# A double commutant theorem for Murray–von Neumann algebras

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Edited by I. M. Singer, MIT, Boxborough, MA, and approved March 6, 2012 (received for review February 14, 2012)

## Abstract

Murray–von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra of the Murray–von Neumann algebra associated with a finite von Neumann algebra is the Murray–von Neumann algebra , where is a maximal abelian self-adjoint subalgebra of and, in addition, is . We also prove that the Murray–von Neumann algebra with the center of is the center of the Murray–von Neumann algebra . Von Neumann’s celebrated double commutant theorem characterizes von Neumann algebras as those for which , where , the commutant of , is the set of bounded operators on the Hilbert space that commute with all operators in . At the end of this article, we present a double commutant theorem for Murray–von Neumann algebras.

In ref. 1, J. von Neumann initiates the subject of von Neumann algebras (which he refers to as “rings of operators”) and proves what is the most fundamental theorem of the subject, his celebrated “double commutant theorem.” That theorem is, in effect, an infinite-dimensional version of Schur’s lemma. The complex-Hilbert-space setting in which this theorem of von Neumann’s is proved is the most “classical” venue available for such an infinite-dimensional version. Von Neumann’s purpose in proving this result was twofold. On the one hand, Schur’s lemma is basic to the study of representation of finite groups; von Neumann was preparing the way for an extension of that study to infinite groups. The “rings of operators” introduced by von Neumann were to (and they do) play the role of a (complex) group algebra of the group. There are other possibilities for such a group algebra that are useful for different purposes. As is often the case when a concept from finite-dimensional algebra is extended to infinite dimensions, that concept “ramifies,” splits into several distinct concepts, subtly different from one another, especially when analysis and topology make an important appearance. It was von Neumann’s hope that his group algebra would “coalesce” large families of groups (that is, be the same algebra for each group of the family) and, therefore, provide a simpler target for classification than the groups themselves. A sample instance of this coalescing occurs when considering “locally finite” groups, those for which each finite subset generates a finite subgroup. One important example of a locally finite group is given by the group of permutations of the integers each of which moves at most a finite set of integers. This example describes a group that has an additional important feature: Each of its conjugacy classes, with the exception of that of the group identity, is infinite. We refer to such groups as “i.c.c. groups.” The von Neumann group algebras of i.c.c. groups have centers consisting just of scalar multiples of the identity operator, the so-called “factors.” The locally finite, i.c.c. groups all have the same group algebra (up to isomorphism). This group algebra is a key von Neumann algebra, the “hyperfinite factor of type II_{1}.”

F. J. Murray and J. von Neumann undertook a thorough study of factors (2⇓⇓–5) classifying them into various types. Each of the i.c.c. groups has a von Neumann algebra that is (a factor) of type II_{1}. The free, noncommutative groups on two or more generators are among these i.c.c. groups. To this day, we do not know whether the II_{1}-factor group algebras of the free groups on two and three generators are or are not isomorphic to one another.

On the other hand, at the time von Neumann was proving his double commutant theorem and introducing his “rings of operators” on Hilbert spaces, quantum theorists were seeking a workable mathematical model that would encompass the ad hoc quantum assumptions with which they altered the strictly classical (Hamiltonian–Newtonian) analysis of the atomic and subatomic physical systems they were examining. Planck’s formula for radiation associated with a full, black-body radiator, Einstein’s photo-electric effect, and Bohr’s remarkable derivation of the visible (Balmer) energy spectrum from his quantized version of the Rutherford (planetary) model of the hydrogen atom are among the first and most basic instances of these quantum assumptions. During this period, Heisenberg discovered his fundamental commutation relation (see also ref. 6), *QP* - *PQ* = *iℏI*, where *Q* and *P* play the role of position and “conjugate” momentum of a particle in the physical system and *ℏ* is *h*/2*π*, where *h* is Planck’s experimentally determined constant. Heisenberg’s relation carries with it the declaration that the mathematical model being sought must be “noncommutative” (so, *QP* = *PQ* has been ruled out) and (finite) matrices would not be suitable (as the “trace” functional on the matrices makes clear). Now, von Neumann was aware of these developments and aware as well that Hilbert spaces and the linear operators on them, especially families of self-adjoint operators, with the algebraic structure such families inherit from the usual product and addition of (everywhere-defined) operators, provided an especially hospitable environment for modeling quantum mechanical systems and supplying them with the necessary degree of noncommutativity.

