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Difference between revisions of "Factor representation"

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A [[Linear representation|linear representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f0380701.png" /> of a group or an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f0380702.png" /> on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f0380703.png" /> such that the [[Von Neumann algebra|von Neumann algebra]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f0380704.png" /> generated by the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f0380705.png" /> is a [[Factor|factor]]. If this factor is of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f0380706.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f0380707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f0380708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f0380709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f03807010.png" /> etc.), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f03807011.png" /> is called a factor representation of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038070/f03807013.png" />, etc.
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A [[Linear representation|linear representation]]  $  \pi $
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of a group or an algebra $  X $
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on a Hilbert space $  H $
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such that the [[Von Neumann algebra|von Neumann algebra]] on $  H $
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generated by the family $  \pi ( X) $
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is a [[Factor|factor]]. If this factor is of type $  \textrm{ I } $(
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respectively, $  \textrm{ II } $,  
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$  \textrm{ III } $,  
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$  \textrm{ II } _ {1} $,  
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$  \textrm{ II } _  \infty  $
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etc.), then $  \pi $
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is called a factor representation of type $  \textrm{ I } $,  
 +
etc.

Latest revision as of 19:38, 5 June 2020


A linear representation $ \pi $ of a group or an algebra $ X $ on a Hilbert space $ H $ such that the von Neumann algebra on $ H $ generated by the family $ \pi ( X) $ is a factor. If this factor is of type $ \textrm{ I } $( respectively, $ \textrm{ II } $, $ \textrm{ III } $, $ \textrm{ II } _ {1} $, $ \textrm{ II } _ \infty $ etc.), then $ \pi $ is called a factor representation of type $ \textrm{ I } $, etc.

How to Cite This Entry:
Factor representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Factor_representation&oldid=46899
This article was adapted from an original article by A. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article