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Two unitary representations (cf. [[Unitary representation|Unitary representation]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q0764801.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q0764802.png" /> (or symmetric representations of a symmetric algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q0764803.png" />) in Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q0764804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q0764805.png" />, respectively, satisfying one of the following four equivalent conditions: 1) there exist unitarily-equivalent representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q0764806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q0764807.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q0764808.png" /> is a multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q0764809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648010.png" /> is a multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648011.png" />; 2) the non-zero subrepresentations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648012.png" /> are not disjoint from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648013.png" />, and the non-zero subrepresentations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648014.png" /> are not disjoint from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648015.png" />; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648016.png" /> is unitarily equivalent to a subrepresentation of some multiple representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648018.png" /> that has unit central support; or 4) there exists an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648019.png" /> of the [[Von Neumann algebra|von Neumann algebra]] generated by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648020.png" /> onto the von Neumann algebra generated by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648022.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648023.png" />. Unitarily-equivalent representations are quasi-equivalent representations; irreducible quasi-equivalent representations (cf. [[Irreducible representation|Irreducible representation]]) are unitarily equivalent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648025.png" /> are quasi-equivalent representations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648026.png" /> is a [[Factor representation|factor representation]], then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648027.png" />; a factor representation and a non-zero subrepresentation of it are quasi-equivalent representations; two factor representations are either disjoint or quasi-equivalent. The notion of quasi-equivalent representations leads to that of a quasi-dual object and a quasi-spectrum for locally compact groups and symmetric algebras, respectively.
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Two unitary representations (cf. [[Unitary representation|Unitary representation]]) $  \pi _ {1} , \pi _ {2} $
 +
of a group $  X $(
 +
or symmetric representations of a symmetric algebra $  X $)  
 +
in Hilbert spaces $  H _ {1} $
 +
and $  H _ {2} $,  
 +
respectively, satisfying one of the following four equivalent conditions: 1) there exist unitarily-equivalent representations $  \rho _ {1} $
 +
and $  \rho _ {2} $
 +
such that $  \rho _ {1} $
 +
is a multiple of $  \pi _ {1} $
 +
and $  \rho _ {2} $
 +
is a multiple of $  \pi _ {2} $;  
 +
2) the non-zero subrepresentations of $  \pi _ {1} $
 +
are not disjoint from $  \pi _ {2} $,  
 +
and the non-zero subrepresentations of $  \pi _ {2} $
 +
are not disjoint from $  \pi _ {1} $;  
 +
3) $  \pi _ {2} $
 +
is unitarily equivalent to a subrepresentation of some multiple representation $  \rho _ {1} $
 +
of $  \pi _ {1} $
 +
that has unit central support; or 4) there exists an isomorphism $  \Phi $
 +
of the [[Von Neumann algebra|von Neumann algebra]] generated by the set $  \pi _ {1} ( X) $
 +
onto the von Neumann algebra generated by the set $  \pi _ {2} ( X) $
 +
such that $  \Phi ( \pi _ {1} ( x) ) = \pi _ {2} ( x) $
 +
for all $  x \in X $.  
 +
Unitarily-equivalent representations are quasi-equivalent representations; irreducible quasi-equivalent representations (cf. [[Irreducible representation|Irreducible representation]]) are unitarily equivalent. If $  \pi _ {1} $
 +
and $  \pi _ {2} $
 +
are quasi-equivalent representations and $  \pi _ {1} $
 +
is a [[Factor representation|factor representation]], then so is $  \pi _ {2} $;  
 +
a factor representation and a non-zero subrepresentation of it are quasi-equivalent representations; two factor representations are either disjoint or quasi-equivalent. The notion of quasi-equivalent representations leads to that of a quasi-dual object and a quasi-spectrum for locally compact groups and symmetric algebras, respectively.
  
 
====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/q/q076/q076480/q07648028.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/q/q076/q076480/q07648028.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Two representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648030.png" /> (of a group or algebra) with representation spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648032.png" />, respectively, are said to be disjoint is there is no non-zero intertwining operator between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648034.png" />. Here, an intertwining operator between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648036.png" /> is a continuous linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648037.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648038.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076480/q07648039.png" />.
+
Two representations $  \pi $
 +
and $  \pi  ^  \prime  $(
 +
of a group or algebra) with representation spaces $  H $
 +
and $  H  ^  \prime  $,  
 +
respectively, are said to be disjoint is there is no non-zero intertwining operator between $  \pi $
 +
and $  \pi  ^  \prime  $.  
 +
Here, an intertwining operator between $  \pi $
 +
and $  \pi  ^  \prime  $
 +
is a continuous linear operator $  T : H \rightarrow H  ^  \prime  $
 +
such that $  T \pi ( x) = \pi  ^  \prime  T ( x) $
 +
for all $  x $.

Latest revision as of 08:09, 6 June 2020


Two unitary representations (cf. Unitary representation) $ \pi _ {1} , \pi _ {2} $ of a group $ X $( or symmetric representations of a symmetric algebra $ X $) in Hilbert spaces $ H _ {1} $ and $ H _ {2} $, respectively, satisfying one of the following four equivalent conditions: 1) there exist unitarily-equivalent representations $ \rho _ {1} $ and $ \rho _ {2} $ such that $ \rho _ {1} $ is a multiple of $ \pi _ {1} $ and $ \rho _ {2} $ is a multiple of $ \pi _ {2} $; 2) the non-zero subrepresentations of $ \pi _ {1} $ are not disjoint from $ \pi _ {2} $, and the non-zero subrepresentations of $ \pi _ {2} $ are not disjoint from $ \pi _ {1} $; 3) $ \pi _ {2} $ is unitarily equivalent to a subrepresentation of some multiple representation $ \rho _ {1} $ of $ \pi _ {1} $ that has unit central support; or 4) there exists an isomorphism $ \Phi $ of the von Neumann algebra generated by the set $ \pi _ {1} ( X) $ onto the von Neumann algebra generated by the set $ \pi _ {2} ( X) $ such that $ \Phi ( \pi _ {1} ( x) ) = \pi _ {2} ( x) $ for all $ x \in X $. Unitarily-equivalent representations are quasi-equivalent representations; irreducible quasi-equivalent representations (cf. Irreducible representation) are unitarily equivalent. If $ \pi _ {1} $ and $ \pi _ {2} $ are quasi-equivalent representations and $ \pi _ {1} $ is a factor representation, then so is $ \pi _ {2} $; a factor representation and a non-zero subrepresentation of it are quasi-equivalent representations; two factor representations are either disjoint or quasi-equivalent. The notion of quasi-equivalent representations leads to that of a quasi-dual object and a quasi-spectrum for locally compact groups and symmetric algebras, respectively.

References

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

Comments

Two representations $ \pi $ and $ \pi ^ \prime $( of a group or algebra) with representation spaces $ H $ and $ H ^ \prime $, respectively, are said to be disjoint is there is no non-zero intertwining operator between $ \pi $ and $ \pi ^ \prime $. Here, an intertwining operator between $ \pi $ and $ \pi ^ \prime $ is a continuous linear operator $ T : H \rightarrow H ^ \prime $ such that $ T \pi ( x) = \pi ^ \prime T ( x) $ for all $ x $.

How to Cite This Entry:
Quasi-equivalent representations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-equivalent_representations&oldid=48381
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article