mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, specifically

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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...

, the von Neumann bicommutant theorem relates the closure of a set of

bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...

s on a

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

in certain topologies to the

bicommutant In algebra, the bicommutant of a subset ''S'' of a semigroup (such as an algebra or a group) is the commutant of the commutant of that subset. It is also known as the double commutant or second commutant and is written S^. The bicommutant is part ...

of that set. In essence, it is a connection between the

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

ic and topological sides of

operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operato ...

. The formal statement of the theorem is as follows: :Von Neumann bicommutant theorem. Let be an

consisting of bounded operators on a Hilbert space , containing the identity operator, and closed under taking

adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...

s. Then the closures of in the

weak operator topology In functional analysis, the weak operator topology, often abbreviated WOT,Ilijas Farah, Combinatorial Set Theory of C*-algebras' (2019), p. 80. is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional ...

and the

strong operator topology In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...

are equal, and are in turn equal to the

of . This algebra is called the

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. It is a special type of C*-algebra. Von Neumann al ...

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

norm topology In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informal ...

then it is a

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of contin ...

, but not necessarily a von Neumann algebra. One such example is the C*-algebra of

compact operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...

s (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 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 . The theorem can be applied to show that any primitive ring can be ...

Proof

Let be a Hilbert space and the bounded operators on . Consider a self-adjoint unital

subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear opera ...

of (this means that contains the adjoints of its members, and the identity operator on ). The theorem is equivalent to the combination of the following three statements: :(i) :(ii) :(iii) where the and subscripts stand for closures in the

weak Weak may refer to: Songs * Weak (AJR song), "Weak" (AJR song), 2016 * Weak (Melanie C song), "Weak" (Melanie C song), 2011 * Weak (SWV song), "Weak" (SWV song), 1993 * Weak (Skunk Anansie song), "Weak" (Skunk Anansie song), 1995 * "Weak", a son ...

and

strong Strong may refer to: Education * The Strong, an educational institution in Rochester, New York, United States * Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas * Strong School, New Haven, Connecticut, United ...

operator topologies, respectively.

Proof of (i)

For any and in , the map ''T'' → <''Tx'', ''y''> is continuous in the weak operator topology, by its definition. Therefore, for any fixed operator , so is the map :

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

Let ''S'' be any subset of , and ''S''′ its

commutant In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

. For any operator in ''S''′, this function is zero for all ''O'' in ''S''. For any not in ''S''′, it must be nonzero for some ''O'' in ''S'' and some ''x'' and ''y'' in . By its continuity there is an open neighborhood of 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 ; since it contains , 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 in , every open neighborhood of in the weak operator topology is also open in the strong operator topology and therefore contains a member of ; therefore is also a member of .

Proof of (iii)

Fix . We must show that , i.e. for each ''h'' ∈ ''H'' and any , there exists ''T'' in with . Fix ''h'' in . The cyclic subspace is invariant under the action of any ''T'' in . Its closure in the norm of ''H'' is a closed linear subspace, with corresponding

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it we ...

: ''H'' → in ''L''(''H''). In fact, this ''P'' is in , as we now show. :Lemma. . :Proof. Fix . As , it is the limit of a sequence with in . For any , is also in , and by the continuity of , this sequence converges to . So , and hence ''PTPx'' = ''TPx''. Since ''x'' was arbitrary, we have ''PTP'' = ''TP'' for all in . :Since is closed under the adjoint operation and ''P'' is

self-adjoint In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a^*). Definition Let \mathcal be a *-algebra. An element a \in \mathcal is called self-adjoint if The set of self-adjoint elements ...

, for any we have ::

\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

:So ''TP'' = ''PT'' for all , meaning ''P'' lies in . By definition of the

, we must have ''XP'' = ''PX''. Since is unital, , and so . Hence . So for each , there exists ''T'' in with , i.e. is in the strong operator closure of .

Non-unital case

A C*-algebra acting on H is said to act ''non-degenerately'' if for ''h'' in , implies . In this case, it can be shown using an

approximate identity In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element. Definition A right approximate ...

in that the identity operator ''I'' lies in the strong closure of . Therefore, the conclusion of the bicommutant theorem holds for .

References