Editing Gelfand–Naimark theorem (section)

==Details==
The Gelfand–Naimark representation π is the Hilbert space analogue of the [[direct sum of representations]] π<sub>''f''</sub> of ''A'' where ''f'' ranges over the set of [[State (functional analysis)#Pure states|pure states]] of A and π<sub>''f''</sub> 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 ''H''<sub>''f''</sub> by

:<math> \pi(x) [\bigoplus_{f} H_f] = \bigoplus_{f} \pi_f(x)H_f.</math>

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 functional]]s, there is a state ''f'' on ''A'' such that ''f''(''z'') ≥ 0 for all non-negative z in ''A'' and ''f''(&minus;''x''* ''x'') < 0.  Consider the GNS representation π<sub>''f''</sub> with [[cyclic vector]] ξ. Since

:<math>
\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}
</math>

it follows that π<sub>''f''</sub> (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
:<math> \|x\|_{\operatorname{C}^*} = \sup_f \sqrt{f(x^* x)} </math>
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 space (linear algebra)|quotient vector space]] ''A'' / '''I''' is  an involutive algebra and the norm

:<math> \| \cdot \|_{\operatorname{C}^*} </math>

[[factorization|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 [[Banach *-algebra]] ''A'' having an approximate identity:
:<math> \sup_{f \in \operatorname{State}(A)} f(x^*x) = \sup_{f \in \operatorname{PureState}(A)} f(x^*x). </math>
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*-algebra]]s of isometries.

'''Remark'''. The [[Gelfand representation]] or  [[Gelfand isomorphism]] for a commutative C*-algebra with unit <math>A</math> is an isometric *-isomorphism from <math>A</math> 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.