The simplest examples of such families are the self-adjoint elements in subalgebras of , the family of all bounded operators on the Hilbert space , that contain *A*^{∗}, the adjoint of the operator *A*, when they contain *A*. Such subalgebras are said to be self-adjoint (“closed under the adjoint operation”). Among the self-adjoint subalgebras of , the most useful for the purposes of modeling quantum mechanics are those closed in with respect to some of the “natural” topologies on , the “norm topology,” corresponding to the topology on arising from the metric ‖*A* - *B*‖ as the distance between *A* and *B* in and the “strong-operator topology” corresponding to convergence of sequences (nets) on vectors in (that is, *A*_{n} → *A* in this topology when *A*_{n}*x* → *Ax*, for each *x* in ). Those self-adjoint algebras closed in the norm topology are called “*C*^{∗}-algebras,” and those closed in the strong-operator topology are called “von Neumann algebras.” In both cases, we include the condition that *I*, the identity operator on is in the algebra.

In ref. 3, Murray and von Neumann discover that unbounded operators closely associated with (we say “affiliated with” in a technical definition) a certain class of von Neumann algebras, the “finite” von Neumann algebras (again, a technical term), have remarkable domain properties that allow virtually unlimited algebraic manipulation with such operators. We study some aspects of the structure of such families of affiliated operators in this article, notably, commutativity.

In Section 2, we provide a background discussion of unbounded operators, with some of the technical details we shall need, affiliated operators, and some of their related spectral theory, and some of the algebraic properties of families of affiliated operators. Some of the technical results on commutativity, maximal abelian subalgebras, and central elements in the algebra of affiliated operators that serve as the basis for our main results, the commutant theorems in Section 4, appear in Section 3.

## 2 Murray–von Neumann Algebras

We use Section 2.7 and Section 5.6 in refs. 7⇓–9 as our basic reference for results in the theory of unbounded operators as well as for much of our notation and terminology.

### 2.1 Basic Results on Unbounded Operators.

Let *T* be a linear mapping of the Hilbert space into the Hilbert space . We denote by “” the domain of *T*. Note that is a linear submanifold of (not necessarily closed). We associate a graph with *T*, where . We say that *T* is closed when is closed. The closed graph theorem tells us that if *T* is defined on all of , then is closed if and only if *T* is bounded. The unbounded operators *T* we consider will usually be densely defined; that is, is dense in . We say that *T*_{0} extends (or is an extension of) *T* and write *T*⊆*T*_{0} when and *T*_{0}*x* = *Tx* for each *x* in . If , the closure of the graph of *T* is the graph of a linear transformation , clearly is the “smallest” closed extension of *T*, we say that *T* is preclosed (or closable) and refer to as the closure of *T*. From the point of view of calculations with an unbounded operator *T*, it is often much easier to study its restriction to a dense linear manifold in its domain than to study *T* itself. If *T* is closed and , we say that is a core for *T*. Each dense linear manifold in corresponds to a core for *T*.

If *T* is a linear transformation with dense in the Hilbert space and range contained in the Hilbert space , we define a mapping *T*^{∗}, the adjoint of *T*, as follows. Its domain consists of those vectors *y* in such that, for some vector *z* in , for all *x* in . For such *y*, *T*^{∗}*y* is *z*. If *T* = *T*^{∗}, we say that *T* is self-adjoint. (Note that the formal relation , familiar from the case of bounded operators, remains valid in the present context only when and .)

If *T* is densely defined, then *T*^{∗} is closed. If *T*_{0} is an extension of *T*, then *T*^{∗} is an extension of .

If *T* is a densely defined linear transformation from the Hilbert space to the Hilbert space , then

if

*T*is preclosed, ;*T*is preclosed if and only if is dense in ;if

*T*is preclosed, ;if

*T*is closed,*T*^{∗}*T*+*I*is one-to-one with range and positive inverse of bound not exceeding 1.

We say that *T* is symmetric when is dense in and for all *x* and *y* in . Equivalently, *T* is symmetric when *T*⊆*T*^{∗}. (Since *T*^{∗} is closed and , in this case, *T* is preclosed if it is symmetric. If *T* is self-adjoint, *T* is both symmetric and closed.)

