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''disjoint representations''
 
''disjoint representations''
  
Unitary representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d0333101.png" /> of a certain group or, correspondingly, symmetric representations of a certain algebra with an involution which satisfy the following equivalent conditions: 1) the unique bounded linear operator from the representation space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d0333102.png" /> into the representation space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d0333103.png" /> is equal to zero; or 2) no non-zero subrepresentations of the representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d0333104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d0333105.png" /> are equivalent. The concept of disjoint representations is fruitful in the study of factor representations; in particular, a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d0333106.png" /> is a factor representation if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d0333107.png" /> cannot be represented as the direct sum of two non-zero disjoint representations. Any two factor representations are either disjoint or else one of them is equivalent to a subrepresentation of the other (and, in the latter case, the representations are quasi-equivalent). The concept of disjoint representations plays an important role in the decomposition of a representation into a direct integral: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d0333108.png" /> is a representation in a separable Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d0333109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331010.png" /> is the [[Von Neumann algebra|von Neumann algebra]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331011.png" /> generated by the operators of the representation, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331012.png" /> is the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331013.png" />, then
+
Unitary representations $  \pi _ {1} , \pi _ {2} $
 +
of a certain group or, correspondingly, symmetric representations of a certain algebra with an involution which satisfy the following equivalent conditions: 1) the unique bounded linear operator from the representation space of $  \pi _ {1} $
 +
into the representation space of $  \pi _ {2} $
 +
is equal to zero; or 2) no non-zero subrepresentations of the representations $  \pi _ {1} $
 +
and $  \pi _ {2} $
 +
are equivalent. The concept of disjoint representations is fruitful in the study of factor representations; in particular, a representation $  \pi $
 +
is a factor representation if and only if $  \pi $
 +
cannot be represented as the direct sum of two non-zero disjoint representations. Any two factor representations are either disjoint or else one of them is equivalent to a subrepresentation of the other (and, in the latter case, the representations are quasi-equivalent). The concept of disjoint representations plays an important role in the decomposition of a representation into a direct integral: If $  \pi $
 +
is a representation in a separable Hilbert space $  H $,  
 +
$  \mathfrak A $
 +
is the [[Von Neumann algebra|von Neumann algebra]] on $  H $
 +
generated by the operators of the representation, and $  Z $
 +
is the centre of $  \mathfrak A $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331014.png" /></td> </tr></table>
+
$$
 +
= \int\limits ^  \oplus  H ( l)  d \mu ( l)
 +
$$
  
is the decomposition of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331015.png" /> into the direct integral of Hilbert spaces, which corresponds to the decomposition
+
is the decomposition of the space $  H $
 +
into the direct integral of Hilbert spaces, which corresponds to the decomposition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331016.png" /></td> </tr></table>
+
$$
 +
\pi  = \int\limits ^  \oplus  \pi ( l)  d \mu ( l) ,
 +
$$
  
and if also the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331017.png" /> corresponds to the algebra of diagonalizable operators, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331018.png" /> is a factor representation for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331019.png" />, and the representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331020.png" /> are pairwise disjoint for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331021.png" />. There is a simple connection between the disjointness of two representations of a separable locally compact group (or of a separable algebra with an involution) and the mutual singularity of the representatives of canonical classes of measures on the quasi-spectrum of the group (algebra) corresponding to these representations.
+
and if also the algebra $  Z $
 +
corresponds to the algebra of diagonalizable operators, then $  \pi ( l) $
 +
is a factor representation for almost-all $  l $,  
 +
and the representations $  \pi ( l) $
 +
are pairwise disjoint for almost-all $  l $.  
 +
There is a simple connection between the disjointness of two representations of a separable locally compact group (or of a separable algebra with an involution) and the mutual singularity of the representatives of canonical classes of measures on the quasi-spectrum of the group (algebra) corresponding to these representations.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331022.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331022.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Arveson,  "An invitation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331023.png" />-algebras" , Springer  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Arveson,  "An invitation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331023.png" />-algebras" , Springer  (1976)</TD></TR></table>

Revision as of 19:36, 5 June 2020


disjoint representations

Unitary representations $ \pi _ {1} , \pi _ {2} $ of a certain group or, correspondingly, symmetric representations of a certain algebra with an involution which satisfy the following equivalent conditions: 1) the unique bounded linear operator from the representation space of $ \pi _ {1} $ into the representation space of $ \pi _ {2} $ is equal to zero; or 2) no non-zero subrepresentations of the representations $ \pi _ {1} $ and $ \pi _ {2} $ are equivalent. The concept of disjoint representations is fruitful in the study of factor representations; in particular, a representation $ \pi $ is a factor representation if and only if $ \pi $ cannot be represented as the direct sum of two non-zero disjoint representations. Any two factor representations are either disjoint or else one of them is equivalent to a subrepresentation of the other (and, in the latter case, the representations are quasi-equivalent). The concept of disjoint representations plays an important role in the decomposition of a representation into a direct integral: If $ \pi $ is a representation in a separable Hilbert space $ H $, $ \mathfrak A $ is the von Neumann algebra on $ H $ generated by the operators of the representation, and $ Z $ is the centre of $ \mathfrak A $, then

$$ H = \int\limits ^ \oplus H ( l) d \mu ( l) $$

is the decomposition of the space $ H $ into the direct integral of Hilbert spaces, which corresponds to the decomposition

$$ \pi = \int\limits ^ \oplus \pi ( l) d \mu ( l) , $$

and if also the algebra $ Z $ corresponds to the algebra of diagonalizable operators, then $ \pi ( l) $ is a factor representation for almost-all $ l $, and the representations $ \pi ( l) $ are pairwise disjoint for almost-all $ l $. There is a simple connection between the disjointness of two representations of a separable locally compact group (or of a separable algebra with an involution) and the mutual singularity of the representatives of canonical classes of measures on the quasi-spectrum of the group (algebra) corresponding to these representations.

References

[1] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)

Comments

References

[a1] W. Arveson, "An invitation to -algebras" , Springer (1976)
How to Cite This Entry:
Disjunctive representations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjunctive_representations&oldid=46743
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article