Von Neumann bicommutant theorem

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 Template:Mvar, 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

Let Template:Mvar be a Hilbert space and L(H) the bounded operators on Template:Mvar. 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 Template:Mvar).

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

(i) cl_W(M) ⊆ M′′

(ii) cl_S(M) ⊆ cl_W(M)

(iii) M′′ ⊆ cl_S(M)

where the Template:Mvar and Template:Mvar subscripts stand for closures in the weak and strong operator topologies, respectively.

Proof of (i)

For any Template:Mvar and Template:Mvar in Template:Mvar, the map T → <Tx, y> is continuous in the weak operator topology, by its definition. Therefore, for any fixed operator Template:Mvar, so is the map

$T\to ⟨(OT-TO)x,y⟩=⟨Tx,O^{*}y⟩-⟨TOx,y⟩$

Let S be any subset of L(H), and S′ its commutant. For any operator Template:Mvar in S′, this function is zero for all O in S. For any Template:Mvar not in S′, it must be nonzero for some O in S and some x and y in Template:Mvar. By its continuity there is an open neighborhood of Template:Mvar for the weak operator topology on which it is nonzero, and which therefore is also not in S′. Hence any commutant S′ is closed in the weak operator topology. In particular, so is M′′; since it contains M, it also contains its weak operator closure.

Proof of (ii)

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

Proof of (iii)

Fix X ∈ M′′. We must show that X ∈ cl_S(M), i.e. for each h ∈ H and any ε > 0, there exists T in M with ||Xh − Th|| < ε.

Fix h in Template:Mvar. The cyclic subspace Mh = {Mh : M ∈ M} 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 Template:Mvar : H → cl(Mh) in L(H). In fact, this P is in M′, as we now show.

Lemma. P ∈ M′.

Proof. Fix x ∈ H. As Px ∈ cl(Mh), it is the limit of a sequence Template:Mvar with Template:Mvar in M. For any T ∈ M, Template:Mvar is also in Mh, and by the continuity of Template:Mvar, this sequence converges to Template:Mvar. So TPx ∈ cl(Mh), and hence PTPx = TPx. Since x was arbitrary, we have PTP = TP for all Template:Mvar in M.

Since M is closed under the adjoint operation and P is self-adjoint, for any x, y ∈ H we have

$⟨x,TPy⟩=⟨x,PTPy⟩=⟨(PTP{)}^{*}x,y⟩=⟨P{T}^{*}Px,y⟩=⟨T^{*}Px,y⟩=⟨Px,Ty⟩=⟨x,PTy⟩$

So TP = PT for all T ∈ M, meaning P lies in M′.

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

Non-unital case

A C*-algebra M acting on H is said to act non-degenerately if for h in Template:Mvar, 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

• W.B. Arveson, An Invitation to C*-algebras, Springer, New York, 1976.
• M. Takesaki, Theory of Operator Algebras I, Springer, 2001, 2nd printing of the first edition 1979.

Further reading

• Jacob Lurie's lecture notes on a von Neumann algebra at https://www.math.ias.edu/~lurie/261y.html