Spectrum of a C*-algebra

In mathematics, 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 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-dimensional spaces. As explained below, the spectrum  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 dual object for any locally compact group. This dual object is suitable for formulating a Fourier transform and a Plancherel theorem for unimodular 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 groups 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 Mn(C) consists of a single point. The topology of  can be defined in several equivalent ways. We first define it in terms of the primitive spectrum . The primitive spectrum of A is the set of primitive ideals Prim(A) of A, where a primitive ideal is the kernel of a non-zero irreducible *-representation. The set of primitive ideals is a topological space with the hull-kernel topology (or Jacobson topology). This is defined as follows: If X is a set of primitive ideals, its hull-kernel closure is Hull-kernel closure is easily shown to be an idempotent operation, that is and it can be shown to satisfy the Kuratowski closure axioms.