dimension of a von Neumann algebra
Is there any dimension on a von Neumann algebra? Is there any relationship between finite von Neumann algebras and finite dimensional von Neumann algebras?
von-neumann-algebras
asked Dec 12 '18 at 14:52
S MS M
62
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Is there any dimension on a von Neumann algebra? Is there any relationship between finite von Neumann algebras and finite dimensional von Neumann algebras?
von-neumann-algebras
asked Dec 12 '18 at 14:52
S MS M
62
add a comment |
Is there any dimension on a von Neumann algebra? Is there any relationship between finite von Neumann algebras and finite dimensional von Neumann algebras?
von-neumann-algebras
asked Dec 12 '18 at 14:52
S MS M
62
Is there any dimension on a von Neumann algebra? Is there any relationship between finite von Neumann algebras and finite dimensional von Neumann algebras?
von-neumann-algebras
von-neumann-algebras
asked Dec 12 '18 at 14:52
S MS M
62
asked Dec 12 '18 at 14:52
S MS M
62
asked Dec 12 '18 at 14:52
S MS M
62
asked Dec 12 '18 at 14:52
S MS M
62
asked Dec 12 '18 at 14:52
S MS M
62
62
add a comment |
add a comment |
1 Answer
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A von Neumann algebra is a vector space, so it has a dimension in that sense, but it is meaningless (actually, it is meaningless for any Banach space as the dimension is either finite or uncountably infinite).
Any finite-dimensional von Neumann algebra is finite. But there are (many) finite, infinite-dimensional von Neumann algebras: precisely those of type II$_1$.
answered Dec 12 '18 at 18:45
Martin ArgeramiMartin Argerami
124k1176175
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
A von Neumann algebra is a vector space, so it has a dimension in that sense, but it is meaningless (actually, it is meaningless for any Banach space as the dimension is either finite or uncountably infinite).
Any finite-dimensional von Neumann algebra is finite. But there are (many) finite, infinite-dimensional von Neumann algebras: precisely those of type II$_1$.
answered Dec 12 '18 at 18:45
Martin ArgeramiMartin Argerami
124k1176175
add a comment |
A von Neumann algebra is a vector space, so it has a dimension in that sense, but it is meaningless (actually, it is meaningless for any Banach space as the dimension is either finite or uncountably infinite).
Any finite-dimensional von Neumann algebra is finite. But there are (many) finite, infinite-dimensional von Neumann algebras: precisely those of type II$_1$.
answered Dec 12 '18 at 18:45
Martin ArgeramiMartin Argerami
124k1176175
add a comment |
A von Neumann algebra is a vector space, so it has a dimension in that sense, but it is meaningless (actually, it is meaningless for any Banach space as the dimension is either finite or uncountably infinite).
Any finite-dimensional von Neumann algebra is finite. But there are (many) finite, infinite-dimensional von Neumann algebras: precisely those of type II$_1$.
answered Dec 12 '18 at 18:45
Martin ArgeramiMartin Argerami
124k1176175
A von Neumann algebra is a vector space, so it has a dimension in that sense, but it is meaningless (actually, it is meaningless for any Banach space as the dimension is either finite or uncountably infinite).
Any finite-dimensional von Neumann algebra is finite. But there are (many) finite, infinite-dimensional von Neumann algebras: precisely those of type II$_1$.
answered Dec 12 '18 at 18:45
Martin ArgeramiMartin Argerami
124k1176175
answered Dec 12 '18 at 18:45
Martin ArgeramiMartin Argerami
124k1176175
answered Dec 12 '18 at 18:45
Martin ArgeramiMartin Argerami
124k1176175
answered Dec 12 '18 at 18:45
Martin ArgeramiMartin Argerami
124k1176175
124k1176175
add a comment |
add a comment |
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