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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 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 ''Â'' is also naturally a topological space; this generalizes 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 M''n''(C) consists of a single point. == Primitive spectrum == The topology of ''Â'' 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 an 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. As a consequence, it can be shown that there is a unique topology τ on Prim(''A'') such that the closure of a set ''X'' with respect to τ is identical to the hull-kernel closure of ''X''. Since unitarily equivalent representations have the same kernel, the map π ↦ ker(π) factors through a surjective map : We use the map ''k'' to define the topology on ''Â'' as follows: Definition. The open sets of ''Â'' are inverse images ''k''−1(''U'') of open subsets ''U'' of Prim(''A''). This is indeed a topology. The hull-kernel topology is an analogue for non-commutative rings of the Zariski topology for commutative rings. The topology on ''Â'' induced from the hull-kernel topology has other characterizations in terms of states of ''A''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Spectrum of a C*-algebra」の詳細全文を読む スポンサード リンク
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