Algebraic Primitive Spectra
Since a C*-algebra A is a ring, we can also consider the set of primitive ideals of A, where A is regarded algebraically. For a ring an ideal is primitive if and only if it is the annihilator of a simple module. It turns out that for a C*-algebra A, an ideal is algebraically primitive if and only if it is primitive in the sense defined above.
Theorem. Let A be a C*-algebra. Any algebraically irreducible representation of A on a complex vector space is algebraically equivalent to a topologically irreducible *-representation on a Hilbert space. Topologically irreducible *-representations on a Hilbert space are algebraically isomorphic if and only if they are unitarily equivalent.
This is the Corollary of Theorem 2.9.5 of the Dixmier reference.
If G is a locally compact group, the topology on dual space of the group C*-algebra C*(G) of G is called the Fell topology, named after J. M. G. Fell.
Read more about this topic: Spectrum Of A C*-algebra
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