With a bout of illness last week I only got to finish off Beltonhttp://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf and start Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

At present I am speedily going through Chapter 1: Elementary Spectral Theory. This has all been done in Belton but I like Murphy’s lucid and no-frills approach. In places Murphy takes a different approach to Belton (e.g. the proof that the spectrum is non-empty establishes the differentiability of the map $\mathbb{C}\backslash \sigma(a)\rightarrow A$, $\lambda\mapsto (a-\lambda)^{-1}$ without recourse to the resolvent). This quick revision will continue until 1.4 Compact and Fredholm Operators – which is a new topic for me. In terms of progress, this starts on p.18; I’m presently on p.9. The entire book weighs in at 245 pages and realistically I certainly wouldn’t expect to be finished before Christmas.