In Kadison's article he says that: "Since the commutant of an irreducible representation consists of scalars, two such are either unitarily equivalent or disjoint." Ref:”https://msp.org/pjm/1960/10-2/pjm-v10-n2-p12-s.pdf”
That the commutant of irreducible representations consist of $\mathbb{C}\mathbb{1}$ I got as a result of considering the subspace generated by the commutant of a irreducible representation $\pi$ (of a Banach $*$-algebra $\mathscr{A}$), then using the irreducible property of $\pi$, the fact that $\pi(\mathscr{A})'$ is a von Neumann algebra and that every von Neumann algebra is generated by its lattice of projectors (the projectors of $\pi(\mathscr{A})'$ were proven to be $0$ and $\mathbb{1}$ only). I am really terrible at representation theory, so for me that phase seems really disconnected, if two representations have as commutant $\mathbb{C}\mathbb{1}$ why are they either unitarily equivalent or disjoint (i.e. have no unitarily equivalent sub-representations)?