Question
I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed book). Maybe the problem is, that I do not understand, how those algebras are constructed. Could somebody explain the second way of the construction (page 91 (85))? I am talking about the part starting:
2) The second is to consider the larger algebra $B \supset C(Y)$ of all $2 \times 2$ matrices: $$ f = \begin{pmatrix} f_{aa} & f_{ab} \\ f_{ba} & f_{bb} \\ \end{pmatrix}. $$ ...
I do not really understand, how to construct this matrix in general. For example - what is $f_{aa}, f_{ab},\ldots$? How are those constructed/represented e.g. in the example $2.\beta$ The dual of the infinite dihedral group (pages 92-93 (87-88))?
Many thanks.
[Update 2]
I have already asked this question here (about two days ago, without any answer yet), but maybe this site is more suitable for this type of question (although maybe only the same people will see it)? I am not sure, so I might delete one of them later (I really don't want to spam every question I don't get answered immediately.
[Update]
I would be glad to receive any tips or guesses. I am aware, that this is broad topic, I am not looking for precise and rigorous explanation.