An important fact in the theory of central simple algebras is the double centralizer theorem, which says: if $k$ is a field, $A$ is a $k$-algebra, $V$ is a faithful semisimple $A$-algebra, then $C(C(A)) = A$, where $C(A)$ is the centralizer of $A$ in $End_k(V)$.
Taking the centralizer is an inclusion-reversing operation. If we consider the algebra $M_n(k)$ of $n\times n$ matrices over a field $k$, identfying $k$ with the scalar matrices and $\Delta$ with the diagonal matrices, we get $C(k)$ = $M_n(k)$, $C(M_n(k)) = k$, and $C(\Delta) = \Delta$.
(1) This suggests that the centralizer operation is something like a duality between $k$-subalgebras. Is there any truth to this?
(2) Will there always be a subalgebra like $\Delta$ which is its own centralizer?
I'm not exactly sure what conditions to put on $k, A$, and $V$, so I'm open to flexibility on that.
(3) Does a double centralizer result hold in the context of groups? (i.e. $C_G(C_G(H)) = H$?). If not in general, for what groups does this hold?