If *A*⊆*T* with *A* self-adjoint and *T* symmetric, then *A*⊆*T*⊆*T*^{∗}, so that *T*^{∗}⊆*A*^{∗} = *A*⊆*T*⊆*T*^{∗} and *A* = *T*. It follows that *A* has no proper symmetric extension. That is, a self-adjoint operator is maximal symmetric.

If *T* is a closed symmetric operator on the Hilbert space , the following assertions are equivalent:

T is self-adjoint;

T

^{∗}± iI have (0) as null space;T ± iI have as range;

T ± iI have ranges dense in .

If *T* is a closed linear operator with domain dense in a Hilbert space and with range in , then where *N*(*T*) and *R*(*T*) denote the projections whose ranges are, respectively, the null space of *T* and the closure of the range of *T*.

Suppose that *A* and *B* are linear operators with their domains dense in a Hilbert space and their ranges in . Then *A*^{∗} + *B*^{∗}⊆(*A* + *B*)^{∗} if *A* + *B* is densely defined, and *B*^{∗}*A*^{∗}⊆(*AB*)^{∗} if *AB* is densely defined.

If *A* and *C* are densely defined preclosed operators and *B* is a bounded (everywhere-defined) operator such that *A* = *BC*, then *A*^{∗} = *C*^{∗}*B*^{∗}.

There is an extension of the polar decomposition for bounded operators to the case of a closed densely defined linear operator from one Hilbert space to another.

If *T* is a closed densely defined linear transformation from one Hilbert space to another, there is a partial isometry *V* with initial space the closure of the range of (*T*^{∗}*T*)^{1/2} and final space the closure of the range of *T* such that *T* = *V*(*T*^{∗}*T*)^{1/2} = (*T*^{∗}*T*)^{1/2}*V*. Restricted to the closures of the ranges of *T*^{∗} and *T*, respectively, *T*^{∗}*T* and *TT*^{∗} are unitarily equivalent (and *V* implements this equivalence). If *T* = *WH*, where *H* is a positive operator and *W* is a partial isometry with initial space the closure of the range of *H*, then *H* = (*T*^{∗}*T*)^{1/2} and *W* = *V*.

### 2.2 Affiliated Operators and Some Spectral Theory.

We say that a closed densely defined operator *T* is *affiliated* with a von Neumann algebra and write when *U*^{∗}*TU* = *T* for each unitary operator *U* commuting with . (Note that the equality, *U*^{∗}*TU* = *T*, is to be understood in the strict sense that *U*^{∗}*TU* and *T* have the same domain and formal equality holds for the transforms of vectors in that domain. As far as the domains are concerned, the effect is that *U* transforms onto itself.)

If *T* is a closed densely defined operator with core and *U*^{∗}*TUx* = *Tx* for each *x* in and each unitary operator *U* commuting with a von Neumann algebra , then .

If *A* is a bounded self-adjoint operator acting on a Hilbert space and is an abelian von Neumann algebra containing *A*, there is a family {*E*_{λ}} of projections in (indexed by ), called the spectral resolution of *A*, such that

*E*_{λ}= 0 if*λ*< -‖*A*‖, and*E*_{λ}=*I*if ‖*A*‖ ≤*λ*;*E*_{λ}≤*E*_{λ′}if*λ*≤*λ*^{′};*E*_{λ}= ∧_{λ′>λ}*E*_{λ′};*AE*_{λ}≤*λE*_{λ}and*λ*(*I*-*E*_{λ}) ≤*A*(*I*-*E*_{λ}) for each*λ*;in the sense of norm convergence of approximating Riemann sums; and

*A*is the norm limit of finite linear combinations with coefficients in sp(*A*), the spectrum of*A*, of orthogonal projections*E*_{λ′}-*E*_{λ}.

{*E*_{λ}} is said to be a resolution of the identity if {*E*_{λ}} satisfies (ii), (iii), , and .

With the abelian von Neumann algebra isomorphic to *C*(*X*) and *X* an extremely disconnected compact Hausdorff space, if *f* and *e*_{λ} in *C*(*X*) correspond to *A* and *E*_{λ} in , then *e*_{λ} is the characteristic function of the largest clopen subset *X*_{λ} on which *f* takes values not exceeding *λ*.

The spectral theory described above can be extended to unbounded self-adjoint operators. We associate an unbounded spectral resolution with each of them.

