The bicommutant 
of a subset 
, where 
is a complex Hilbert
space and 
is the vector space of continuous linear operators on it, is the commutant
of the commutant 
of 
, i.e., 
. In particular, the bicommutant is the collection
of all elements in 
that commute with all elements
of 
:
 |
Across the literature on the subject, the set 
is sometimes denoted 
, a reference to the fact that a linear
operator between normed spaces is continuous iff it is bounded (Royden
and Fitzpatrick 2010, p. 257). Likewise, the bicommutant is sometimes called
the double commutant.
The notions of commutant and bicommutant are fundamental to the study of von Neumann algebras (Dixmier 1981).
