In a passage in a proof, there is the following sequence of implications which I didn't quite understand: "The projector P in a representation $\pi$ of a Banach $*$-algebra $\mathscr{A}$, is its own central carrier" => "in particular P lies in the center of the commutant" => "and therefore in $\overline{\pi(\mathscr{A})}^S$" (where $\overline{\,\,\,\,}^S $ is the strong operator topology closure).
I am certain that that last step is related with the fact that $\pi(\mathscr{A})'$ is a von Neumann algebra (a result that I didn't know about and that was the subject of a previous question).
I am really bad at understanding representation theory, the majority of its results are enunciated as phrases instead of formal symbolic expressions, and since I don't usually work with representation theory I have to go back to the definition of each term write it out and try to see if I can manipulate those symbolic expressions using those hypothesis to get the result, but I am not accustomed with the usual way specialists in representation theory think about the subject.
Here is the proof in question, it comes from Kadison and Glimm's 1960 paper "UNITARY OPERATORS IN C*-ALGEBRAS": https://msp.org/pjm/1960/10-2/pjm-v10-n2-p12-p.pdf
The references are just references to definitions of partial isometry and properties of the lowest upper bound of a sequence of projectors that are orthogonal to each other which allows one to go from the central carrier of $P_\alpha$ to each $P_alpha$.
Note: the $B^-$ notation means $\overline{B}^s$ as the very autors point out to in the beginning of the paper.