If *A* is a self-adjoint operator acting on a Hilbert space , *A* is affiliated with some abelian von Neumann algebra . There is a resolution of the identity {*E*_{λ}} in such that is a core for *A*, where *F*_{n} = *E*_{n} - *E*_{-n}, and for each *x* in and all *n*, in the sense of norm convergence of approximating Riemann sums.

Since *A* is self-adjoint, from Proposition 6, *A* + *iI* and *A* - *iI* have range and null space (0); in addition, they have inverses, say, *T*_{+} and *T*_{-}, that are everywhere-defined with bound not exceeding 1. Let be an abelian von Neumann algebra containing *I*, *T*_{+} and *T*_{-}. If *U* is a unitary operator in , for each *x* in , *Ux* = *UT*_{+}(*A* + *iI*)*x* = *T*_{+}*U*(*A* + *iI*)*x* so that (*A* + *iI*)*Ux* = *U*(*A* + *iI*)*x*; and *U*^{-1}(*A* + *iI*)*U* = *A* + *iI*. Thus *U*^{-1}*AU* = *A* and . In particular, *A* is affiliated with the abelian von Neumann algebra generated by *I*, *T*_{+} and *T*_{-}. Because is abelian, is isomorphic to *C*(*X*) with *X* an extremely disconnected compact Hausdorff space. Let *g*_{+} and *g*_{-} be the functions in *C*(*X*) corresponding to *T*_{+} and *T*_{-}. Let *f*_{+} and *f*_{-} be the functions defined as the reciprocals of *g*_{+} and *g*_{-}, respectively, at those points where *g*_{+} and *g*_{-} do not vanish. Then, *f*_{+} and *f*_{-} are continuous where they are defined on *X*, as is the function *f* defined by *f* = (*f*_{+} + *f*_{-})/2. In a formal sense, *f* is the function that corresponds to *A*. Let *X*_{λ} be the largest clopen set on which *f* takes values not exceeding *λ*. Let *e*_{λ} be the characteristic function of *X*_{λ} and *E*_{λ} be the projection in corresponding to *e*_{λ}. In this case, {*E*_{λ}} satisfies *E*_{λ} ≤ *E*_{λ′} if *λ* ≤ *λ*^{′}, *E*_{λ} = ∧_{λ′>λ}*E*_{λ′}, ∨_{λ}*E*_{λ} = *I* and ∧_{λ}*E*_{λ} = 0. That is, we have constructed a resolution of the identity {*E*_{λ}}. This resolution is unbounded if *f*∉*C*(*X*). Let *F*_{n} = *E*_{n} - *E*_{-n}, the spectral projection corresponding to the interval [-*n*,*n*] for each positive integer *n*. Then *AF*_{n} is bounded and self-adjoint. Moreover, is a core for *A*. From the spectral theory of bounded self-adjoint operators, , for each *x* in and all *n*. If , Interpreted as an improper integral, we write

The abelian von Neumann algebra generated by *T*_{+} and *T*_{-} in the above discussion is the smallest von Neumann algebra with which the self-adjoint operator *A* is affiliated. We refer to as the von Neumann algebra generated by *A*.

If {*E*_{λ}} is a resolution of the identity on a Hilbert space and is an abelian von Neumann algebra containing {*E*_{λ}}, there is a self-adjoint operator *A* affiliated with such that for each *x* in and all *n*, where *F*_{n} = *E*_{n} - *E*_{-n}; and {*E*_{λ}} is the spectral resolution of *A*.

If *A* is a closed operator on the Hilbert space , {*E*_{λ}} is a resolution of the identity on , is a core for *A*, where *F*_{n} = *E*_{n} - *E*_{-n}, and for each *x* in and all *n*, then *A* is self-adjoint and {*E*_{λ}} is the spectral resolution of *A*.

If *A* is a closed operator acting on the Hilbert space and *CA*⊆*AC* for each *C* in a self-adjoint subset of , then *TA*⊆*AT* for each *T* in the von Neumann algebra generated by .

If *BA*⊆*AB* and , where *A* is a self-adjoint operator and *B* is a closed operator on the Hilbert space , then *E*_{λ}*B*⊆*BE*_{λ} for each *E*_{λ} in the spectral resolution {*E*_{λ}} of *A*.

We say that a closed densely defined operator *A* is *normal* when the two self-adjoint operators *A*^{∗}*A* and *AA*^{∗} are equal.

