be really interested to know why it fails, what should be modified to get the limit. (With my limited experience of C*-algebras, there often seems to be difficulties with the choice of a norm)
(I would also appreciate help for the tags if the question is also of interest in other fields)
category-theory operator-algebras c-star-algebras
asked Jun 20 '15 at 16:41
Noix07Noix07
1,241922
$endgroup$
add a comment |
$begingroup$
(I think that the term "inverse limit" is used when the index set is directed)
- To begin with, I'd like to know if projective limits of C*-algebras (in the category of C*-algebras) always exist, and if not, what are some sufficient conditions for their existence.
A rapid look at various articles in internet shows that people have been interested in projective limits of C*-algebras but within categories of topological algebras, e.g. Operator Algebra and its Applications: Vol. 1, David E. Evans, Masamichi Takesaki p.130-131
- In that particular excerpt, it says that the usual construction (which they recall equation ($*$)) does not yield the projective limit. I would be really interested to know why it fails, what should be modified to get the limit. (With my limited experience of C*-algebras, there often seems to be difficulties with the choice of a norm)
(I would also appreciate help for the tags if the question is also of interest in other fields)
category-theory operator-algebras c-star-algebras
asked Jun 20 '15 at 16:41
Noix07Noix07
1,241922
$endgroup$
(I think that the term "inverse limit" is used when the index set is directed)
- To begin with, I'd like to know if projective limits of C*-algebras (in the category of C*-algebras) always exist, and if not, what are some sufficient conditions for their existence.
A rapid look at various articles in internet shows that people have been interested in projective limits of C*-algebras but within categories of topological algebras, e.g. Operator Algebra and its Applications: Vol. 1, David E. Evans, Masamichi Takesaki p.130-131
- In that particular excerpt, it says that the usual construction (which they recall equation ($*$)) does not yield the projective limit. I would be really interested to know why it fails, what should be modified to get the limit. (With my limited experience of C*-algebras, there often seems to be difficulties with the choice of a norm)
(I would also appreciate help for the tags if the question is also of interest in other fields)
category-theory operator-algebras c-star-algebras
category-theory operator-algebras c-star-algebras
asked Jun 20 '15 at 16:41
Noix07Noix07
1,241922
asked Jun 20 '15 at 16:41
Noix07Noix07
1,241922
asked Jun 20 '15 at 16:41
Noix07Noix07
1,241922
asked Jun 20 '15 at 16:41
Noix07Noix07
1,241922
asked Jun 20 '15 at 16:41
Noix07Noix07
1,241922
1,241922
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The category of $C^*$-algebras is complete. The limit of a diagram $(A_i,lVert - rVert)_i in I$ of $C^*$-algebras has as underlying $*$-algebra the $*$-subalgebra of the $*$-algebra $prod_i in I A_i$ whose elements $x=(x_i)_i in I$ are subject to two conditions: First, the usual matching condition: For edges $i to j$ the map $A_i to A_j$ should map $x_i$ to $x_j$. Second, a boundedness condition: The set $lVert x_i rVert : i in I$ is bounded. We then define $lVert x rVert := sup_i in I lVert x_i rVert$. It is straight forward to check that this is a $C^*$-norm. The only nontrivial fact is that the norm is complete. But you can simply copy the usual proof that $ell^infty$ is complete. It is also easy to check the universal property, using the fact that $*$-homomorphisms between $C^*$-algebras are of norm $leq 1$.
answered Jun 20 '15 at 17:10
Martin BrandenburgMartin Brandenburg
109k13165335
$endgroup$
1
$begingroup$
Thanks!. I found a reference for $mathbfC*-Alg$ complete and cocomplete : math.washington.edu/~warner/C-star.pdf statement 3.15 p.30 (of the document, strange numeration of the pages)
$endgroup$
– Noix07
Jun 21 '15 at 13:58
1
$begingroup$
comment: in case of inverse limit, people conventionally map $irightarrow j$ to $A_jrightarrow A_i$
$endgroup$
– Noix07
Jun 21 '15 at 14:04
add a comment |
$begingroup$
The category of $C^*$-algebras is complete like Martin Brandenburg explained. However, given an inverse system of $C^*$-algebras, it is often more convenient to consider its limit in the category of all topological $*$-algebras. The limit of a diagram $(A_i,lVert - rVert)_i in I$ of $C^*$-algebras is then the topological $*$-subalgebra of the topological $*$-algebra $prod_i in I A_i$ (in the product topology) whose elements $x=(x_i)_i in I$ are subject only to the usual matching condition but not the boundedness condition. See the following paper:
http://jot.theta.ro/jot/archive/1988-019-001/1988-019-001-010.pdf
for more detail.
