Von Neumann bicommutant theorem

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In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory.

The formal statement of the theorem is as follows:

Von Neumann bicommutant theorem. Let M be an algebra consisting 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 called 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 operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are 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.

Proof[edit]

Let H be a Hilbert space and L(H) the bounded operators on H. Consider a self-adjoint unital subalgebra M of L(H) (this means that M contains the adjoints of its members, and the identity operator on H).

The theorem is equivalent to the combination of the following three statements:

(i) clW(M) ⊆ M′′
(ii) clS(M) ⊆ clW(M)
(iii) M′′ ⊆ clS(M)

where the W and S subscripts stand for closures in the weak and strong operator topologies, respectively.

Proof of (i)[edit]

By definition of the weak operator topology, for any x and y in H, the map T → <Tx, y> is continuous in this topology. Therefore, for any operator O (and by substituting once yOy and once xOx), so is the map

Let S be any subset of L(H), and S′ its commutant. For any operator T not in S′, <OTx, y> - <TOx, y> is nonzero for some O in S and some x and y in H. By the continuity of the abovementioned mapping, there is an open neighborhood of T in the weak operator topology for which this is nonzero, therefore this open neighborhood is also not in S′. Thus S′ is closed in the weak operator, i.e. S′ is weakly closed. Thus every commutant is weakly closed, and so is M′′; since it contains M, it also contains its weak closure.

Proof of (ii)[edit]

This follows directly from the weak operator topology being coarser than the strong operator topology: for every point x in clS(M), every open neighborhood of x in the weak operator topology is also open in the strong operator topology and therefore contains a member of M; therefore x is also a member of clW(M).

Proof of (iii)[edit]

Fix XM′′. We must show that X ∈ clS(M), i.e. for each hH and any ε > 0, there exists T in M with ||XhTh|| < ε.

Fix h in H. The cyclic subspace Mh = {Mh : MM} is invariant under the action of any T in M. Its closure cl(Mh) in the norm of H is a closed linear subspace, with corresponding orthogonal projection P : Hcl(Mh) in L(H). In fact, this P is in M, as we now show.

Lemma. PM.
Proof. Fix xH. As Px ∈ cl(Mh), it is the limit of a sequence Onh with On in M. For any TM, TOnh is also in Mh, and by the continuity of T, this sequence converges to TPx. So TPx ∈ cl(Mh), and hence PTPx = TPx. Since x was arbitrary, we have PTP = TP for all T in M.
Since M is closed under the adjoint operation and P is self-adjoint, for any x, yH we have
So TP = PT for all TM, meaning P lies in M.

By definition of the bicommutant, we must have XP = PX. Since M is unital, hMh, and so h = Ph. Hence Xh = XPh = PXh ∈ cl(Mh). So for each ε > 0, there exists T in M with ||XhTh|| < ε, i.e. X is in the strong operator closure of M.

Non-unital case[edit]

A C*-algebra M acting on H is said to act non-degenerately if for h in H, Mh = {0} implies h = 0. In this case, it can be shown using an approximate identity in M that the identity operator I lies in the strong closure of M. Therefore, the conclusion of the bicommutant theorem holds for M.

References[edit]

  • W.B. Arveson, An Invitation to C*-algebras, Springer, New York, 1976.