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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.

For abelian C*-algebras we were able to completely determine the structure of the algebra in terms of the character space, that is, in terms of the one-dimensional representations. For the non-abelian case this is quite inadequate, and we have to look at representations of arbitrary dimension. There is a deep inter-relationship between the representations and the positive linear functionals of a C*-algebra. Representations will be defined and some aspects of this inter-relationship investigated in the next section. In this section we establish the basis properties of positive linear functionals.

** I’m getting the impression that the bra-ket notation is more useful for linear ON THE LEFT!**

An *approximate unit *for a C*-algebra is an increasing net of positive elements in the closed unit ball of such that for all .

## Example

Let be a Hilbert space with infinite orthonormal basis . The C*-algebra is now non-unital. If is the projection onto , then the increasing sequence is an approximate unit for . It will suffice to show that if , since is dense in . Now if , there exist , such that:

.

Hence,

.

Since for all , therefore for each $k$:

.

Hence, .

# Question 1

*Let be a Banach algebra such that for all the implication*

* or *

*holds. Let , be linear mappings from to itself such that for all ,*

*, , and .*

*Show that and are necessarily continuous.*

# Question 2

*Let be a unital C*-algebra.*

## (a)

*If are positive elements of , show that .*

### Solution (Wills)

For elements of a unital algebra :

If then so that

Now if , for any , . Hence and the result follows (note that need not be hermitian)

Let be a compact Hausdorff space and a Hilbert Space. A *spectral measure *relative to is a map from the -algebra of all Borel sets of to the set of projections in such that

- , ;
- for all Borel sets of ;
- for all , the function , is a regular Borel complex measure on .

A Borel measure is a measure defined on Borel sets. If every Borel set in is both outer and inner regular, then is called regular. A measurable is inner and outer regular if

, and

Denote by the Banach space of all regular Borel complex measures on , and by the C*-algebra of all bounded Borel-measurable complex-valued functions on (I assume with respect to the Borel -algebra on ).

**This starts on P.58 rather than p.53. I underline my own extra explanations, calculations, etc. The notation varies on occasion from Murphy to notation which I prefer.**

is always a Hilbert space. is the set of unit vectors.

If is a finite-rank projection on , then the C*-algebra is finite dimensional. To see this, write , where . If , then

Hence, , () (*), and therefore is finite dimensional.

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