Gelfand–Naimark theorem
Encyclopedia
In mathematics
, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operator
s on a Hilbert space
. This result was proven by Israel Gelfand
and Mark Naimark
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 algebra of operators.
The Gelfand–Naimark representation π is the direct sum of representations πf
of A where f ranges over the set of pure states of A and πf is the irreducible representation associated to f by the GNS construction. Thus the Gelfand–Naimark representation acts on
the Hilbert direct sum of the Hilbert spaces Hf by
Note that π(x) is a bounded linear operator 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 functionals, 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 πf with cyclic vector ξ. Since
it follows that πf ≠ 0. Injectivity of π follows.
The construction of Gelfand–Naimark representation depends only on the GNS construction and therefore it is meaningful for any C*-algebra A having an approximate identity
. In general 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
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 B*-algebra 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 of A with the weak* topology.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operator
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
s on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
. This result was proven by Israel Gelfand
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand was a Soviet mathematician who made major contributions to many branches of mathematics, including group theory, representation theory and functional analysis...
and Mark Naimark
Mark Naimark
Mark Aronovich Naimark was a Soviet mathematician.He was born in Odessa, Russian Empire into a Jewish family and died in Moscow, USSR...
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 algebra of operators.
The Gelfand–Naimark representation π is the direct sum of representations πf
of A where f ranges over the set of pure states of A and πf is the irreducible representation associated to f by the GNS construction. Thus the Gelfand–Naimark representation acts on
the Hilbert direct sum of the Hilbert spaces Hf by
Note that π(x) is a bounded linear operator 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 functionals, 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 πf with cyclic vector ξ. Since
it follows that πf ≠ 0. Injectivity of π follows.
The construction of Gelfand–Naimark representation depends only on the GNS construction and therefore it is meaningful for any C*-algebra A having an approximate identity
Approximate identity
In functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring that acts as a substitute for an identity element....
. In general it will not be a faithful representation
Faithful representation
In mathematics, especially in the area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ.In more abstract language, this means...
. 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
Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....
A / I is an involutive algebra and the norm
factor
Factorization
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...
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 B*-algebra 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
Gelfand representation
In mathematics, the Gelfand representation in functional analysis has two related meanings:* a way of representing commutative Banach algebras as algebras of continuous functions;...
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 of A with the weak* topology.