Centralizer of a subalgebra ($C_A(B)$)

For a subalgebra $B$ of an associative algebra $A$, its centralizer isIt is a subalgebra of $A$. When $A$ is a group algebra and $B$ is the group algebra of a subgroup $H$, its elements are exactly the linear combinations with coefficients constant on the conjugation orbits of $H$.

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 unital star-subalgebra $A$ of the bounded operators on a complex Hilbert space, its strong operator topology and weak operator topology closures equal its double commutant of an operator algebra. To approximate $T\in A^{\prime \prime }$ on finitely many vectors, take their tuple in $H^m$. The projection onto its diagonal-$A$ cyclic subspace has matrix entries in $A^{\prime }$, so it commutes with the diagonal action of $T$. Therefore the image tuple under $T$ lies in that cyclic subspace and can be approximated by a single $a\in A$.

For a unital star-subalgebra $A\subseteq B(H)$, its unit ball is dense in the unit ball of $A^{\prime \prime }$ in the strong operator topology. The same holds for self-adjoint unit balls and for positive unit balls, and the full unit balls may be approximated strongly together with their adjoints. Passing first to the norm closure allows continuous functional calculus; resolvent identities prove strong continuity of continuous functions vanishing at infinity, even for unbounded approximating nets of self-adjoint operators. A cutoff function supplies the norm bound, and a two-by-two self-adjoint matrix removes the self-adjoint restriction.

A unital subalgebra $A\subseteq \mathrm{End}(V)$ closed under the adjoint operator, on a finite-dimensional real or complex inner product space, equals its double commutant of an operator algebra. Here is a proof that does not require a classification of algebras. For $N=\dim V$, choose a basis $(v_1,\dots ,v_N)$ and let $W=\{(a{v}_1,\dots ,a{v}_N):a\in A\}\subseteq V^N$. The diagonal action of $A$ preserves $W$ and $W^{\perp }$, so the orthogonal projection onto $W$ commutes with that action. Each matrix entry of this projection belongs to $A^{\prime }$. Hence the diagonal action of $b\in A^{\prime \prime }$ commutes with the projection and preserves $W$. Since $1\in A$, the tuple $(v_1,\dots ,v_N)$ belongs to $W$, and therefore $(b{v}_1,\dots ,b{v}_N)=(a{v}_1,\dots ,a{v}_N)$ for one $a\in A$. This implies $b=a$. Without the unital hypothesis the conclusion is $A^{\prime \prime }=A+\mathbb{F}1$.

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.