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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 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 generalizes the notion of the [[spectrum of a ring]].
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| 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.
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| == Primitive spectrum ==
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| The [[topology]] of ''Â'' can be defined in several equivalent ways. We first define it in terms of the '''primitive spectrum''' .
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| The primitive spectrum of ''A'' is the set of [[primitive ideal]]s 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
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| :<math> \overline{X} = \{\rho \in \operatorname{Prim}(A): \rho \supseteq \bigcap_{\pi \in X} \pi\}. </math>
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| Hull-kernel closure is easily shown to be an [[idempotent]] operation, that is
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| :<math> \overline{\overline{X}} = \overline{X},</math>
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| 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''.
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| Since unitarily equivalent representations have the same kernel, the map π {{mapsto}} ker(π) factors through a [[surjective]] map
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| :<math> \operatorname{k}: \hat{A} \rightarrow \operatorname{Prim}(A). </math>
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| We use the map ''k'' to define the topology on ''Â'' as follows:
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| '''Definition'''. The open sets of ''Â'' are inverse images ''k''<sup>−1</sup>(''U'') of open subsets ''U'' of Prim(''A''). This is indeed a topology.
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| The hull-kernel topology is an analogue for non-commutative rings of the [[Zariski topology]] for commutative rings.
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| The topology on ''Â'' induced from the hull-kernel topology has other characterizations in terms of [[state (functional analysis)|state]]s of ''A''.
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| == Examples ==
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| === Commutative C*-algebras ===
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| [[File:3-dim commut algebra, subalgebras, ideals.svg|thumb|left|224px|3-dimensional commutative C*-algebra and its ideals. Each of 8 ideals corresponds to a closed subset of discrete 3-points space (or to an open complement). Primitive ideals correspond to closed [[singleton (mathematics)|singletons]]. See details at the image description page.]]
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| The spectrum of a commutative C*-algebra ''A'' coincides with the [[Gelfand transformation|usual dual]] of ''A'' (not to be confused with the [[Banach space|dual]] ''A''' of the Banach space ''A''). In particular, suppose ''X'' is a [[compact space|compact]] [[Hausdorff space]]. Then there is a [[natural transformation|natural]] [[homeomorphism]]
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| :<math> \operatorname{I}: X \cong \operatorname{Prim}( \operatorname{C}(X)).</math>
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| This mapping is defined by
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| : <math> \operatorname{I}(x) = \{f \in \operatorname{C}(X): f(x) = 0 \}.</math>
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| I(''x'') is a closed maximal ideal in C(''X'') so is in fact primitive. For details of the proof, see the Dixmier reference. For a commutative C*-algebra,
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| :<math> \hat{A} \cong \operatorname{Prim}(A).</math>
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| === The C*-algebra of bounded operators ===
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| Let ''H'' be a separable [[Hilbert space]]. ''L''(''H'') has two norm-closed *-ideals: ''I''<sub>0</sub> = {0} and the ideal ''K'' = ''K''(''H'') of compact operators. Thus as a set, Prim(''L''(''H'')) = {''I''<sub>0</sub>, ''K''}. Now
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| * {''K''} is a closed subset of Prim(''L''(''H'')).
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| * The closure of {''I''<sub>0</sub>} is Prim(''L''(''H'')).
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| Thus Prim(''L''(''H'')) is a non-Hausdorff space.
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| The spectrum of ''L''(''H'') on the other hand is much larger. There are many inequivalent irreducible representations with kernel ''K''(''H'') or with kernel {0}.
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| === Finite dimensional C*-algebras ===
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| Suppose ''A'' is a finite dimensional C*-algebra. It is known ''A'' is isomorphic to a finite direct sum of full matrix algebras:
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| :<math> A \cong \bigoplus_{e \in \operatorname{min}(A)} A e, </math>
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| where min(''A'') are the minimal central projections of ''A''. The spectrum of ''A'' is canonically isomorphic to min(''A'') with the [[discrete topology]]. For finite dimensional C*-algebras, we also have the isomorphism
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| :<math> \hat{A} \cong \operatorname{Prim}(A).</math>
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| == Other characterizations of the spectrum ==
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| The hull-kernel topology is easy to describe abstractly, but in practice for C*-algebras associated to [[locally compact]] [[topological group]]s, other characterizations of the topology on the spectrum in terms of positive definite functions are desirable.
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| In fact, the topology on ''Â'' is intimately connected with the concept of [[weak containment]] of representations as is shown by the following:
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| '''Theorem'''. Let ''S'' be subset of ''Â''. Then the following are equivalent for an irreducile representation π
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| # The equivalence class of π in ''Â'' is in the closure of ''S''
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| # Every state associated to π, that is one of the form
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| ::<math> f_\xi(x) = \langle \xi \mid \pi(x) \xi \rangle </math>
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| :with ||ξ||=1, is the weak limit of states associated to representations in ''S''.
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| The second condition means exactly that π is weakly contained in ''S''.
