I have continued to work through Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

Before the Christmas break I finished off the chapter 1 exercises.

Chapter 2: C*-Algebras and Hilbert Space Operators.

2.1 C*-Algebras

Initially we defined a C*-algebra, $A$, as a complete normed algebra, together with a conjugate-linear involution * that satisfies the C*-equation:

$\|a^*a\|=\|a\|^2$, $\forall\,a\in A$

Self-adjoint or Hermitian elements are defined to have the property $a^*=a$. As a consequence of this, and the C*-equation, the spectral radius of a self-adjoint element, $\nu(a)$, is equal to its norm, $\|a\|$. As a corollary of this, of all the norms that can be put on the *-algebra, only one makes it into a C*-algebra – i.e. satisfying the C*-equation.

In the previous chapter we have seen that an algebra, $A$, can be unitised to form a new algebra, $\tilde{A}$, which contains $A$ as a subspace. In general, the norm got by extending the norm on $A$ to a norm on $\tilde{A}$ does not make $\tilde{A}$ into a C*-algebra. However Theorem 2.1.6 shows that there does exist a (unique) norm on $\tilde{A}$ making it a C*-algebra. In many examples we may now assume that a general C*-algebra is unital – replacing it with the unique unitisation, $\tilde{A}$, if necessary.

One such result which depends on this fact is that the the spectrum of a self-adjoint element is real.

A central result in this chapter is that all abelian C*-algebras are $C_0(X)$, for some locally compact Hausdorff space, $X$. In fact $X$ is the character space $\Phi(A)$ (as with Belton, this is via the Gelfand transformation). This identification allows the development of the powerful functional calculus. Briefly, if $a$ is a normal element of a C*-algebra $A$, ($a^*a=aa^*$), and $z$ is the inclusion map from $\sigma(a)\rightarrow \mathbb{C}$, then there exists a unique *-homomorphism $\varphi:C(\sigma(a))\rightarrow A$ such that $\varphi(z)=a$. This unique *-homorphism is called the functional calculus at $a$. This particular section ended with the Belton result that if $X$ is a compact Hausdorff space, $\Phi(C(X))\cong X$ (via $x\mapsto \delta_x$).

2.2 Positive Elements of C*-Algebras

This section introduces a partial order on $A_{\text{SA}}$ (the set of self-adjoint elements of $A$). Namely, an element $a\in A_{\text{SA}}$ is positive if $\sigma(a)\subset \mathbb{R}^+$. The partial order is defined in the obvious way.

As a consequence of the Gelfand transformation and the functional calculus, we can show that positive elements of a C*-algebra possess unique positive square roots.  Another prominent result is that for an arbitrary element $a\in A$$a^*a$ is positive.

2.3 Operators and Sesquilinear Forms

As a first move, we prove that bounded operators on Hilbert spaces have adjoints. Next projections are examined and partial isometries are examined. This leads onto the polar decomposition theorem. Namely, if $T$ is a continuous linear operator on a Hilbert space $H$, there exists a unique partial isometry $S$ such that $T=S|T|$; where $|T|=(T^*T)^{1/2}$. The rest of the section focusses on the connection between operators and sesquilinear forms.

2.4 Compact Hilbert Space Operators

At first this chapter looks at some of the basic properties of these objects – e.g. if $T$ is compact so are $|T|$ and $T^*$. Thus $K(H)$ is self-adjoint and thus a C*-algebra (it is a closed ideal in $B(H)$). We see that normal compact operators are diagonalisable.

We look at the finite rank operators, $F(H)$ and see that they are dense in $K(H)$. Next the operator $x\otimes y$ is examined:

$(x\otimes y)(z)=\langle z,y\rangle x$

These are rank-one, and the $x\otimes x$ are rank-one projections if $x$ is a unit vector. This leads on to the fact that $F(H)$ is linearly spanned by these rank-one projections.

This is a synopsis of what I covered up until recently (up to p.56). As an experiment I am attempting to do my study of Murphy by way of fully presenting the details on this webpage. I am unsure of whether or not this is too time consuming. Presently I am on page 63 and I will have to cover the rest of the chapter material (10 pages) in one day or similar if I am going to consider this tactic feasible.