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.