Editing Spectrum of a C*-algebra (section)

{{Short description|Mathematical concept}}
In mathematics, the '''spectrum of a [[C*-algebra]]''' or '''dual of a C*-algebra''' ''A'', denoted ''Â'', is the set of [[unitary representation|unitary equivalence]] classes of  [[irreducible representation|irreducible]] *-representations of ''A''. A [[*-representation]] π 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 π(''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-[[dimension]]al [[space (mathematics)|spaces]]. As explained below, the spectrum ''Â'' is also naturally a [[topological space]]; this is similar to the notion of the [[spectrum of a ring]].

One of the most important applications of this concept is to provide a notion of [[duality (mathematics)|dual]] object for any [[locally compact group]].  This dual object is suitable for formulating a [[Fourier transform]] and a [[Plancherel theorem]] for [[unimodular group|unimodular]] [[separable space|separable]] locally compact groups of type I and a decomposition theorem for arbitrary representations of separable locally compact groups of type I.  The resulting duality theory for locally compact groups is however much weaker than the [[Tannaka–Krein duality]] theory for [[compact topological group]]s or [[Pontryagin duality]] for locally compact ''abelian'' groups, both of which are complete invariants. That the dual is not a complete invariant is easily seen as the  dual of any finite-dimensional full matrix algebra M<sub>''n''</sub>('''C''') consists of a single point.