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| The [[GNS construction]] is a recipe for associating states of a C*-algebra ''A'' to representations of ''A''. By one of the basic theorems associated to the GNS construction, a state ''f'' is [[pure state|pure]] if and only if the associated representation π<sub>''f''</sub> is irreducible. Moreover, the mapping κ: PureState(''A'') → ''Â'' defined by ''f'' {{mapsto}} π<sub>''f''</sub> is a surjective map.
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| From the previous theorem one can easily prove the following;
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| '''Theorem''' The mapping
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| : <math> \kappa: \operatorname{PureState}(A) \rightarrow \hat{A} </math> | |
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| given by the GNS construction is continuous and open.
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| === The space Irr<sub>''n''</sub>(''A'') ===
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| There is yet another characterization of the topology on ''Â'' which arises by considering the space of representations as a topological space with an appropriate pointwise convergence topology. More precisely, let ''n'' be a cardinal number and let ''H''<sub>''n''</sub> be the canonical Hilbert space of dimension ''n''.
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| Irr<sub>''n''</sub>(''A'') is the space of irreducible *-representations of ''A'' on ''H''<sub>''n''</sub> with the point-weak topology. In terms of convergence of nets, this topology is defined by π<sub>''i''</sub> → π if and only if
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| :<math> \langle \pi_i(x) \xi \mid \eta \rangle \rightarrow \langle \pi(x) \xi \mid \eta \rangle \quad \forall \xi, \eta \in H_n \ x \in A. </math>
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| It turns out that this topology on Irr<sub>''n''</sub>(''A'') is the same as the point-strong topology, i.e. π<sub>''i''</sub> → π if and only if
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| :<math> \pi_i(x) \xi \rightarrow \pi(x) \xi \quad \mbox{ normwise } \forall \xi \in H_n \ x \in A. </math>
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| '''Theorem'''. Let ''Â''<sub>''n''</sub> be the subset of ''Â'' consisting of equivalence classes of representations whose underlying Hilbert space has dimension ''n''. The canonical map Irr<sub>''n''</sub>(''A'') → ''Â''<sub>''n''</sub> is continuous and open. In particular, ''Â''<sub>''n''</sub> can be regarded as the quotient topological space of Irr<sub>''n''</sub>(''A'') under unitary equivalence.
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| '''Remark'''. The piecing together of the various ''Â''<sub>''n''</sub> can be quite complicated.
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| == Mackey Borel structure ==
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| ''Â'' is a topological space and thus can also be regarded as a [[Borel set|Borel space]]. A famous conjecture of [[G. Mackey]] proposed that a ''separable'' locally compact group is of type I if and only if the Borel space is standard, i.e. is isomorphic (in the category of Borel spaces) to the underlying Borel space of a [[Polish space|complete separable metric space]]. Mackey called Borel spaces with this property '''smooth'''. This conjecture was proved by [[James Glimm]] for separable C*-algebras in the 1961 paper listed in the references below.
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| '''Definition'''. A non-degenerate *-representation π of a separable C*-algebra ''A'' is a '''factor representation''' if and only if the center of the von Neumann algebra generated by π(''A'') is one-dimensional. A C*-algebra ''A'' is of type I if and only if any separable factor representation of ''A'' is a finite or countable multiple of an irreducible one.
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| Examples of separable locally compact groups ''G'' such that C*(''G'') is of type I are [[connected space|connected]] (real) [[nilpotent]] [[Lie group]]s and connected real [[semi-simple]] Lie groups. Thus the [[Heisenberg group]]s are all of type I. Compact and abelian groups are also of type I.
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| '''Theorem'''. If ''A'' is separable, ''Â'' is smooth if and only if ''A'' is of type I.
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| The result implies a far-reaching generalization of the structure of representations of separable type I C*-algebras and correspondingly of separable locally compact groups of type I.
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| == Algebraic primitive spectra ==
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| Since a C*-algebra ''A'' is a [[ring (mathematics)|ring]], we can also consider the set of [[primitive ideal]]s of ''A'', where ''A'' is regarded algebraically. For a ring an ideal is primitive if and only if it is the [[Annihilator (ring theory)|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.
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| '''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.
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| This is the Corollary of Theorem 2.9.5 of the Dixmier reference.
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| If ''G'' is a locally compact group, the topology on dual space of the [[group algebra|group C*-algebra]] C*(''G'') of ''G'' is called the '''Fell topology''', named after [[J. M. G. Fell]].
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| == References ==
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| * J. Dixmier, ''Les C*-algèbres et leurs représentations'', Gauthier-Villars, 1969.
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| * J. Glimm, ''Type I C*-algebras'', Annals of Mathematics, vol 73, 1961.
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| * G. Mackey, ''The Theory of Group Representations'', The University of Chicago Press, 1955.
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| {{Functional Analysis}}
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| [[Category:C*-algebras]]
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| [[Category:Spectral theory]]
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