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An additive functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t0935603.png" /> on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t0935604.png" /> of positive elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t0935605.png" /> that takes values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t0935606.png" />, is homogeneous with respect to multiplication by positive numbers and satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t0935607.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t0935608.png" />. A trace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t0935609.png" /> is said to be finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356011.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356012.png" />, and semi-finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356015.png" />. The finite traces on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356016.png" /> are the restrictions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356017.png" /> of those positive linear functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356020.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356021.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356022.png" /> be a trace on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356023.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356024.png" /> be the set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356026.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356027.png" /> be the set of linear combinations of products of pairs of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356028.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356030.png" /> are self-adjoint two-sided ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356031.png" />, and there is a unique linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356033.png" /> that coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356034.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356035.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356036.png" /> be a lower semi-continuous semi-finite trace on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356037.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356038.png" />. Then the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356039.png" /> defines a Hermitian form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356040.png" />, with respect to which the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356041.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356042.png" /> into itself is continuous for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356043.png" />. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356044.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356045.png" /> be the completion of the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356046.png" /> with respect to the scalar product defined by the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356047.png" />. By passing to the quotient space and subsequent completion, the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356048.png" /> determine certain operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356049.png" /> on the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356050.png" />, and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356051.png" /> is a representation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356052.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356053.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356054.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356055.png" /> establishes a one-to-one correspondence between the set of lower semi-continuous semi-finite traces on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356056.png" /> and the set of representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356057.png" /> with a trace, defined up to quasi-equivalence.
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====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356058.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>
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'' $  A $''
  
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An additive functional  $  f $
 +
on the set  $  A  ^ {+} $
 +
of positive elements of  $  A $
 +
that takes values in  $  [ 0, + \infty ] $,
 +
is homogeneous with respect to multiplication by positive numbers and satisfies the condition  $  f ( xx  ^ {*} ) = f ( x  ^ {*} x) $
 +
for all  $  x \in A $.
 +
A trace  $  f $
 +
is said to be finite if  $  f ( x) < + \infty $
 +
for all  $  x \in A  ^ {+} $,
 +
and semi-finite if  $  f ( x) = \sup \{ {f ( y) } : {y \in A, y \leq  x, f( y) < + \infty } \} $
 +
for all  $  x \in A  ^ {+} $.
 +
The finite traces on  $  A $
 +
are the restrictions to  $  A  ^ {+} $
 +
of those positive linear functionals  $  \phi $
 +
on  $  A $
 +
such that  $  \phi ( xy) = \phi ( yx) $
 +
for all  $  x, y \in A $.
 +
Let  $  f $
 +
be a trace on  $  A $,
 +
let  $  \mathfrak N _ {f} $
 +
be the set of elements  $  x \in A $
 +
such that  $  f ( xx  ^ {*} ) < + \infty $,
 +
and let  $  \mathfrak M _ {f} $
 +
be the set of linear combinations of products of pairs of elements of  $  \mathfrak N _ {f} $.
 +
Then  $  \mathfrak N _ {f} $
 +
and  $  \mathfrak M _ {f} $
 +
are self-adjoint two-sided ideals of  $  A $,
 +
and there is a unique linear functional  $  \phi $
 +
on  $  \mathfrak M _ {f} $
 +
that coincides with  $  f $
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on  $  \mathfrak M _ {f} \cap A  ^ {+} $.
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Let  $  f $
 +
be a lower semi-continuous semi-finite trace on a  $  C  ^ {*} $-
 +
algebra  $  A $.
 +
Then the formula  $  s ( x, y) = \phi ( y  ^ {*} x) $
 +
defines a Hermitian form on  $  \mathfrak N _ {f} $,
 +
with respect to which the mapping  $  \lambda _ {f} ( x):  x \mapsto xy $
 +
of  $  \mathfrak N _ {f} $
 +
into itself is continuous for any  $  x \in A $.
 +
Put  $  N _ {f} = \{ {x \in \mathfrak N _ {f} } : {s ( x, x) = 0 } \} $,
 +
and let  $  H _ {f} $
 +
be the completion of the quotient space  $  \mathfrak N _ {f} /N _ {f} $
 +
with respect to the scalar product defined by the form  $  s $.
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By passing to the quotient space and subsequent completion, the operators  $  \lambda _ {f} ( x) $
 +
determine certain operators  $  \pi _ {f} ( x) $
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on the Hilbert space  $  H _ {f} $,
 +
and the mapping  $  x \mapsto \pi _ {f} ( x) $
 +
is a representation of the  $  C  ^ {*} $-
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algebra  $  A $
 +
in  $  H _ {f} $.
 +
The mapping  $  f \mapsto \pi _ {f} $
 +
establishes a one-to-one correspondence between the set of lower semi-continuous semi-finite traces on  $  A $
 +
and the set of representations of  $  A $
 +
with a trace, defined up to quasi-equivalence.
  
