I'm trying to get a better understanding of the double commutant theorem in the context of group representations.
Googling for "double commutant theorem" I mostly get results about $*$-algebras and von Neumann's bicommutant theorem. For example the Wikipedia page, and these pdf notes.
However, I've also seen in other references the double commutant theorem discussed as a way to get a canonical decomposition of a vector space in terms of irreps of a finite group. In particular, in this blog post, as well as in section II of arXiv:2212.14340.
I'm not very well-versed on $*$-algebra things. Are these two approaches one and the same? Does the discussion in terms of $*$-algebras reduces to the one in terms of irreps of some group, or are they different results? Also, what are some standard references discussing the double commutant theorem from the group theory perspective?