Commutant of an operator algebra ($A\prime ={\mathrm{End}}_A(T)$)

The commutant of $A\subseteq \mathrm{End}(T)$ is $A\prime =\{b:ba=ab\text{ for all }a\in A\}$. This centralizer is an operator algebra. The double-centralizer theorem for semisimple operator algebras and Schur–Weyl duality use this meaning, which is distinct from the centralizer of a single element of a group.

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.