In this section we introduce the important GNS construction and prove that every C*-algebra can be regarded as a C*-subalgebra of for some Hilbert space . It is partly due to this concrete realisation of the C*-algebras that their theory is so accessible in comparison with more general Banach spaces.

A *representation *of a C*-algebra is a pair where is a Hilbert space and is a *-homomorphism. We say is *faithful *if is injective.

If is a family of representations of , their *direct sum *is the representation got by setting setting , and for all and . It is readily verified that is indeed a representation of . If for each non-zero element there is an index such that , then is faithful.

Recall now that if is a pre-Hilbert space (not closed necessarily), then there is a unique inner product of the Banach space completion of extending the inner product of and having its associated norm the norm of . We call endowed with this inner product the *Hilbert space completion of .*

With each positive linear functional, there is associated a representation. Suppose that is a positive linear functional on a C*-algebra . Setting

,

it is easy to check (using Th.3.3.7 — closure is got by seeing that is the kernel of the continuous map ) that is a closed left ideal of and that the map:

; ,

is a well defined inner product on (all straightforward calculations). We denote by the Hilbert space completion of .

If , define an operator by setting:

.

The inequality holds since we have

.

The latter inequality is by Th. 3.3.7. The operator has a unique extension to a bounded operator on . The map

; ,

is a *-homomorphism (an easy exercise — needed to show, where is the inner product on :

.

).

The representation of is the *Gelfand-Naimark-Segal representation* (or *GNS representation*) associated to .

If is non-zero, we define its *universal representation *to be the direct sum of all the representations , where ranges over .

# Theorem 3.4.1 (Gelfand-Naimark)

*If is a C*-algebra, then it has a faithful representation. Specifically, its universal representation is faithful.*

## Proof

Let be the universal representation of and suppose that is an element of such that . By Th. 3.3.6 (if is a normal element of there exists a state such that ) there is a state on such that . Hence, if , then (since , so The first thing to note is that is positive. I can show that for a homomorphism in an algebra, is nilpotent if and only if is nilpotent.) Hence, , and is injective

The Gelfand-Naimark theorem is one of those results that are used all of the time. For the present we give just two applications.

The first application is to matrix algebras. If is an algebra, denotes the algebra of all matrices with entries in (with the usual operations). If is a *-algebra, so is , where the involution is given by the conjugate transpose.

If is a *-homomorphism between *-algebras, its *inflation *is the *-homomorphism

, .

If is a Hilbert space, we write for the orthogonal sum of copies of . If , we define by setting

,

for all . It is readily verified that the map

, ,

is a *-isomomorphism. We call the *canonical *-isomorphism of onto , *and useit to identify these two algebras. If is an operator in such that where , we call the *operator matrix of . *We define a norm on making it a C*-algebra by setting . The following inequalities for are easy to verify and are often useful:

.

# Theorem 3.4.2

*If is a C*-algebra, then there is a unique norm on making it a C*-algebra.*

## Proof

Let the pair be the universal representation of , so the *-homomorphism is injective. We define a norm on making it a C*-algebra by setting for all (completeness can be easily checked using the inequalities preceding this theorem). Uniqueness is given by Corollary 2.1.2 (the C*-algebra norm is unique)

### Remark

If is a C*-algebra and , then

.

These inequalities follow from the corresponding inequalities in .

Matrix algebras play a fundamental role in the K-theory of C*-algebras. The idea is to study not just the algebra but simultaneously all of the matrix algebras over also.

Whereas it seems that the only way of showing that matrix algebras over general C*-algebras are themselves normable as C*-algebras is to use the Gelfand-Naimark representation, for our second application of this representation alternative proofs exist, but the proof given here is very natural.

# Theorem 3.4.3

*Let be a self-adjoint element of the C*-algebra . Then iff for all positive linear functionals on .*

## Proof

The forward implication is straightforward. Suppose conversely that for all positive linear functionals on . Let be the universal representation of , and let . Then the linear functional

, ,

is positive, so l that is, . Since this is true for all , and since is self-adjoint, therefore is a positive operator on . Hence , so , because the map is a *-isomorphism

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