symmetric monoidal (∞,1)-category of spectra
In an associative algebra $A$, the commutant of a set $B \subset A$ of elements of $A$ is the set
of elements in $A$ that commute with all elements in $B$.
The operation of taking a commutant is a contravariant map $P(A) \to P(A)$ that is adjoint to itself in the sense of Galois connections. In other words, we have for any two subsets $B, C \subseteq A$ the equivalence
Hence $B \subseteq B''$ and also $B' = B'''$.