Gelfand–Naimark–Segal construction
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In
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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, a discipline within
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, given a
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
''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 map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
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 Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел ...
,
Mark Naimark Mark Aronovich Naimark (russian: Марк Ароно́вич Наймарк) (5 December 1909 – 30 December 1978) was a Soviet mathematician who made important contributions to functional analysis and mathematical physics. Life Naimark was b ...
, and Irving Segal.


States and representations

A *-representation of a
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
''A'' on a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''H'' is a
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
ping π 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 ...
s on ''H'' such that * π is a
ring homomorphism In ring theory, a branch of abstract algebra, 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 such that ''f'' is: :addition preservi ...
which carries
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
on ''A'' into involution on operators * π is
nondegenerate In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. T ...
, 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, i.e. π maps the identity of ''A'' to the identity operator on ''H''. A
state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * ''Our S ...
on a C*-algebra ''A'' is a
positive linear functional In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space (V, \leq) is a linear functional f on V so that for all positive elements v \in V, that is v \geq 0, it holds that f(v) \geq 0. In ot ...
''f'' of norm 1. If ''A'' has a multiplicative unit element this condition is equivalent to ''f''(1) = 1. For a representation π of a C*-algebra ''A'' on a Hilbert space ''H'', an element ξ 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,W ...
is cyclic. However, non-zero vectors in a cyclic representation may fail to be cyclic.


The GNS construction

Let π be a *-representation of a C*-algebra ''A'' on the Hilbert space ''H'' and ξ be a unit norm cyclic vector for π. 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. :Theorem. Given a state ρ of ''A'', there is a *-representation π of ''A'' acting on a Hilbert space ''H'' with distinguished unit cyclic vector ξ such that \rho(a)=\langle \pi(a) \xi, \xi \rangle for every ''a'' in ''A''. :Proof. :1) Construction of the Hilbert space ''H'' :Define on ''A'' a semi-definite
sesquilinear form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
:: \langle a, b \rangle =\rho(b^*a), \; a, b \in A. :By the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...
, the degenerate elements, ''a'' in ''A'' satisfying ρ(''a* a'')= 0, form a vector subspace ''I'' of ''A''. By a C*-algebraic argument, one can show that ''I'' is a
left ideal In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers pre ...
of ''A'' (known as the left kernel of ρ). In fact, it is the largest left ideal in the null space of ρ. The quotient space of ''A'' by the vector subspace ''I'' is an inner product space with the inner product defined by\langle a+I,b+I\rangle :=\rho(b^*a),\; a,b\in A. The
Cauchy completion In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ...
of ''A''/''I'' in the norm induced by this inner product is a Hilbert space, which we denote by ''H''. :2) Construction of the representation π :Define the action π of ''A'' on ''A''/''I'' by π(''a'')(''b''+''I'') = ''ab''+''I'' of ''A'' on ''A''/''I''. The same argument showing ''I'' is a left ideal also implies that π(''a'') is a bounded operator on ''A''/''I'' and therefore can be extended uniquely to the completion. Unravelling the definition of the
adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
of an operator on a Hilbert space, π turns out to be *-preserving. This proves the existence of a *-representation π. :3) Identifying the unit norm cyclic vector ξ :If ''A'' has a multiplicative identity 1, then it is immediate that the equivalence class ξ in the GNS Hilbert space ''H'' containing 1 is a cyclic vector for the above representation. If ''A'' is non-unital, take an
approximate identity In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element. Definition A right approximate ...
for ''A''. Since positive linear functionals are bounded, the equivalence classes of the net converges to some vector ξ in ''H'', which is a cyclic vector for π. :It is clear from the definition of the inner product on the GNS Hilbert space ''H'' that the state ρ can be recovered as a vector state on ''H''. This proves the theorem. 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. :Theorem. Given a state ρ of ''A'', let π, π' be *-representations of ''A'' on Hilbert spaces ''H'', ''H''' respectively each with unit norm cyclic vectors ξ ∈ ''H'', ξ' ∈ ''H''' such that \rho(a) = \langle \pi(a) \xi, \xi \rangle = \langle \pi'(a) \xi ', \xi ' \rangle for all a \in A. Then π, π' are unitarily equivalent *-representations i.e. there is a unitary operator ''U'' from ''H'' to ''H''' such that π'(''a'') = Uπ(''a'')U* for all ''a'' in ''A''. The operator ''U'' that implements the unitary equivalence maps π(''a'')ξ to π'(''a'')ξ' for all ''a'' in ''A''.


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 an ...
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 Φ is the universal representation of a C*-algebra ''A'', the closure of Φ(''A'') in the
weak operator topology In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is ...
is called the
enveloping von Neumann algebra In operator algebras, the enveloping von Neumann algebra of a C*-algebra is a von Neumann algebra that contains all the operator-algebraic information about the given C*-algebra. This may also be called the ''universal'' enveloping von Neumann alg ...
of ''A''. It can be identified with the double dual ''A**''.


Irreducibility

Also of significance is the relation between
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
*-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 . :Theorem. The set of states of a C*-algebra ''A'' with a unit element is a compact
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
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 In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proo ...
. In the unital commutative case, for the C*-algebra ''C''(''X'') of continuous functions on some compact ''X'',
Riesz–Markov–Kakutani representation theorem In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for who introduced it for continuo ...
says that the positive functionals of norm ≤ 1 are precisely the Borel positive measures on ''X'' with total mass ≤ 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 arbitrar ...
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 μ is irreducible if and only if μ is an extremal state. This is in fact true for C*-algebras in general. :Theorem. Let ''A'' be a C*-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 In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or ...
of the convex set of positive linear functionals on ''A'' of norm ≤ 1. 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 of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
of π(''A''), denoted by π(''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 ''Tg'' in π(''A'')' with 0 ≤ ''T'' ≤ 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 π is unitarily equivalent to a subrepresentation of π''g'' ⊕ π''g' ''. This shows that π is irreducible if and only if any such π''g'' is unitarily equivalent to π, i.e. ''g'' is a scalar multiple of ''f'', which proves the theorem. Extremal states are usually called
pure states In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
. 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 In mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C*-algebra ''A'' as a compositio ...
characterizing
completely positive map In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition. Definition Let A and B be C*-algebras. A linear m ...
s 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 (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
only for the specific case of the non-relativistic Schrödinger-Heisenberg theory.


See also

* Cyclic and separating vector * KSGNS construction


References

*
William Arveson William B. Arveson (22 November 1934 – 15 November 2011) was a mathematician specializing in operator algebras who worked as a professor of Mathematics at the University of California, Berkeley. Biography Arveson obtained his Ph.D. from UCLA i ...
, ''An Invitation to C*-Algebra'', Springer-Verlag, 1981 *
Kadison, Richard Richard Vincent Kadison (July 25, 1925 – August 22, 2018)F ...
, ''Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory'', American Mathematical Society. . *
Jacques Dixmier Jacques Dixmier (born 24 May 1924) is a French mathematician. He worked on operator algebras, especially C*-algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace and the Dixmier mapping. Biograph ...
, ''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 Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
'', 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)
* ;Inline references: {{DEFAULTSORT:Gelfand-Naimark-Segal construction Functional analysis C*-algebras Axiomatic quantum field theory ru:Алгебраическая квантовая теория