An operator *A* is normal if and only if it is affiliated with an abelian von Neumann algebra. If *A* is normal, there is a smallest von Neumann algebra with which *A* is affiliated. The algebra is abelian.

### 2.3 The Algebra of Affiliated Operators.

Let be a Hilbert space. Two projections *E* and *F* acting on are said to be orthogonal if *EF* = 0. If the range of *F* is contained in the range of *E* (equivalently, *EF* = *F*), we say that *F* is a subprojection of *E* and write *F* ≤ *E*. Let be a von Neumann algebra acting on . Suppose that *E* and *F* are nonzero projections in . We say that *E* is a minimal projection in if *F* ≤ *E* implies *F* = *E*. Murray and von Neumann conceived the idea of comparing the “sizes” of projections in a von Neumann algebra in the following way: *E* and *F* are said to be equivalent (modulo or relative to ), written *E* ∼ *F*, when *V*^{∗}*V* = *E* and *VV*^{∗} = *F* for some *V* in . (Such an operator *V* is called a partial isometry with initial projection *E* and final projection *F*.) We write *E*≾*F* when *E* ∼ *F*_{0} and *F*_{0} ≤ *F* and *E*≺*F* when *E* is, in addition, not equivalent to *F*. It is apparent that ∼ is an equivalence relation on the projections in . In addition, ≾ is a partial ordering of the equivalence classes of projections in , and it is a nontrivial and crucially important fact that this partial ordering is a total ordering when is a factor. (Factors are von Neumann algebras whose centers consist of scalar multiples of the identity operator.) Murray and von Neumann also define infinite and finite projections in this framework modeled on the set-theoretic approach. The projection *E* in is infinite (relative to ) when *E* ∼ *F* < *E*, and finite otherwise. We say that the von Neumann algebra is finite when the identity operator *I* is finite.

Throughout the rest of this subsection, denotes a finite von Neumann algebra acting on a Hilbert space , and denotes the family of operators affiliated with .

In ref. 10, the following are proved.

Suppose that operators *S* and *T* are affiliated with , then:

*S*+*T*is densely defined, preclosed and its closure, denoted by , is affiliated with ;*ST*is densely defined, preclosed and its closure, denoted by , is affiliated with .

Suppose that operators *A*, *B* and *C* are affiliated with , then that is, the associative law holds for the multiplication .

Suppose that operators *A*, *B* and *C* are affiliated with , then that is, the distributive laws hold for the multiplication relative to the addition .

Suppose that operators *A* and *B* are affiliated with , then where * is the usual adjoint operation on operators (possibly unbounded).

Therefore, , provided with the operations (addition) and (multiplication), is a * algebra (with unit *I*). Recall, is finite (and must be) as a von Neumann algebra for this to be valid.

We use “” to denote the * algebra (). We call *the Murray–von Neumann algebra* associated with .

## 3 Commutativity and Affiliation

If is a finite von Neumann algebra acting on a Hilbert space , *H* and *K* are self-adjoint operators in , and , are the spectral resolutions of *H*, *K*, respectively, then if and only if for each *λ* in , and if and only if *E*_{λ}*F*_{λ′} = *F*_{λ′}*E*_{λ} for all *λ* and *λ*^{′} in .

Suppose, first, that . We show that *H* and *K* are affiliated with some abelian von Neumann subalgebra of . For this, we prove that is a normal operator affiliated with . To see this, we observe that from the properties of as an associative * algebra. (See Section 3.2 in ref. 10.) From Theorem 20, “generates” an abelian von Neumann algebra with which it is affiliated as is . Thus *H* and *K* are affiliated with . The spectral resolutions and lie in . Since is abelian, *E*_{λ}*F*_{λ′} = *F*_{λ′}*E*_{λ}, , and , for all *λ* and *λ*^{′} in .

Suppose, now, that *E*_{λ}*F*_{λ′} = *F*_{λ′}*E*_{λ} for all *λ* and *λ*^{′} in . Then generates an abelian von Neumann algebra with which each of *H* and *K* are affiliated (Theorem 13, Lemma 15, Lemma 16). From Theorem 5.6.12 and Theorem 5.6.15 in refs. 7⇓–9, is abelian. Hence , , and .

Finally, if for all *λ* in , then *E*_{λ}*F*_{λ′} = *F*_{λ′}*E*_{λ} for all *λ* and *λ*^{′} in , from what we have just proved, with *E*_{λ} in place of *H*. Thus, again from what we have just proved, .

