Double-centralizer theorem for semisimple operator algebras ($A^{\prime \prime }=A$)

For a finite-dimensional semisimple algebra acting faithfully on $T$, write $T={⨁}_i{S}_i\otimes M_i$. Then the algebra acts by the full matrix algebra on each simple factor, while its commutant acts by the full matrix algebra on each multiplicity factor. Taking the commutant twice recovers the original image. This is the algebraic basis of Schur–Weyl duality.