answered Jul 17 '15 at 14:37
Ilan BarneaIlan Barnea
1213
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The category of $C^*$-algebras is complete. The limit of a diagram $(A_i,lVert - rVert)_i in I$ of $C^*$-algebras has as underlying $*$-algebra the $*$-subalgebra of the $*$-algebra $prod_i in I A_i$ whose elements $x=(x_i)_i in I$ are subject to two conditions: First, the usual matching condition: For edges $i to j$ the map $A_i to A_j$ should map $x_i$ to $x_j$. Second, a boundedness condition: The set $lVert x_i rVert : i in I$ is bounded. We then define $lVert x rVert := sup_i in I lVert x_i rVert$. It is straight forward to check that this is a $C^*$-norm. The only nontrivial fact is that the norm is complete. But you can simply copy the usual proof that $ell^infty$ is complete. It is also easy to check the universal property, using the fact that $*$-homomorphisms between $C^*$-algebras are of norm $leq 1$.
answered Jun 20 '15 at 17:10
Martin BrandenburgMartin Brandenburg
109k13165335
$endgroup$
1
$begingroup$
Thanks!. I found a reference for $mathbfC*-Alg$ complete and cocomplete : math.washington.edu/~warner/C-star.pdf statement 3.15 p.30 (of the document, strange numeration of the pages)
$endgroup$
– Noix07
Jun 21 '15 at 13:58
1
$begingroup$
comment: in case of inverse limit, people conventionally map $irightarrow j$ to $A_jrightarrow A_i$
$endgroup$
– Noix07
Jun 21 '15 at 14:04
add a comment |
$begingroup$
The category of $C^*$-algebras is complete. The limit of a diagram $(A_i,lVert - rVert)_i in I$ of $C^*$-algebras has as underlying $*$-algebra the $*$-subalgebra of the $*$-algebra $prod_i in I A_i$ whose elements $x=(x_i)_i in I$ are subject to two conditions: First, the usual matching condition: For edges $i to j$ the map $A_i to A_j$ should map $x_i$ to $x_j$. Second, a boundedness condition: The set $lVert x_i rVert : i in I$ is bounded. We then define $lVert x rVert := sup_i in I lVert x_i rVert$. It is straight forward to check that this is a $C^*$-norm. The only nontrivial fact is that the norm is complete. But you can simply copy the usual proof that $ell^infty$ is complete. It is also easy to check the universal property, using the fact that $*$-homomorphisms between $C^*$-algebras are of norm $leq 1$.
answered Jun 20 '15 at 17:10
Martin BrandenburgMartin Brandenburg
109k13165335
$endgroup$
1
$begingroup$
Thanks!. I found a reference for $mathbfC*-Alg$ complete and cocomplete : math.washington.edu/~warner/C-star.pdf statement 3.15 p.30 (of the document, strange numeration of the pages)
$endgroup$
– Noix07
Jun 21 '15 at 13:58
1
$begingroup$
comment: in case of inverse limit, people conventionally map $irightarrow j$ to $A_jrightarrow A_i$
$endgroup$
– Noix07
Jun 21 '15 at 14:04
add a comment |
$begingroup$
The category of $C^*$-algebras is complete. The limit of a diagram $(A_i,lVert - rVert)_i in I$ of $C^*$-algebras has as underlying $*$-algebra the $*$-subalgebra of the $*$-algebra $prod_i in I A_i$ whose elements $x=(x_i)_i in I$ are subject to two conditions: First, the usual matching condition: For edges $i to j$ the map $A_i to A_j$ should map $x_i$ to $x_j$. Second, a boundedness condition: The set $lVert x_i rVert : i in I$ is bounded. We then define $lVert x rVert := sup_i in I lVert x_i rVert$. It is straight forward to check that this is a $C^*$-norm. The only nontrivial fact is that the norm is complete. But you can simply copy the usual proof that $ell^infty$ is complete. It is also easy to check the universal property, using the fact that $*$-homomorphisms between $C^*$-algebras are of norm $leq 1$.
answered Jun 20 '15 at 17:10
Martin BrandenburgMartin Brandenburg
109k13165335
$endgroup$
The category of $C^*$-algebras is complete. The limit of a diagram $(A_i,lVert - rVert)_i in I$ of $C^*$-algebras has as underlying $*$-algebra the $*$-subalgebra of the $*$-algebra $prod_i in I A_i$ whose elements $x=(x_i)_i in I$ are subject to two conditions: First, the usual matching condition: For edges $i to j$ the map $A_i to A_j$ should map $x_i$ to $x_j$. Second, a boundedness condition: The set $lVert x_i rVert : i in I$ is bounded. We then define $lVert x rVert := sup_i in I lVert x_i rVert$. It is straight forward to check that this is a $C^*$-norm. The only nontrivial fact is that the norm is complete. But you can simply copy the usual proof that $ell^infty$ is complete. It is also easy to check the universal property, using the fact that $*$-homomorphisms between $C^*$-algebras are of norm $leq 1$.
