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STATES AND REPRESENTATIONS



  • π is a Ring Homomorphism which carries Involution on ''A'' into involution on operators

  • π is Nondegenerate , that is the space of vectors π(''x'') ξ is dense as ''x'' ranges through ''A'' and ξ ranges through ''H''. Note that if ''A'' has an identity, nondegeneracy means exactly π is Unit-preserving .


  • -algebra ''A'' is a Positive Linear Functional ''f'' of norm 1. If ''A'' has a multiplicative unit element this condition is equivalent to ''f''(1) = 1.


  • -algebra ''A'' with a unit element is a compact Convex Set under the weak
    topology. In general, (regardless of whether or not ''A'' has a unit element) the set of positive functionals of norm ≤ 1 is a compact convex set.


Both of these results follow immediately from the Banach-Alaoglu Theorem .

  • -algebra ''A'' on a Hilbert space ''H'' has Cyclic Vector ξ if and only if the set of vectors

    • :\{\pi(x)\xi:x\in A\}

is norm dense in ''H''. Any non-zero vector of an irreducible representation is cyclic. However, non-zero vectors in a cyclic representation may fail to be cyclic.

Note to reader: In our definition of inner product, the conjugate linear argument is the first argument and the linear argument is the second argument. This is done for reasons of compatibility with the physics literature. Thus the order of arguments in some of the constructions below is exactly the opposite from those in many mathematics textbooks.

  • -algebra. If π is a
    representation of

    • ''A'' on the Hilbert space ''H'' with cyclic vector ξ having norm 1. Then

: x \mapsto \langle \xi, \pi(x)\xi angle
  • -representations π, π' each with unit norm cyclic vectors ξ, ξ' and having the same associated states, then π, π' are unitarily equivalent representations; moreover, the unitary operator ''U'' that implements the unitary equivalence can be chosen to map ξ to ξ'.


  • -representation π of ''A'' with distinguished cyclic vector ξ such that its associated state is ρ

    • : ho(x)=\langle \xi, \pi(x) \xi angle

for every ''x'' in ''A''.

The construction proceeds as follows: Assume ''A'' has a unit element. ''A'' can be equipped with a ''singular'' Inner Product
  • y)

  • x'')=0. The vector space ''I'' is actually a Left Ideal of ''A'', so the elements of ''A'' act on ''A''/''I'' as operators on the left. ''H'' is then taken to be the Cauchy Completion of ''A''/''I'', equipped with the quotient norm. The cyclic vector ξ is the image of 1 in ''A''/''I''.

  • -algebra ''A''1 obtained from ''A'' by adjoining a multiplicative identity. Any state ''f'' on ''A'' extends uniquely to a state ''f''1 on ''A''1. Apply the previous construction to ''f''1.


  • -algebras as algebras of operators.



IRREDUCIBILITY


  • -representations and extreme points of the convex set of states. A representation π on ''H'' is irreducible if and only if there are no closed subspaces of ''H'' which are invariant under all the operators π(''x'')

    • other than ''H'' itself and the trivial subspace {0}.


  • -algebra. If π is a
    representation of

    • ''A'' on the Hilbert space ''H'' with unit norm cyclic vector ξ, then

π is irreducible if and only if the corresponding state ''f'' is an Extreme Point of the convex set of positive linear functionals on ''A'' of norm ≤ 1.

To prove this result one notes that given a self-adjoint operator ''T'' on ''H'' which commutes with all the operators π(x), and is such that
0 ≤ ''T'' ≤ 1 in the operator order,
: f_T(x) = \langle T \xi, \pi(x) T\xi angle
is a positive linear functional on ''A'' (not in general a state) dominated by ''f''. This map is easily shown to be a bijection. Now the representation π is irreducible if and only if the only bounded operators which commute with all the π(x) are scalar multiples of the identity. Thus a necessary and sufficient condition π be irreducible is that the set of states dominated by ''f'' consist only of scalar multiples of ''f''. This condition on ''f'' can be shown to be equivalent to ''f'' being an extreme point in the set of positive linear functionals of norm ≤ 1.

Extremal states are usually called Pure States . Note that a state is a pure state if and only if it is extremal in the convex set of states.

  • -algebras are valid more generally in the context of B
    algebra     s with approximate identity.



GENERALIZATIONS


The Stinespring Factorization Theorem characterizing Completely Positive Map s is an important generalization of the GNS construction.


REFERENCES