The first part of the preceding proof requires us to find a way to move from a family of commuting elements in to a closely associated family of operators in . The natural associated family is the set of projections in the various spectral resolutions. However, our problem is precisely that of showing that the spectral resolutions, for self-adjoint elements of that happen to commute (algebraically), commute with one another. Of course, the process for connecting a self-adjoint operator to its spectral projections is a vital part of what we must use. That process is “analytic” in nature and calls for a certain amount of “backing and filling” if conclusions are to be drawn from it. We have seemingly avoided that process—but we haven’t really done that. What we have done is to take advantage of Theorem 20, which gives us special information about a normal operator, that we have constructed from two commuting, self-adjoint operators *H* and *K* in with the well-functioning algebraic equipment Murray and von Neumann have left us. (See Section 3 in ref. 10.) At the same time, Theorem 20 makes use of the special circumstances a normal operator provides to apply Lemma 18, which moves us from *H* and *K* to their respective resolutions, but all in an abelian framework (supplied by Theorem 20). Still, the “analysis,” effecting the shift from *H* and *K* to their resolutions, is hidden. It has been shifted in Lemma 18 to Remark 14 and from there to Theorem 13 where it appears in full force through the introduction of the operators *T*_{+} and *T*_{-}, bounded, everywhere-defined, and inverse to *H* + *iI* and *H* - *iI*, respectively. Once again, the analysis is partially hidden, because we pass to the von Neumann algebra generated by *T*_{+} and *T*_{-}, with which *H* is affiliated (on simple set-theoretic and mapping grounds). Now, that algebra contains the spectral resolution of *H*, essentially by virtue of von Neumann’s double commutant theorem, a large and very powerful approximation theorem, the guiding theorem of this article.

From this discussion of the pieces of our proof of Proposition 26, we can see a direct route to the proof that if we are willing to enter the proof of Theorem 13 and use *T*_{+} and *T*_{-}, now inverses to *H* + *iI* and *H* - *iI*, respectively. In , we have from our assumption that . Thus , and . Similarly, and *K* commutes with each element of the von Neumann algebra generated by *T*_{+} and *T*_{-} (Lemma 17). In particular, , for each *λ* in . From this same conclusion, with *E*_{λ} in place of *H*, we have that *E*_{λ}*F*_{λ′} = *F*_{λ′}*E*_{λ}, for all *λ* and *λ*^{′} in .

Let be a finite von Neumann algebra acting on a Hilbert space . Suppose that is a self-adjoint, abelian subset of . Then there is an abelian von Neumann subalgebra of with which every element in is affiliated.

Since is a self-adjoint family, it will suffice to show that the “real” and “imaginary” parts of each element in are affiliated with some one abelian algebra. This follows immediately from the preceding proposition because that proposition assures us that all the spectral resolutions of this family of self-adjoint operators commute with one another. Thus the family of projections in all the spectral resolutions is a commuting self-adjoint family in and generates an abelian subalgebra of with which all the operators in are affiliated.

Let be a finite von Neumann algebra acting on a Hilbert space and be a maximal abelian self-adjoint (masa) subalgebra of . Let be . Then is a maximal abelian self-adjoint subalgebra of . In addition, .

Suppose *H* is a self-adjoint operator in with spectral resolution {*E*_{λ}}. If *K* is a self-adjoint operator in , *K* commutes with *H*. From Proposition 26, *K* commutes with {*E*_{λ}}. Since is a masa, for every *λ*. Hence all spectral resolutions of self-adjoint operators in lie in . Suppose *A* is a self-adjoint operator in commuting with . Then, again, from Proposition 26, *A* commutes with each self-adjoint operator *H* in (since *A* commutes with the spectral resolution of *H*). By maximality (and self-adjointness), *A* is in , hence in . Hence is a masa in . [To see that is a strong-operator-closed algebra, let *A* be an operator in that is a strong-operator limit of a net of operators {*A*_{j}} in . Then any *B* in commutes with the operators in the net. Since *A*_{j} → *A* in the strong-operator topology, *A*_{j}*Bx* → *ABx* for every . At the same time, *A*_{j}*Bx* = *BA*_{j}*x* → *BAx*, from the continuity (boundedness) of *B*. Thus *A* commutes with every *B* in and hence *A* commutes with . From maximality of , *A* is in . Hence, is a strong-operator-closed subalgebra of .]

