Editing Spectrum of a C*-algebra

{{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.

== 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 ideal]]s 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

:<math> \overline{X} = \left \{\rho \in \operatorname{Prim}(A): \rho \supseteq \bigcap_{\pi \in X} \pi \right \}. </math>

Hull-kernel closure is easily shown to be an [[idempotent]] operation, that is

:<math> \overline{\overline{X}} = \overline{X},</math>

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

:<math> \operatorname{k}: \hat{A} \to \operatorname{Prim}(A). </math>

We use the map ''k'' to define the topology on ''Â'' as follows:

'''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.

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 [[state (functional analysis)|state]]s of ''A''.

== Examples ==

=== Commutative C*-algebras ===
[[File:3-dim commut algebra, subalgebras, ideals.svg|thumb|right|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.]]

The spectrum of a commutative C*-algebra ''A'' coincides with the [[Gelfand transformation|Gelfand 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]]

:<math> \operatorname{I}: X \cong \operatorname{Prim}( \operatorname{C}(X)).</math>

This mapping is defined by

: <math>  \operatorname{I}(x) = \{f \in \operatorname{C}(X): f(x) = 0 \}.</math>

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,

:<math> \hat{A} \cong \operatorname{Prim}(A).</math>

=== The C*-algebra of bounded operators ===
Let ''H'' be a separable infinite-dimensional [[Hilbert space]]. ''L''(''H'') has two norm-closed *-ideals: ''I''<sub>0</sub>&nbsp;=&nbsp;{0} and the ideal ''K''&nbsp;=&nbsp;''K''(''H'') of compact operators.  Thus as a set, Prim(''L''(''H'')) =&nbsp;{''I''<sub>0</sub>,&nbsp;''K''}. Now

* {''K''} is a closed subset of Prim(''L''(''H'')).
* The closure of {''I''<sub>0</sub>} is Prim(''L''(''H'')).

Thus Prim(''L''(''H'')) is a non-Hausdorff space.

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&nbsp;{0}.

=== Finite-dimensional C*-algebras ===
Suppose ''A'' is a finite-dimensional C*-algebra.  It is known ''A'' is isomorphic to a finite direct sum of full matrix algebras:

:<math> A \cong \bigoplus_{e \in \operatorname{min}(A)} Ae, </math>

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

:<math> \hat{A} \cong \operatorname{Prim}(A).</math>

== Other characterizations of the spectrum ==
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.

In fact, the topology on ''Â'' is intimately connected with the concept of [[weak containment]] of representations as is shown by the following:

:'''Theorem'''. Let ''S'' be a subset of ''Â''. Then the following are equivalent for an irreducible representation π;
:# The equivalence class of π in ''Â'' is in the closure of ''S''
:# Every state associated to π, that is one of the form
:::<math> f_\xi(x) = \langle \xi  \mid \pi(x) \xi \rangle </math>
::with ||ξ|| = 1, is the weak limit of states associated to representations in ''S''.

The second condition means exactly that π is weakly contained in ''S''.

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'' ↦ π<sub>''f''</sub> is a surjective map.

From the previous theorem one can easily prove the following;

:'''Theorem''' The mapping
::<math> \kappa: \operatorname{PureState}(A) \to \hat{A} </math>
:given by the GNS construction is continuous and open.

=== The space Irr<sub>''n''</sub>(''A'') ===
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''.

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

:<math>\langle \pi_i(x) \xi \mid \eta \rangle \to \langle \pi(x) \xi \mid \eta \rangle \quad \forall \xi, \eta \in H_n \ x \in A. </math>

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

:<math> \pi_i(x) \xi \to \pi(x) \xi \quad \mbox{ normwise } \forall \xi \in H_n \ x \in A. </math>

:'''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.

'''Remark'''.  The piecing together of the various ''Â<sub>n</sub>'' can be quite complicated.

== Mackey–Borel structure ==
''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.

'''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.

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.

:'''Theorem'''. If ''A'' is separable, ''Â'' is smooth if and only if ''A'' is of type I.

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.

== Algebraic primitive spectra  ==
{{see also|Primitive spectrum}}
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.

:'''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.

This is the Corollary of Theorem 2.9.5 of the Dixmier reference.

If ''G'' is a locally compact group, the topology on dual space of the [[group algebra of a locally compact group|group C*-algebra]] C*(''G'') of ''G'' is called the '''Fell topology''', named after [[J. M. G. Fell]].

== References ==
{{Reflist}}
* J. Dixmier, ''C*-Algebras'', North-Holland, 1977 (a translation of ''Les C*-algèbres et leurs représentations'')
* J.  Dixmier, ''Les C*-algèbres et leurs représentations'', Gauthier-Villars, 1969.
* J. Glimm, ''Type I C*-algebras'', Annals of Mathematics, vol 73, 1961.
* G. Mackey, ''The Theory of Group Representations'', The University of Chicago Press, 1955.

{{Functional Analysis}}
{{SpectralTheory}}

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[[Category:C*-algebras]]
[[Category:Spectral theory]]