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 Template:Math 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 Template:Math in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant Template:Math of Template:Math.

This algebra is called the von Neumann algebra generated by Template:Math.

There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If Template:Math 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.

ProofEdit

Let Template:Mvar be a Hilbert space and Template:Math the bounded operators on Template:Mvar. Consider a self-adjoint unital subalgebra Template:Math of Template:Math (this means that Template:Math 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) Template:Math

(ii) Template:Math

(iii) Template:Math

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

Proof of (i)Edit

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

<math>T \to \langle (OT - TO)x, y\rangle = \langle Tx, O^*y\rangle - \langle TOx, y\rangle </math>

Let S be any subset of Template:Math, 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 Template:Math; since it contains Template:Math, it also contains its weak operator closure.

Proof of (ii)Edit

This follows directly from the weak operator topology being coarser than the strong operator topology: for every point Template:Mvar in Template:Math, 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 Template:Math; therefore Template:Mvar is also a member of Template:Math.

Proof of (iii)Edit

Fix Template:Math. We must show that Template:Math, i.e. for each h ∈ H and any Template:Math, there exists T in Template:Math with Template:Math.

Fix h in Template:Mvar. The cyclic subspace Template:Math} is invariant under the action of any T in Template:Math. Its closure Template:Math in the norm of H is a closed linear subspace, with corresponding orthogonal projection Template:Mvar : H → Template:Math in L(H). In fact, this P is in Template:Math, as we now show.

Lemma. Template:Math.

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

Since Template:Math is closed under the adjoint operation and P is self-adjoint, for any Template:Math we have

<math>\langle x,TPy\rangle = \langle x,PTPy\rangle = \langle (PTP)^*x,y\rangle = \langle PT^*Px,y\rangle = \langle T^*Px,y\rangle = \langle Px,Ty\rangle = \langle x,PTy\rangle</math>

So TP = PT for all Template:Math, meaning P lies in Template:Math.

By definition of the bicommutant, we must have XP = PX. Since Template:Math is unital, Template:Math, and so Template:Math. Hence Template:Math. So for each Template:Math, there exists T in Template:Math with Template:Math, i.e. Template:Mvar is in the strong operator closure of Template:Math.

Non-unital caseEdit

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

ReferencesEdit

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.