Gelfand–Naimark–Segal construction

In functional analysis, a discipline within mathematics, given a -algebra , the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic -representations of and certain linear functionals on (called states). The correspondence is shown by an explicit construction of the -representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal.

States and representations

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A  -representation of a  -algebra   on a Hilbert space   is a mapping   from   into the algebra of bounded operators on   such that

  •   is a ring homomorphism which carries involution on   into involution on operators
  •   is nondegenerate, that is the space of vectors     is dense as   ranges through   and   ranges through  . Note that if   has an identity, nondegeneracy means exactly   is unit-preserving, i.e.   maps the identity of   to the identity operator on  .

A state on a  -algebra   is a positive linear functional   of norm  . If   has a multiplicative unit element this condition is equivalent to  .

For a representation   of a  -algebra   on a Hilbert space  , an element   is called a cyclic vector if the set of vectors

 

is norm dense in  , in which case π is called a cyclic representation. Any non-zero vector of an irreducible representation is cyclic. However, non-zero vectors in a general cyclic representation may fail to be cyclic.

The GNS construction

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Let   be a  -representation of a  -algebra   on the Hilbert space   and   be a unit norm cyclic vector for  . Then   is a state of  .

Conversely, every state of   may be viewed as a vector state as above, under a suitable canonical representation.

Theorem.[1] — Given a state   of  , there is a  -representation   of   acting on a Hilbert space   with distinguished unit cyclic vector   such that   for every   in  .

Proof
  1. Construction of the Hilbert space  

    Define on   a semi-definite sesquilinear form  

    By the triangle inequality, the degenerate elements,   in   satisfying  , form a vector subspace   of  . By a  -algebraic argument, one can show that   is a left ideal of   (known as the left kernel of  ). In fact, it is the largest left ideal in the null space of ρ. The quotient space of   by the vector subspace   is an inner product space with the inner product defined by , which is well-defined due to the Cauchy–Schwarz inequality. The Cauchy completion of   in the norm induced by this inner product is a Hilbert space, which we denote by  .
  2. Construction of the representation  
    Define the action   of   on   by   of   on  . The same argument showing   is a left ideal also implies that   is a bounded operator on   and therefore can be extended uniquely to the completion. Unravelling the definition of the adjoint of an operator on a Hilbert space,   turns out to be  -preserving. This proves the existence of a  -representation  .
  3. Identifying the unit norm cyclic vector  

    If   has a multiplicative identity  , then it is immediate that the equivalence class   in the GNS Hilbert space   containing   is a cyclic vector for the above representation. If   is non-unital, take an approximate identity   for  . Since positive linear functionals are bounded, the equivalence classes of the net   converges to some vector   in  , which is a cyclic vector for  .

    It is clear from the definition of the inner product on the GNS Hilbert space   that the state   can be recovered as a vector state on  . This proves the theorem.

The method used to produce a  -representation from a state of   in the proof of the above theorem is called the GNS construction. For a state of a  -algebra  , the corresponding GNS representation is essentially uniquely determined by the condition,   as seen in the theorem below.

Theorem.[2] — Given a state   of  , let  ,   be  -representations of   on Hilbert spaces  ,   respectively each with unit norm cyclic vectors  ,   such that   for all  . Then  ,   are unitarily equivalent  -representations i.e. there is a unitary operator   from   to   such that   for all   in  . The operator   that implements the unitary equivalence maps   to   for all   in  .

Significance of the GNS construction

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The GNS construction is at the heart of the proof of the Gelfand–Naimark theorem characterizing  -algebras as algebras of operators. A  -algebra has sufficiently many pure states (see below) so that the direct sum of corresponding irreducible GNS representations is faithful.

The direct sum of the corresponding GNS representations of all states is called the universal representation of  . The universal representation of   contains every cyclic representation. As every  -representation is a direct sum of cyclic representations, it follows that every  -representation of   is a direct summand of some sum of copies of the universal representation.

If   is the universal representation of a  -algebra  , the closure of   in the weak operator topology is called the enveloping von Neumann algebra of  . It can be identified with the double dual  .

Irreducibility

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Also of significance is the relation between irreducible  -representations and extreme points of the convex set of states. A representation π on   is irreducible if and only if there are no closed subspaces of   which are invariant under all the operators   other than   itself and the trivial subspace  .

Theorem — The set of states of a  -algebra   with a unit element is a compact convex set under the weak-  topology. In general, (regardless of whether or not   has a unit element) the set of positive functionals of norm   is a compact convex set.