We show, next, that is . Suppose *H* is a self-adjoint operator in . Then the spectral resolution of *H* lies in hence in . Thus every self-adjoint operator in commutes with this spectral resolution, hence with *H* (Proposition 26). By maximality of , *H* is in . Thus .

Suppose *H* is a self-adjoint operator in . Its spectral resolution is in , and hence since *H* is affiliated with the von Neumann algebra generated by its spectral resolution. Thus .

If is a finite von Neumann algebra acting on a Hilbert space with center , then is the center, , of .

If *C* is in , then *C* commutes with every projection in , hence, with every spectral resolution in , and therefore, with every self-adjoint element in . Thus *C* lies in *C* and .

From the properties of the adjoint operation, with *C*, now, in and *A* in , Since is a self-adjoint algebra, *C*^{∗} is in . Thus is a self-adjoint algebra. Hence it suffices to show that each self-adjoint element in is in in order to show . For this, if *A* is a self-adjoint operator in , it commutes with all elements of . From Proposition 26, the spectral resolution of *A* commutes with all elements of . Therefore, the spectral resolution of *A* lies in and *A* is in . Thus .

To show that , note first, that is a self-adjoint family. Hence it suffices to show that each self-adjoint *A* in is in . In this case, the spectral resolution of *A* lies in , hence in . Since the spectral resolution of *A* commutes with every self-adjoint operator in , *A* commutes with every self-adjoint operator in and hence . Thus .

If *A* is a closed, densely defined operator on a Hilbert space and *B* is a bounded, everywhere-defined operator on , we say that *A* and *B* commute when *BA*⊆*AB*.

Toward understanding what we should mean by “the commutant of ,” we prove the proposition that follows.

Suppose *H* and *K* are self-adjoint operators (possibly unbounded) acting on a Hilbert space with spectral resolutions and , respectively. Then the following conditions are equivalent:

*E*_{λ}*K*⊆*KE*_{λ}, for all*λ*in ; that is*E*_{λ}and*K*commute;*E*_{λ}*F*_{λ′}=*F*_{λ′}*E*_{λ}, for all*λ*and*λ*^{′}in ;*H*and*K*are affiliated with the (abelian) von Neumann algebra generated by and ;*F*_{λ}*H*⊆*HF*_{λ}, for all*λ*in .

(i) → (ii) From Lemma 18, with *E*_{λ} in place of *B* and *K* in place of *A* in that lemma. Since *F*_{λ′}*E*_{λ} is defined on all of , *F*_{λ′}*E*_{λ} = *E*_{λ}*F*_{λ′}, for all *λ* and *λ*^{′} in .

(ii) → (iii) Since *E*_{λ}*F*_{λ′} = *F*_{λ′}*E*_{λ} for all *λ* and *λ*^{′} in , generates an abelian von Neumann algebra on with which *H* and *K* are affiliated.

(iii) → (iv) Since is abelian, is abelian from Theorem 5.6.12 and Theorem 5.6.15 in refs. 7⇓–9. As *H* and *F*_{λ} are in , . Now, since *H* is self-adjoint (hence, closed) and *F*_{λ} is bounded.

By symmetry, (i) and (iv) are the same condition, so that we have proved the equivalence of (i), (ii), (iii) and (iv).

Suppose *T* is a self-adjoint operator acting on a Hilbert space and *B* is a self-adjoint, everywhere-defined and bounded operator acting on with spectral resolution . Then the following conditions are equivalent:

*E*_{λ}*T*⊆*TE*_{λ}for all*λ*in ;*BT*⊆*TB*.

(i) → (ii) It follows from Lemma 17 with *A*, *C*, *T* and of that lemma replaced by *T*, *E*_{λ}, *B* and , respectively.

(ii) → (i) From Lemma 18, *F*_{λ}*B*⊆*BF*_{λ} for all *λ* in , where is the spectral resolution of *T*. From the preceding proposition, *E*_{λ}*T*⊆*TE*_{λ}.

Suppose *S* and *T* are self-adjoint operators (possibly unbounded) acting on a Hilbert space . We say that *S* commutes with *T* if *S* commutes with the spectral resolution of *T*.

From Proposition 32, *S* commutes with *T* if and only if *T* commutes with *S* (if and only if their spectral resolutions commute), in which case we also say that *S* and *T* commute.

