symmetric monoidal (∞,1)-category of spectra
The bicommutant theorem (as known as the double commutant theorem, or von Neumann’s double commutant theorem) is the following result:
Let be a sub--algebra of the algebra of bounded linear operators on a Hilbert space . Then is a von Neumann algebra (and therefore also a -algebra) in if and only if , where denotes the commutant of .
Notice that the condition of being a von Neumann algebra (being closed in the weak operator topology; “weak” here can be replaced by “strong”, “ultrastrong”, or “ultraweak” as described in operator topology), which is a topological condition, is by this result equivalent to an algebraic condition (being equal to its bicommutant).