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Difference between revisions of "Spectrum of a C*-algebra"

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The set of unitary equivalence classes of irreducible representations of the [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086560/s0865602.png" />-algebra]]. The spectrum can be topologized if one declares that the closure of a subset is the family of all (equivalence classes of) representations whose kernels contain the intersection of the kernels of all the representations of this subset. For a commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086560/s0865603.png" />-algebra, the resulting topological space coincides with the space of characters (which is homeomorphic to the space of maximal ideals, cf. [[Character of a C*-algebra|Character of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086560/s0865604.png" />-algebra]]; [[Maximal ideal|Maximal ideal]]). In the general case, the spectrum of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086560/s0865605.png" />-algebra is the basis for decomposing its representations into direct integrals of irreducible representations.
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The set of unitary equivalence classes of irreducible representations of the [[C*-algebra|$C^*$-algebra]]. The spectrum can be topologized if one declares that the closure of a subset is the family of all (equivalence classes of) representations whose kernels contain the intersection of the kernels of all the representations of this subset. For a commutative $C^*$-algebra, the resulting topological space coincides with the space of characters (which is homeomorphic to the space of maximal ideals, cf. [[Character of a C*-algebra|Character of a $C^*$-algebra]]; [[Maximal ideal|Maximal ideal]]). In the general case, the spectrum of a $C^*$-algebra is the basis for decomposing its representations into direct integrals of irreducible 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/s/s086/s086560/s0865606.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Dixmier,  "$C^*$ algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
This topology on the spectrum of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086560/s0865607.png" />-algebra is called the hull-kernel topology, or Jacobson topology.
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This topology on the spectrum of a $C^*$-algebra is called the hull-kernel topology, or Jacobson topology.
  
 
====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/s/s086/s086560/s0865608.png" />-algebras" , Springer  (1976)  pp. Chapts. 3–4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.K. Pedersen,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086560/s0865609.png" />-algebras and their automorphism groups" , Acad. Press  (1979)  pp. §4.1</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Arveson,  "An invitation to $C^*$-algebras" , Springer  (1976)  pp. Chapts. 3–4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.K. Pedersen,  "$C^*$-algebras and their automorphism groups" , Acad. Press  (1979)  pp. §4.1</TD></TR></table>

Revision as of 15:19, 1 May 2014

The set of unitary equivalence classes of irreducible representations of the $C^*$-algebra. The spectrum can be topologized if one declares that the closure of a subset is the family of all (equivalence classes of) representations whose kernels contain the intersection of the kernels of all the representations of this subset. For a commutative $C^*$-algebra, the resulting topological space coincides with the space of characters (which is homeomorphic to the space of maximal ideals, cf. Character of a $C^*$-algebra; Maximal ideal). In the general case, the spectrum of a $C^*$-algebra is the basis for decomposing its representations into direct integrals of irreducible representations.

References

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


Comments

This topology on the spectrum of a $C^*$-algebra is called the hull-kernel topology, or Jacobson topology.

References

[a1] W. Arveson, "An invitation to $C^*$-algebras" , Springer (1976) pp. Chapts. 3–4
[a2] G.K. Pedersen, "$C^*$-algebras and their automorphism groups" , Acad. Press (1979) pp. §4.1
How to Cite This Entry:
Spectrum of a C*-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectrum_of_a_C*-algebra&oldid=32060
This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article