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]
After two days of without activity I've asked the same question here. I might delete one of them later, do not want to spam.
[Update 1]
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 (although that would be also appreciated).