Gelfand–Naimark–Segal Construction
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functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, a discipline within
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, given a C^*-algebra A, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of A and certain
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
s on A (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 Irving Ezra Segal (1918–1998) was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is often referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for h ...
.


States and representations

A *-representation of a C^*-algebra A on a
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
H is a
map A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...
ping \pi from A into the algebra of
bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...
s on H such that * \pi is a
ring homomorphism In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
which carries
involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...
on A into involution on operators * \pi is nondegenerate, that is the space of vectors \pi (x) \xi is dense as x ranges through A and \xi ranges through H. Note that if A has an identity, nondegeneracy means exactly \pi is unit-preserving, i.e. \pi maps the identity of A to the identity operator on H. A
state State most commonly refers to: * State (polity), a centralized political organization that regulates law and society within a territory **Sovereign state, a sovereign polity in international law, commonly referred to as a country **Nation state, a ...
on a C^*-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. For a representation \pi of a C^*-algebra A on a Hilbert space H, an element \xi is called a cyclic vector if the set of vectors :\ is norm dense in H, in which case π is called a cyclic representation. Any non-zero vector of an
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
is cyclic. However, non-zero vectors in a general cyclic representation may fail to be cyclic.


The GNS construction

Let \pi be a *-representation of a C^*-algebra A on the Hilbert space H and \xi be a unit norm cyclic vector for \pi. Then a \mapsto \langle \pi(a) \xi, \xi\rangle is a state of A. Conversely, every state of A may be viewed as a vector state as above, under a suitable canonical representation. The method used to produce a *-representation from a state of A in the proof of the above theorem is called the GNS construction. For a state of a C^*-algebra A, the corresponding GNS representation is essentially uniquely determined by the condition, \rho(a) = \langle \pi(a) \xi, \xi \rangle 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 In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra ''A'' is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 ...
characterizing C^*-algebras as algebras of operators. A C^*-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 A. The universal representation of A contains every cyclic representation. As every *-representation is a direct sum of cyclic representations, it follows that every *-representation of A is a direct summand of some sum of copies of the universal representation. If \Phi is the universal representation of a C^*-algebra A, the closure of \Phi(A) in the weak operator topology is called the enveloping von Neumann algebra of A. It can be identified with the double dual A^.


Irreducibility

Also of significance is the relation between irreducible *-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 \pi(x) other than H itself and the trivial subspace \. Both of these results follow immediately from the Banach–Alaoglu theorem. In the unital commutative case, for the C^*-algebra C(X) of continuous functions on some compact X, Riesz–Markov–Kakutani representation theorem says that the positive functionals of norm \leq 1 are precisely the Borel positive measures on X with total mass \leq 1. It follows from
Krein–Milman theorem In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). This theorem generalizes to infinite-dimensional spaces and to arbitra ...
that the extremal states are the Dirac point-mass measures. On the other hand, a representation of C(X) is irreducible if and only if it is one-dimensional. Therefore, the GNS representation of C(X) corresponding to a measure \mu is irreducible if and only if \mu is an extremal state. This is in fact true for C^*-algebras in general. To prove this result one notes first that a representation is irreducible if and only if the
commutant In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...
of \pi(A), denoted by \pi(A)', consists of scalar multiples of the identity. Any positive linear functionals g on A dominated by f is of the form g(x^*x) = \langle \pi(x) \xi, \pi(x) T_g \, \xi \rangle for some positive operator T_g in \pi(A)' with 0 \leq T \leq 1 in the operator order. This is a version of the
Radon–Nikodym theorem In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurab ...
. For such g, one can write f as a sum of positive linear functionals: f = g + g'. So \pi is unitarily equivalent to a subrepresentation of \pi_g \oplus \pi_. This shows that π is irreducible if and only if any such \pi_g is unitarily equivalent to \pi, i.e. g is a scalar multiple of f, 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 C^*-algebras are valid more generally in the context of B^*-algebras 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'' representations of a C^*-algebra. In quantum theory this means that the C^*-algebra is generated by the observables. This, as Segal pointed out, had been shown earlier by
John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
only for the specific case of the non-relativistic Schrödinger-Heisenberg theory.


See also

* Cyclic and separating vector * KSGNS construction


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 mathematics and physics, deformation quantization roughly amounts to finding a (quantum) algebra whose classical limit is a given (classical) algebra such as a Lie algebra or a Poisson algebra. In physics Intuitively, a deformation of a math ...
'', 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

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