The Gelfand–Naimark representation π is the Hilbert space analogue of the
direct sum of representations π
f of
A where
f ranges over the set of
pure states of A and π
f 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 : \pi(x) [\bigoplus_{f} H_f] = \bigoplus_{f} \pi_f(x)H_f. 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)
f with
cyclic vector ξ. Since : \begin{align} \|\pi_f(x) \xi\|^2 & = \langle \pi_f(x) \xi \mid \pi_f(x) \xi \rangle = \langle \xi \mid \pi_f(x^*) \pi_f(x) \xi \rangle \\[6pt] & = \langle \xi \mid \pi_f(x^* x) \xi \rangle= f(x^* x) > 0, \end{align} 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 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 : \|x\|_{\operatorname{C}^*} = \sup_f \sqrt{f(x^* x)} 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 : \| \cdot \|_{\operatorname{C}^*}
factors 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 A having an approximate identity: : \sup_{f \in \operatorname{State}(A)} f(x^*x) = \sup_{f \in \operatorname{PureState}(A)} f(x^*x). 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 A is an isometric *-isomorphism from A 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==