## 4 Main Theorem

If is a family of self-adjoint operators (possibly unbounded) acting on a Hilbert space , we call the set of all self-adjoint operators that commute with all the operators in the *self-adjoint commutant* of (written, “sa-commutant”).

Let be a von Neumann algebra. Suppose is the family of all self-adjoint operators affiliated with . Then the double sa-commutant of coincides with .

We prove first that , the sa-commutant of , is the set of all self-adjoint operators affiliated with . (So, is the set of self-adjoint elements in when is finite.) To see this, choose a self-adjoint operator *H*^{′} affiliated with . Then the spectral resolution of *H*^{′} is in . If *H* is in , its spectral resolution is in , and hence, commutes with the spectral resolution of *H*^{′}. By denition, *H*^{′} commutes with *H*. Since *H* is an arbitrary element of *S*, *H*^{′} is in .

If *K*^{′} is in , then *K*^{′} commutes, in particular, with each self-adjoint operator in and hence with every operator in . To see this, with *B* in , *B* = *B*_{1} + *iB*_{2} where *B*_{1} and *B*_{2} are self-adjoint operators in . Since *B*_{1}*K*^{′}⊆*K*^{′}*B*_{1}, and *B*_{2}*K*^{′}⊆*K*^{′}*B*_{2}, Since is a von Neumann algebra, from von Neumann’s double commutant theorem, . It follow that *K*^{′} commutes with every operator in the commutant of and hence, by denition, *K*^{′} is affiliated with .

We, now, apply what we have just proved to (the family of all self-adjoint operators affiliated with ). The sa-commutant of is the set of all self-adjoint operators affiliated with . Therefore, the double sa-commutant of coincides with .

With a finite von Neumann algebra acting on a Hilbert space , the *commutant* of is the set of closed, densely defined operators *C*^{′} on that commute with each self-adjoint operator *H* in , that is, *E*_{λ}*C*^{′}⊆*C*^{′}*E*_{λ}, for each *E*_{λ} in the spectral resolution of *H*.

With a finite von Neumann algebra acting on a Hilbert space , if and only if *BC*^{′}⊆*C*^{′}*B* for each *B* in .

Suppose, first, that . Then *EC*^{′}⊆*C*^{′}*E* for each projection *E* in . Since the set of projections in is a self-adjoint family that generates the von Neumann algebra , *BC*^{′}⊆*C*^{′}*B* for each *B* in from Lemma 17.

If *BC*^{′}⊆*C*^{′}*B* for each *B* in , then *EC*^{′}⊆*C*^{′}*E* for each projection *E* in the spectral resolution of a self-adjoint operator. Thus in this case.

If is a finite von Neumann algebra acting on a Hilbert space and , the commutant of , is also finite (as a von Neumann algebra), then , and

Suppose . From Proposition 39, *BC*^{′}⊆*C*^{′}*B*, for each *B* in . In particular, then, *UC*^{′}⊆*C*^{′}*U*, for each unitary operator *U* in . From 5.6(13) in refs. 7⇓–9, *C*^{′} = *U*^{-1}*UC*^{′}⊆*U*^{-1}*C*^{′}*U*, and *UC*^{′}*U*^{-1}⊆*C*^{′}*UU*^{-1} = *C*^{′}. As this is true for each unitary operator *U* in , replacing *U* by *U*^{-1} in the second inclusion, we have that *U*^{-1}*C*^{′}*U*⊆*C*^{′}, as well as, *C*^{′}⊆*U*^{-1}*C*^{′}*U*. Thus *C*^{′} = *U*^{-1}*C*^{′}*U* for each unitary operator *U* in . Hence (since , from the von Neumann double commutant theorem). It follows that and that .

Suppose, next, that . Then *C*^{′}*U* = *UC*^{′} for each unitary operator *U* in . Let *E* be a projection in . Then *E* + *i*(*I* - *E*) (= *U*_{+}) and *E* - *i*(*I* - *E*) (= *U*_{-}) are unitary operators in (). Thus Hence *EC*^{′}⊆*C*^{′}*E*, for each projection *E* in , and . Therefore and . The last line of the statement of this theorem now becomes the completion of this proof.

## Footnotes

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^{1}To whom correspondence should be addressed. E-mail: zheliu{at}sas.upenn.edu.

Author contributions: Z.L. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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