In
mathematics, the Gelfand–Naimark theorem states that an arbitrary
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 continuou ...
''A'' is isometrically *-isomorphic to a C*-subalgebra 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 a
Hilbert space. This result was proven by
Israel Gelfand and
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 ...
in 1943 and was a significant point in the development of the theory of C*-algebras since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an
operator algebra.
Details
The Gelfand–Naimark representation π is the
direct sum of representations
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
π
''f''
of ''A'' where ''f'' ranges over the set of
pure state
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 in t ...
s of A and π
''f'' is the
irreducible representation associated to ''f'' by the
GNS construction
GNS may refer to:
Places
* Binaka Airport, in Gunung Sitoli, Nias Island, Indonesia
* Gainesville station (Georgia), an Amtrak station in Georgia, United States
Companies and organizations
* Gesellschaft für Nuklear-Service, a German nuclear-wa ...
. Thus the Gelfand–Naimark representation acts on
the Hilbert direct sum of the Hilbert spaces ''H''
''f'' by
:
π(''x'') is a
bounded linear 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 ...
since it is the direct sum of a family of operators, each one having norm ≤ , , ''x'', , .
Theorem. The Gelfand–Naimark representation of a C*-algebra is an isometric *-representation.
It suffices to show the map π is
injective, since for *-morphisms of C*-algebras injective implies isometric. Let ''x'' be a non-zero element of ''A''. By the
Krein extension theorem for positive
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, there is a state ''f'' on ''A'' such that ''f''(''z'') ≥ 0 for all non-negative z in ''A'' and ''f''(−''x''* ''x'') < 0. Consider the GNS representation π
''f'' with
cyclic vector
An operator ''A'' on an (infinite dimensional) Banach space or Hilbert space H has a cyclic vector ''f'' if the vectors ''f'', ''Af'', ''A2f'',... span H. Equivalently, ''f'' is a cyclic vector for ''A'' in case the set of all vectors of the form ' ...
ξ. Since
:
it follows that π
''f'' (x) ≠ 0, so π (x) ≠ 0, so π is injective.
The construction of Gelfand–Naimark ''representation'' depends only on the GNS construction and therefore it is meaningful for any
Banach *-algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
''A'' having an
approximate identity. In general (when ''A'' is not a C*-algebra) it will not be a
faithful representation. The closure of the image of π(''A'') will be a C*-algebra of operators called the
C*-enveloping algebra of ''A''. Equivalently, we can define the
C*-enveloping algebra as follows: Define a real valued function on ''A'' by
:
as ''f'' ranges over pure states of ''A''. This is a semi-norm, which we refer to as the ''C* semi-norm'' of ''A''. The set I of elements of ''A'' whose semi-norm is 0 forms a two sided-ideal in ''A'' closed under involution. Thus the
quotient vector space ''A'' / I is an involutive algebra and the norm
:
factor
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, su ...
s through a norm on ''A'' / I, which except for completeness, is a C* norm on ''A'' / I (these are sometimes called pre-C*-norms). Taking the completion of ''A'' / I relative to this pre-C*-norm produces a C*-algebra ''B''.
By the
Krein–Milman theorem one can show without too much difficulty that for ''x'' an element of the
Banach *-algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
''A'' having an approximate identity:
:
It follows that an equivalent form for the C* norm on ''A'' is to take the above supremum over all states.
The universal construction is also used to define
universal C*-algebras of isometries.
Remark. The
Gelfand representation or
Gelfand isomorphism for a commutative C*-algebra with unit
is an isometric *-isomorphism from
to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals, which in the commutative case are precisely the pure states, of ''A'' with the weak* topology.
See also
*
GNS construction
GNS may refer to:
Places
* Binaka Airport, in Gunung Sitoli, Nias Island, Indonesia
* Gainesville station (Georgia), an Amtrak station in Georgia, United States
Companies and organizations
* Gesellschaft für Nuklear-Service, a German nuclear-wa ...
*
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 compositi ...
*
Gelfand–Raikov theorem
*
Tannaka–Krein duality
References
* (als
available from Google Books
* , also available in English from North Holland press, see in particular sections 2.6 and 2.7.
{{DEFAULTSORT:Gelfand-Naimark theorem
Operator theory
Theorems in functional analysis
C*-algebras