Bicommutant
Encyclopedia
In algebra
, the bicommutant of a subset
S of a semigroup
(such as an algebra
or a group
) is the commutant of the commutant of that subset. It is also known as the double commutant or second commutant and is written
.
The bicommutant is particularly useful in operator theory
, due to the von Neumann double commutant theorem, which relates the algebraic and analytic structures of operator algebra
s. Specifically, it shows that if M is a unital, self-adjoint operator algebra in the C*-algebra B(H), for some Hilbert space
H, then the weak closure
, strong closure
and bicommutant of M are equal. This tells us that a unital C*-subalgebra M of B(H) is a von Neumann algebra
if, and only if,
, and that if not, the von Neumann algebra it generates is 
.
The bicommutant of S always contains S. So
. On the other hand, 
. So 
, i.e. the commutant of the bicommutant of S is equal to the commutant of S. By induction, we have:
and
for n > 1.
It is clear that, if S1 and S2 are subsets of a semigroup,
If it is assumed that
and 
(this is the case, for instance, for von Neumann algebra
s), then the above equality gives
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, the bicommutant of a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
S of a semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
(such as an algebra
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
or a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
) is the commutant of the commutant of that subset. It is also known as the double commutant or second commutant and is written
.The bicommutant is particularly useful in operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
, due to the von Neumann double commutant theorem, which relates the algebraic and analytic structures of operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
s. Specifically, it shows that if M is a unital, self-adjoint operator algebra in the C*-algebra B(H), for some Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H, then the weak closure
Weak operator topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number is continuous for any vectors x and y in the Hilbert space.Equivalently, a...
, strong closure
Strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number \|Tx\| is continuous for each vector...
and bicommutant of M are equal. This tells us that a unital C*-subalgebra M of B(H) is a von Neumann algebra
Von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
if, and only if,
, and that if not, the von Neumann algebra it generates is .The bicommutant of S always contains S. So
. On the other hand, . So , i.e. the commutant of the bicommutant of S is equal to the commutant of S. By induction, we have:and
for n > 1.
It is clear that, if S1 and S2 are subsets of a semigroup,
If it is assumed that
and (this is the case, for instance, for von Neumann algebraVon Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
s), then the above equality gives