Spectrum of a C*-algebra
The set of unitary equivalence classes of irreducible representations of the -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
-algebra, the resulting topological space coincides with the space of characters (which is homeomorphic to the space of maximal ideals, cf. Character of a
-algebra; Maximal ideal). In the general case, the spectrum of a
-algebra is the basis for decomposing its representations into direct integrals of irreducible representations.
References
[1] | J. Dixmier, "![]() |
Comments
This topology on the spectrum of a -algebra is called the hull-kernel topology, or Jacobson topology.
References
[a1] | W. Arveson, "An invitation to ![]() |
[a2] | G.K. Pedersen, "![]() |
Spectrum of a C*-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectrum_of_a_C*-algebra&oldid=32060