I have continued my work on Belton http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf.  I finished off exercises 6.7-6.8.

I looked at the section on Characters and Maximal Ideals. Some really nice results in this area. For example, every proper ideal of a commutative, unital complex Banach algebra $A$ contains no invertible elements and is contained in a maximal ideal. I saw that there is a bijection between the set of characters of $A$ and the set of all maximal ideals.

I saw the links between the characters of $A$ and the spectrum of elements of $A$. The Jacobson radical was introduced; and the Gelfand topology was presented. I have done the first three exercises 7.1-3 out of 10.

When this is finished I must present a summary of the different initial topologies and review the various definitions, etc.