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| | + | {{TEX|done}} |
| + | 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, " | + | <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 | + | 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 | + | <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
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