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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 } ,  
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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=14007
This article was adapted from an original article by A. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article