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An involutive subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f0380501.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f0380502.png" /> of linear operators on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f0380503.png" /> that is closed relative to so-called weak convergence of operators and has the property that its centre (that is, the collection of all operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f0380504.png" /> that commute with every operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f0380505.png" />) consists of scalar multiples of the unit operator.
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An involutive subalgebra $\mathfrak A$ of the algebra $B(H,H)$ of linear operators on a Hilbert space $H$ that is closed relative to so-called weak convergence of operators and has the property that its centre (that is, the collection of all operators in $\mathfrak A$ that commute with every operator in $\mathfrak A$) consists of scalar multiples of the unit operator.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f0380506.png" /> is a factor, then for a large supply of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f0380507.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f0380508.png" /> one can define the concept of the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f0380509.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f03805010.png" /> as an invariant that is preserved, not under arbitrary isometries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f03805011.png" />, but only under those in the given factor with additional natural properties (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f03805012.png" />). All factors can be divided into five classes corresponding to the values that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f03805013.png" /> can take, where, for example, for a factor of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f03805014.png" /> it can take any value in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f03805015.png" />.
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If $\mathfrak A$ is a factor, then for a large supply of subspaces $F$ of $H$ one can define the concept of the dimension $\dim_\mathfrak AF$ relative to $\mathfrak A$ as an invariant that is preserved, not under arbitrary isometries $\mathcal F$, but only under those in the given factor with additional natural properties (for example, $\dim_\mathfrak A(F_1\oplus F_2)=\dim_\mathfrak AF_1+\dim_\mathfrak AF_2$). All factors can be divided into five classes corresponding to the values that $\dim_\mathfrak AF$ can take, where, for example, for a factor of class $\Pi_\infty$ it can take any value in $[0,\infty]$.
  
  
  
 
====Comments====
 
====Comments====
An involutive algebra is an algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f03805016.png" /> endowed with an [[Involution|involution]]. For information concerning various types of factors cf. [[Von Neumann algebra|von Neumann algebra]].
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An involutive algebra is an algebra over $\mathbf C$ endowed with an [[Involution|involution]]. For information concerning various types of factors cf. [[Von Neumann algebra|von Neumann algebra]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.K. Pedersen,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f03805017.png" />-algebras and their automorphism groups" , Acad. Press  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038050/f03805018.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.K. Pedersen,  "$C^*$-algebras and their automorphism groups" , Acad. Press  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Dixmier,  "$C^*$ algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>

Revision as of 11:24, 1 August 2014

An involutive subalgebra $\mathfrak A$ of the algebra $B(H,H)$ of linear operators on a Hilbert space $H$ that is closed relative to so-called weak convergence of operators and has the property that its centre (that is, the collection of all operators in $\mathfrak A$ that commute with every operator in $\mathfrak A$) consists of scalar multiples of the unit operator.

If $\mathfrak A$ is a factor, then for a large supply of subspaces $F$ of $H$ one can define the concept of the dimension $\dim_\mathfrak AF$ relative to $\mathfrak A$ as an invariant that is preserved, not under arbitrary isometries $\mathcal F$, but only under those in the given factor with additional natural properties (for example, $\dim_\mathfrak A(F_1\oplus F_2)=\dim_\mathfrak AF_1+\dim_\mathfrak AF_2$). All factors can be divided into five classes corresponding to the values that $\dim_\mathfrak AF$ can take, where, for example, for a factor of class $\Pi_\infty$ it can take any value in $[0,\infty]$.


Comments

An involutive algebra is an algebra over $\mathbf C$ endowed with an involution. For information concerning various types of factors cf. von Neumann algebra.

References

[a1] G.K. Pedersen, "$C^*$-algebras and their automorphism groups" , Acad. Press (1979)
[a2] J. Dixmier, "$C^*$ algebras" , North-Holland (1977) (Translated from French)
How to Cite This Entry:
Factor algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Factor_algebra&oldid=11524
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article