Spectrum of a C*-algebra

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), Chap. 3

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