Notes on universal C$^*$-algebras

In these notes, we explain in details the construction of the universal C$^*$-algebra as done by Blackadar, starting with the construction of the free $*$-algebra. At the end, we explain the extension done by Boava and the author to include relations described using the strong operator topology. As an example, we show that a graph C$^*$-algebra can be defined using infinite sums.