 
====Comments====
 
====Comments====
Cf. also [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093560/t09356059.png" />-algebra]]; [[Trace|Trace]]; [[Quasi-equivalent representations|Quasi-equivalent representations]].
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Cf. also [[C*-algebra| $  C  ^ {*} $-
 +
algebra]]; [[Trace]]; [[Quasi-equivalent representations]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Bratteli,   D.W. Robinson,   "Operator algebras and quantum statistical mechanics" , '''1''' , Springer  (1979)</TD></TR></table>
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<table>
 +
<tr><td valign="top">[1]</td> <td valign="top">  J. Dixmier, "$C ^ { * }$ algebras" , North-Holland  (1977)  (Translated from French)</td></tr>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , '''1''' , Springer  (1979)</td></tr>
 +
</table>

Latest revision as of 13:58, 21 January 2024


$ A $

An additive functional $ f $ on the set $ A ^ {+} $ of positive elements of $ A $ that takes values in $ [ 0, + \infty ] $, is homogeneous with respect to multiplication by positive numbers and satisfies the condition $ f ( xx ^ {*} ) = f ( x ^ {*} x) $ for all $ x \in A $. A trace $ f $ is said to be finite if $ f ( x) < + \infty $ for all $ x \in A ^ {+} $, and semi-finite if $ f ( x) = \sup \{ {f ( y) } : {y \in A, y \leq x, f( y) < + \infty } \} $ for all $ x \in A ^ {+} $. The finite traces on $ A $ are the restrictions to $ A ^ {+} $ of those positive linear functionals $ \phi $ on $ A $ such that $ \phi ( xy) = \phi ( yx) $ for all $ x, y \in A $. Let $ f $ be a trace on $ A $, let $ \mathfrak N _ {f} $ be the set of elements $ x \in A $ such that $ f ( xx ^ {*} ) < + \infty $, and let $ \mathfrak M _ {f} $ be the set of linear combinations of products of pairs of elements of $ \mathfrak N _ {f} $. Then $ \mathfrak N _ {f} $ and $ \mathfrak M _ {f} $ are self-adjoint two-sided ideals of $ A $, and there is a unique linear functional $ \phi $ on $ \mathfrak M _ {f} $ that coincides with $ f $ on $ \mathfrak M _ {f} \cap A ^ {+} $. Let $ f $ be a lower semi-continuous semi-finite trace on a $ C ^ {*} $- algebra $ A $. Then the formula $ s ( x, y) = \phi ( y ^ {*} x) $ defines a Hermitian form on $ \mathfrak N _ {f} $, with respect to which the mapping $ \lambda _ {f} ( x): x \mapsto xy $ of $ \mathfrak N _ {f} $ into itself is continuous for any $ x \in A $. Put $ N _ {f} = \{ {x \in \mathfrak N _ {f} } : {s ( x, x) = 0 } \} $, and let $ H _ {f} $ be the completion of the quotient space $ \mathfrak N _ {f} /N _ {f} $ with respect to the scalar product defined by the form $ s $. By passing to the quotient space and subsequent completion, the operators $ \lambda _ {f} ( x) $ determine certain operators $ \pi _ {f} ( x) $ on the Hilbert space $ H _ {f} $, and the mapping $ x \mapsto \pi _ {f} ( x) $ is a representation of the $ C ^ {*} $- algebra $ A $ in $ H _ {f} $. The mapping $ f \mapsto \pi _ {f} $ establishes a one-to-one correspondence between the set of lower semi-continuous semi-finite traces on $ A $ and the set of representations of $ A $ with a trace, defined up to quasi-equivalence.

Comments

Cf. also $ C ^ {*} $- algebra; Trace; Quasi-equivalent representations.

References

[1] J. Dixmier, "$C ^ { * }$ algebras" , North-Holland (1977) (Translated from French)
[a1] O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , 1 , Springer (1979)
How to Cite This Entry:
Trace on a C*-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trace_on_a_C*-algebra&oldid=16757
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article