answered Jun 20 '15 at 17:10
Martin BrandenburgMartin Brandenburg
109k13165335
answered Jun 20 '15 at 17:10
Martin BrandenburgMartin Brandenburg
109k13165335
answered Jun 20 '15 at 17:10
Martin BrandenburgMartin Brandenburg
109k13165335
answered Jun 20 '15 at 17:10
Martin BrandenburgMartin Brandenburg
109k13165335
109k13165335
1
$begingroup$
Thanks!. I found a reference for $mathbfC*-Alg$ complete and cocomplete : math.washington.edu/~warner/C-star.pdf statement 3.15 p.30 (of the document, strange numeration of the pages)
$endgroup$
– Noix07
Jun 21 '15 at 13:58
1
$begingroup$
comment: in case of inverse limit, people conventionally map $irightarrow j$ to $A_jrightarrow A_i$
$endgroup$
– Noix07
Jun 21 '15 at 14:04
add a comment |
1
$begingroup$
Thanks!. I found a reference for $mathbfC*-Alg$ complete and cocomplete : math.washington.edu/~warner/C-star.pdf statement 3.15 p.30 (of the document, strange numeration of the pages)
$endgroup$
– Noix07
Jun 21 '15 at 13:58
1
$begingroup$
comment: in case of inverse limit, people conventionally map $irightarrow j$ to $A_jrightarrow A_i$
$endgroup$
– Noix07
Jun 21 '15 at 14:04
1
1
$begingroup$
Thanks!. I found a reference for $mathbfC*-Alg$ complete and cocomplete : math.washington.edu/~warner/C-star.pdf statement 3.15 p.30 (of the document, strange numeration of the pages)
$endgroup$
– Noix07
Jun 21 '15 at 13:58
$begingroup$
Thanks!. I found a reference for $mathbfC*-Alg$ complete and cocomplete : math.washington.edu/~warner/C-star.pdf statement 3.15 p.30 (of the document, strange numeration of the pages)
$endgroup$
– Noix07
Jun 21 '15 at 13:58
1
1
$begingroup$
comment: in case of inverse limit, people conventionally map $irightarrow j$ to $A_jrightarrow A_i$
$endgroup$
– Noix07
Jun 21 '15 at 14:04
$begingroup$
comment: in case of inverse limit, people conventionally map $irightarrow j$ to $A_jrightarrow A_i$
$endgroup$
– Noix07
Jun 21 '15 at 14:04
add a comment |
$begingroup$
The category of $C^*$-algebras is complete like Martin Brandenburg explained. However, given an inverse system of $C^*$-algebras, it is often more convenient to consider its limit in the category of all topological $*$-algebras. The limit of a diagram $(A_i,lVert - rVert)_i in I$ of $C^*$-algebras is then the topological $*$-subalgebra of the topological $*$-algebra $prod_i in I A_i$ (in the product topology) whose elements $x=(x_i)_i in I$ are subject only to the usual matching condition but not the boundedness condition. See the following paper:
http://jot.theta.ro/jot/archive/1988-019-001/1988-019-001-010.pdf
for more detail.
answered Jul 17 '15 at 14:37
Ilan BarneaIlan Barnea
1213
$endgroup$
add a comment |
$begingroup$
The category of $C^*$-algebras is complete like Martin Brandenburg explained. However, given an inverse system of $C^*$-algebras, it is often more convenient to consider its limit in the category of all topological $*$-algebras. The limit of a diagram $(A_i,lVert - rVert)_i in I$ of $C^*$-algebras is then the topological $*$-subalgebra of the topological $*$-algebra $prod_i in I A_i$ (in the product topology) whose elements $x=(x_i)_i in I$ are subject only to the usual matching condition but not the boundedness condition. See the following paper:
http://jot.theta.ro/jot/archive/1988-019-001/1988-019-001-010.pdf
for more detail.
answered Jul 17 '15 at 14:37
Ilan BarneaIlan Barnea
1213
$endgroup$
add a comment |
$begingroup$
The category of $C^*$-algebras is complete like Martin Brandenburg explained. However, given an inverse system of $C^*$-algebras, it is often more convenient to consider its limit in the category of all topological $*$-algebras. The limit of a diagram $(A_i,lVert - rVert)_i in I$ of $C^*$-algebras is then the topological $*$-subalgebra of the topological $*$-algebra $prod_i in I A_i$ (in the product topology) whose elements $x=(x_i)_i in I$ are subject only to the usual matching condition but not the boundedness condition. See the following paper:
http://jot.theta.ro/jot/archive/1988-019-001/1988-019-001-010.pdf
for more detail.
answered Jul 17 '15 at 14:37
Ilan BarneaIlan Barnea
1213
$endgroup$
The category of $C^*$-algebras is complete like Martin Brandenburg explained. However, given an inverse system of $C^*$-algebras, it is often more convenient to consider its limit in the category of all topological $*$-algebras. The limit of a diagram $(A_i,lVert - rVert)_i in I$ of $C^*$-algebras is then the topological $*$-subalgebra of the topological $*$-algebra $prod_i in I A_i$ (in the product topology) whose elements $x=(x_i)_i in I$ are subject only to the usual matching condition but not the boundedness condition. See the following paper:
http://jot.theta.ro/jot/archive/1988-019-001/1988-019-001-010.pdf
for more detail.
answered Jul 17 '15 at 14:37
Ilan BarneaIlan Barnea
1213
answered Jul 17 '15 at 14:37
Ilan BarneaIlan Barnea
1213
answered Jul 17 '15 at 14:37
Ilan BarneaIlan Barnea
1213
answered Jul 17 '15 at 14:37
Ilan BarneaIlan Barnea
1213
1213
add a comment |
add a comment |
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