Von Neumann bicommutant theorem
Encyclopedia
In mathematics
, specifically functional analysis
, the von Neumann bicommutant theorem relates the closure
of a set of bounded operator
s on a Hilbert space
in certain topologies
to the bicommutant of that set. In essence, it is a connection between the algebra
ic and topological sides of operator theory
.
The formal statement of the theorem is as follows. Let M be an algebra of bounded operators on a Hilbert space H, containing the identity operator and closed under taking adjoints. Then the closures of M in the weak operator topology
and the strong operator topology
are equal, and are in turn equal to the bicommutant M′′ of M. This algebra is the von Neumann algebra
generated by M.
There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If M is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra. One such example is the C*-algebra of compact operator
s (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are still von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak
, ultrastrong, and *-ultrastrong topologies.
It is related to the Jacobson density theorem
.
As stated above, the theorem claims the following are equivalent:
The adjoint map T → T* is continuous in the weak operator topology. So the commutant S’ of any subset S of L(H) is weakly closed. This gives i) ⇒ ii). Since the weak operator topology is weaker than the strong operator topology, it is also immediate that ii) ⇒ iii). What remains to be shown is iii) ⇒ i). It is true in general that S ⊂ S′′ for any set S, and that any commutant S′ is strongly closed. So the problem reduces to showing M′′ lies in the strong closure of M.
For h in H, consider the smallest closed subspace Mh that contains {Mh| M ∈ M}, and the corresponding orthogonal projection P.
Since M is an algebra, one has PTP = TP for all T in M. Self-adjointness of M further implies that P lies in M′. Therefore for any operator X in M′′, one has XP = PX. Since M is unital, h ∈ Mh, hence Xh∈ Mh and for all ε > 0, there exists T in M with ||Xh - Th|| < ε.
Given a finite collection of vectors h1,...hn, consider the direct sum
The algebra N defined by
is self-adjoint, closed in the strong operator topology, and contains the identity operator. Given a X in M′′, the operator
lies in N′′, and the argument above shows that, all ε > 0, there exists T in M with ||Xh1 - Th1||,...,||Xhn - Thn|| < ε. By definition of the strong operator topology, the theorem holds.
in M that the identity operator I lies in the strong closure of M. Therefore the bicommutant theorem still holds.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the von Neumann bicommutant theorem relates the closure
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
of a set of bounded operator
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
s on a 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...
in certain topologies
Operator topology
In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B of bounded linear operators on a Hilbert space H.-Introduction:...
to the bicommutant of that set. In essence, it is a connection between the algebra
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...
ic and topological sides of 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....
.
The formal statement of the theorem is as follows. Let M be an algebra of bounded operators on a Hilbert space H, containing the identity operator and closed under taking adjoints. Then the closures of M in the weak operator topology
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...
and the strong operator topology
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...
are equal, and are in turn equal to the bicommutant M′′ of M. This algebra is the 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...
generated by M.
There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If M is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra. One such example is the C*-algebra of compact operator
Compact operator on Hilbert space
In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite-rank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using...
s (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are still von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak
Ultraweak topology
In functional analysis, a branch of mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set B of bounded operators on a Hilbert space is the weak-* topology obtained from the predual B* of B, the trace class operators on H...
, ultrastrong, and *-ultrastrong topologies.
It is related to the Jacobson density theorem
Jacobson density theorem
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R....
.
Proof
Let H be a Hilbert space and L(H) the bounded operators on H. Consider a self-adjoint subalgebra M of L(H). Suppose also, M contains the identity operator on H.As stated above, the theorem claims the following are equivalent:
- i) M = M′′.
- ii) M is closed in the weak operator topologyWeak operator topologyIn 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...
. - iii) M is closed in the strong operator topologyStrong operator topologyIn 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...
.
The adjoint map T → T* is continuous in the weak operator topology. So the commutant S’ of any subset S of L(H) is weakly closed. This gives i) ⇒ ii). Since the weak operator topology is weaker than the strong operator topology, it is also immediate that ii) ⇒ iii). What remains to be shown is iii) ⇒ i). It is true in general that S ⊂ S′′ for any set S, and that any commutant S′ is strongly closed. So the problem reduces to showing M′′ lies in the strong closure of M.
For h in H, consider the smallest closed subspace Mh that contains {Mh| M ∈ M}, and the corresponding orthogonal projection P.
Since M is an algebra, one has PTP = TP for all T in M. Self-adjointness of M further implies that P lies in M′. Therefore for any operator X in M′′, one has XP = PX. Since M is unital, h ∈ Mh, hence Xh∈ Mh and for all ε > 0, there exists T in M with ||Xh - Th|| < ε.
Given a finite collection of vectors h1,...hn, consider the direct sum
The algebra N defined by
is self-adjoint, closed in the strong operator topology, and contains the identity operator. Given a X in M′′, the operator
lies in N′′, and the argument above shows that, all ε > 0, there exists T in M with ||Xh1 - Th1||,...,||Xhn - Thn|| < ε. By definition of the strong operator topology, the theorem holds.
Non-unital case
The algebra M is said to be non-degenerate if for all h in H, Mh = {0} implies h = 0. If M is non-degenerate and a sub C*-algebra of L(H), it can be shown using an approximate identityApproximate identity
In functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring that acts as a substitute for an identity element....
in M that the identity operator I lies in the strong closure of M. Therefore the bicommutant theorem still holds.