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

I have finished off my revision of sections 1.2 (The Spectrum and the Spectral Radius) & 1.3 (The Gelfand Representation). Section 1.4 is a new topic for me – Compact and Fredholm Operators. A linear map $T:X\rightarrow Y$ between Banach spaces is compact if $T(B_1^X[0])$ is totally bounded. As a corollary, all linear maps on finite dimensional spaces are compact. The transpose $T^*:Y^*\rightarrow X^*$ has been introduced by Murphy is this chapter, and I have seen that if $T$ is compact, then so is $T^*$. A linear map $T$ is Fredholm if the $T(X)$ and $\text{ker }T$ are finite dimensional. In terms of progress, I am on p.25 of 265.