Wikipedia / Gelfand–Naimark–Segal construction
Wikipedia / Gelfand–Naimark–Segal construction
In functional analysis, a discipline within mathematics, given a C^*-algebra, the Gelfand–Naimark–Segal construction and certain linear functionals (called states-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal.
States and representations
A -representation of a C^*-algebra on a Hilbert space is a mapping into the algebra of bounded operators such that
is a ring homomorphism which carries involution into involution on operators
is nondegenerate.
A state is a positive linear functional.
is called a cyclic vector if the set of vectors
, 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
.
may be viewed as a vector state as above, under a suitable canonical representation.
in the proof of the above theorem is called the GNS construction as seen in the theorem below.
Significance of the GNS construction
The GNS construction is at the heart of the proof of the Gelfand–Naimark theorem-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 is a direct summand of some sum of copies of the universal representation.
in the weak operator topology is called the enveloping von Neumann algebra.
Irreducibility
Also of significance is the relation between irreducible.
Both of these results follow immediately from the Banach–Alaoglu theorem.
, Riesz–Markov–Kakutani representation theorem. It follows from Krein–Milman theorem that the extremal states are the Dirac point-mass measures.
-algebras in general.
To prove this result one notes first that a representation is irreducible if and only if the commutant, consists of scalar multiples of the identity.
in the operator order. This is a version of the Radon–Nikodym theorem.
is unitarily equivalent to a subrepresentation, 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.
-algebras are valid more generally in the context of B^*-algebrass with approximate identity.
Generalizations
The Stinespring factorization theorem characterizing completely positive maps is an important generalization of the GNS construction.
History
Gelfand and Naimark's paper on the Gelfand–Naimark theorem was published in 1943. Segal recognized the construction that was implicit in this work and presented it in sharpened form.
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-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.
See also
References
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. .
Jacques Dixmier, Les C*-algèbres et leurs Représentations, Gauthier-Villars, 1969.English translation:
Thomas Timmermann, An invitation to quantum groups and duality: from Hopf algebras to multiplicative unitaries and beyond, European Mathematical Society, 2008, – Appendix 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, , p.107–134 – section 4. The GNS construction (p. 113)
Shoichiro Sakai, C*-Algebras and W*-Algebras, Springer-Verlag 1971.
Inline references