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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 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''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 .

  • -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


: \overline{X} = \{ ho \in \operatorname{Prim}(A): ho \supseteq \bigcap_{\pi \in X} \pi\}.

Hull-kernel closure is easily shown to be an Idempotent operation, that is

: \overline{\overline{X}} = \overline{X},

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 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 \mapsto ker(π) factors through a Surjective map

: \operatorname{k}: \hat{A} ightarrow \operatorname{Prim}(A).

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 State s of ''A''.


EXAMPLES

  • -algebras



: \operatorname{I}: X \cong \operatorname{Prim}( \operatorname{C}_0(X)).

This mapping is defined by

: \operatorname{I}(x) = \{f \in \operatorname{C}_0(X): f(x) = 0 \}.

  • -algebra,


: \hat{A} \cong \operatorname{Prim}(A).

  • -algebra of bounded operators


  • -ideals: ''I''0={0} and the ideal ''K''=''K''(''H'') of compact operators. Thus as a set, Prim(''L''(''H'')) = {''I''0, ''K''}. Now


  • {''K''} is a closed subset of Prim(''L''(''H'')).


  • The closure of {''I''0} 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 {0}.

  • -algebras


  • -algebra. It is known ''A'' is isomorphic to a finite direct sum of full matrix algebras:


: A \cong \bigoplus_{e \in \operatorname{min}(A)} A e,

  • -algebras, we also have the isomorphism


: \hat{A} \cong \operatorname{Prim}(A).


OTHER CHARACTERIZATIONS OF THE SPECTRUM


  • -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 subset of ''Â''. Then the following are equivalent for an irreducile 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

:: f_\xi(x) = \langle \xi \mid \pi(x) \xi angle