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*Taken from C*-algebras and Operator Theory by Gerald Murphy.*

A useful way of thinking of the theory of C*-algebras is as “non-commutative topology”. This is justified by the correspondence between abelian C*-algebras and locally compact Hausdorff spaces given by the Gelfand representation. The algebras studied in this chapter, von Neumann algebras, are a class of C*-algebras whose study can be thought of as “non-commutative measure theory”. The reason for the analogy in this case is that the abelian von Neumann algebras are (up to isomorphism) of the form , where is a measure space.

The theory of von Neumann algebras is a vast and very well-developed area of the theory of operator algebras. We shall be able only to cover some of the basics. The main results of this chapter are the von Neumann double commutant theorem and the Kaplansky density theorem.

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# Question 1

*Let be normal elements of a C*-algebra **, and ** an element of ** such that **. Show that **, using Fuglede’s theorem and the fact that the element*

*is normal in ** and commutes with*

*.*

*This more general result is called the Putnam-Fuglede theorem.*

## Solution

Fuglede’s theorem states that if is a normal element commuting with some , then also commutes with . Now we can show that using the normality of and . We can also show that and commute. Hence by the theorem and commute. This yields:

.

Taking conjugates:

,

as required

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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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