Spectrum of a C -algebra

High Quality Content by WIKIPEDIA articles! The spectrum of a C -algebra or dual of a C -algebra A, denoted Â, is the set of unitary equivalence classes of irreducible -representations of A. A -representation of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant under all operators (x) with x A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum  is also naturally a topological space; this generalizes the notion of the spectrum of a ring.

Klappentext

High Quality Content by WIKIPEDIA articles! The spectrum of a C-algebra or dual of a C-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible -representations of A. A -representation p of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant under all operators p(x) with x A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum  is also naturally a topological space; this generalizes the notion of the spectrum of a ring.