Both of these results follow immediately from the Banach–Alaoglu theorem.

In the unital commutative case, for the  -algebra   of continuous functions on some compact  , Riesz–Markov–Kakutani representation theorem says that the positive functionals of norm   are precisely the Borel positive measures on   with total mass  . It follows from Krein–Milman theorem that the extremal states are the Dirac point-mass measures.

On the other hand, a representation of   is irreducible if and only if it is one-dimensional. Therefore, the GNS representation of   corresponding to a measure   is irreducible if and only if   is an extremal state. This is in fact true for  -algebras in general.

Theorem — Let   be a  -algebra. If   is a  -representation of   on the Hilbert space   with unit norm cyclic vector  , then   is irreducible if and only if the corresponding state   is an extreme point of the convex set of positive linear functionals on   of norm  .

To prove this result one notes first that a representation is irreducible if and only if the commutant of  , denoted by  , consists of scalar multiples of the identity.

Any positive linear functionals   on   dominated by   is of the form   for some positive operator   in   with   in the operator order. This is a version of the Radon–Nikodym theorem.

For such  , one can write   as a sum of positive linear functionals:  . So   is unitarily equivalent to a subrepresentation of  . This shows that π is irreducible if and only if any such   is unitarily equivalent to  , i.e.   is a scalar multiple of  , which proves the theorem.

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.

The theorems above for  -algebras are valid more generally in the context of  -algebras with approximate identity.

Generalizations

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The Stinespring factorization theorem characterizing completely positive maps is an important generalization of the GNS construction.

History

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Gelfand and Naimark's paper on the Gelfand–Naimark theorem was published in 1943.[3] Segal recognized the construction that was implicit in this work and presented it in sharpened form.[4]

In his paper of 1947 Segal showed that it is sufficient, for any physical system that can be described by an algebra of operators on a Hilbert space, to consider the irreducible representations of a  -algebra. In quantum theory this means that the  -algebra is generated by the observables. This, as Segal pointed out, had been shown earlier by John von Neumann only for the specific case of the non-relativistic Schrödinger-Heisenberg theory.[5]

See also

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References

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  • William Arveson, An Invitation to C*-Algebra, Springer-Verlag, 1981
  • Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191.
  • Jacques Dixmier, Les C*-algèbres et leurs Représentations, Gauthier-Villars, 1969.
    English translation: Dixmier, Jacques (1982). C*-algebras. North-Holland. ISBN 0-444-86391-5.
  • Thomas Timmermann, An invitation to quantum groups and duality: from Hopf algebras to multiplicative unitaries and beyond, European Mathematical Society, 2008, ISBN 978-3-03719-043-2Appendix 12.1, section: GNS construction (p. 371)
  • Stefan Waldmann: On the representation theory of deformation quantization, In: Deformation Quantization: Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, May 31-June 2, 2001 (Studies in Generative Grammar) , Gruyter, 2002, ISBN 978-3-11-017247-8, p. 107–134 – section 4. The GNS construction (p. 113)
  • G. Giachetta, L. Mangiarotti, G. Sardanashvily (2005). Geometric and Algebraic Topological Methods in Quantum Mechanics. World Scientific. ISBN 981-256-129-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
  • Shoichiro Sakai, C*-Algebras and W*-Algebras, Springer-Verlag 1971. ISBN 3-540-63633-1

Inline references

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  1. ^ Kadison, R. V., Theorem 4.5.2, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191
  2. ^ Kadison, R. V., Proposition 4.5.3, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191
  3. ^ I. M. Gelfand, M. A. Naimark (1943). "On the imbedding of normed rings into the ring of operators on a Hilbert space". Matematicheskii Sbornik. 12 (2): 197–217. (also Google Books, see pp. 3–20)
  4. ^ Richard V. Kadison: Notes on the Gelfand–Neimark theorem. In: Robert C. Doran (ed.): C*-Algebras: 1943–1993. A Fifty Year Celebration, AMS special session commemorating the first fifty years of C*-algebra theory, January 13–14, 1993, San Antonio, Texas, American Mathematical Society, pp. 21–54, ISBN 0-8218-5175-6 (available from Google Books, see pp. 21 ff.)
  5. ^ I. E. Segal (1947). "Irreducible representations of operator algebras" (PDF). Bull. Am. Math. Soc. 53 (2): 73–88. doi:10.1090/s0002-9904-1947-08742-5.