Von Neumann Algebras and the Hilbert Algebra Completeness

Introduction

A Hilbert algebra is an algebraic object with a positivity; a von Neumann algebra is an operator object closed in the weak topology. The two are related by a theorem that is both a completion and a reconstruction: the completion of a Hilbert algebra is a von Neumann algebra, in the sense that the algebra generated by the extended left multiplications on the completion is weakly closed, and conversely every von Neumann algebra with a cyclic and separating vector arises this way from a Hilbert algebra, namely from the algebra itself with the form defined by the vector. So the Hilbert algebras are exactly the algebraic skeletons of the standard forms of von Neumann algebras, and the passage in either direction loses nothing.

The completeness has a sharp form. The left regular representation of a Hilbert algebra gives a von Neumann algebra on the completion — not merely a $\ast$-algebra of operators — and the proof identifies the weak closure with a double commutant, using the commutant relation of the regular representations. Conversely, given a von Neumann algebra and a cyclic and separating vector, the pair (algebra, vector) carries a natural Hilbert-algebra structure, and the completion returns the given algebra. The consequence is that the theory of Hilbert algebras and the theory of von Neumann algebras with a faithful normal state are the same theory, and statements may be proved on whichever side is convenient.

This article fixes the completeness theorem, the reconstruction, the standard form, the left Hilbert algebra, and the identification of the three descriptions.

The Hilbert algebra is Hilbert Algebras; its completion and its regular representations are The Completion of a Hilbert Algebra and The Left and the Right Regular Representation; the modular structure is The Modular Operator and Tomita-Takesaki Theory; the states are The GNS Construction; the positivity is Self-Adjoint Elements and the Positive Cone. Those are cited. The algebra is $A$, its completion is $H$, and the von Neumann algebra generated is $\mathcal{M}$.

The Completeness of a Hilbert Algebra

Definition. A Hilbert algebra is complete when its completion $H$ is already the algebra with the form, that is when $A$ is complete for the norm $\langle\cdot,\cdot\rangle^{1/2}$; a left Hilbert algebra is a Hilbert algebra whose multiplication satisfies the extra continuity needed for the reconstruction, in particular the boundedness of the left multiplications together with $\overline{A^{2}} = A$.

Proposition (the left multiplications see the topology). For a Hilbert algebra the left multiplication is bounded with $\|L_x\|\leq\|x\|$, so the representation $x\mapsto\bar L_x$ on the completion is norm-continuous, and the algebraic and operator topologies agree on the algebra.

Proof. Boundedness is an axiom of a Hilbert algebra; the identification of the two topologies follows from $\bar L_x(1) = x$ for a unital algebra.

Remark (why the word completeness). "The completeness of a Hilbert algebra" in the sense of this article is not the metric completeness of a single algebra: it is the property that the operator algebra generated by the algebra is closed, so that no operators are lost in passing from the algebra to its weak closure. The two notions coincide in the finite-dimensional case and must be distinguished otherwise.

The von Neumann Algebra of the Completion

Theorem (the generated algebra is a von Neumann algebra). Let $A$ be a Hilbert algebra with completion $H$. Then the algebra

$$ \mathcal{M} = \overline{\{\bar L_x : x\in A\}}^{\text{weak}} \subseteq B(H) $$

generated by the extended left multiplications is a von Neumann algebra, and it is the strong closure of the image of $A$; moreover $\mathcal{M}$ is a factor exactly when the Hilbert algebra has no nontrivial central elements corresponding to projections.

Proof. The weak closure of a self-adjoint algebra of operators is a von Neumann algebra; self-adjointness is $\bar L_{x^{\dagger}} = \bar L_x^{*}$; and the criterion for a factor is the triviality of the centre, which is computed from the central projections of the algebra.

Theorem (reconstruction). Let $\mathcal{M}$ be a von Neumann algebra on $H$ with a cyclic and separating vector $\xi$, and let

$$ A = \{x\xi : x\in\mathcal{M}\} \text{ with the product } x\xi\cdot y\xi = xy\xi , \qquad (x\xi)^{\dagger} = x^{*}\xi , \qquad \langle x\xi,y\xi\rangle = \langle x\xi,y\xi\rangle_{H} . $$

Then $A$ is a Hilbert algebra whose completion is $H$ and whose generated von Neumann algebra is $\mathcal{M}$.

Proof. The product is well defined because $\xi$ is separating, associativity is that of $\mathcal{M}$, the involution is well defined and involutive, and the adjoint axiom $\langle uv,w\rangle = \langle v,u^{\dagger}w\rangle$ is the self-adjointness of $\mathcal{M}$; the completion is $H$ because $A$ is dense, and the generated algebra is $\mathcal{M}$ because the left multiplications of $A$ are the elements of $\mathcal{M}$.

Corollary (the two theories are one). The Hilbert algebras are exactly the algebras $A = \mathcal{M}\xi$ from a von Neumann algebra $\mathcal{M}$ with a cyclic and separating vector $\xi$; the completion returns $\mathcal{M}$ and the reconstruction returns $A$, and the two constructions are inverse.

Proof. The two theorems, one in each direction.

The Standard Form

Definition. A standard form of a von Neumann algebra $\mathcal{M}$ is a quadruple $(\mathcal{M}, H, J, P)$ with $H$ a Hilbert space on which $\mathcal{M}$ acts faithfully and normally, $J$ an antiunitary involution with $J\mathcal{M}J = \mathcal{M}'$, and $P$ a self-dual cone in $H$ with $J\xi = \xi$ for its vectors and $xJxJ$ preserving it.

Theorem (the standard form exists and is unique). Every von Neumann algebra $\mathcal{M}$ with a cyclic and separating vector has a standard form, obtained from the modular conjugation $J$ of The Modular Operator and Tomita-Takesaki Theory and the positive cone of the algebra; the standard form is unique up to the appropriate isomorphism.

Proof. The modular conjugation supplies $J$ and the commutant identification; the positive cone $P = \overline{\{xJxJ\xi : x\in\mathcal{M}\}}$ is self-dual by the modular theory; uniqueness is the uniqueness of the modular data together with the ordering of the cone.

Proposition (the standard form is a Hilbert algebra object). In a standard form the algebra, the commutant, the conjugation and the cone are all determined by the Hilbert algebra $A = \mathcal{M}\xi$ and its involution; so the standard form is the operator-theoretic avatar of the completion of The Completion of a Hilbert Algebra.

Proof. The algebra is the generated von Neumann algebra of $A$, the commutant is that of the right multiplications, and the conjugation is the closure of the involution.

Worked Cases

Finite Dimensions

For $A = M_n(\mathbb{C})$ with the Hilbert–Schmidt form the completion is $M_n(\mathbb{C})$, the generated von Neumann algebra is the left-action copy of $M_n(\mathbb{C})$, and the standard form has $\mathcal{M}'$ the right-action copy; the cone is the cone of positive matrices.

The Group Algebra

For a finite group $G$ the completion of $\mathbb{C}[G]$ is $\mathbb{C}^{G}$, the generated algebra is the algebra of operators of the left regular representation, which is a factor if and only if every nontrivial conjugacy class of $G$ satisfies the condition of the correspondence; the standard form is the usual one of a finite-dimensional factor.

An Abelian Example

For the abelian algebra $A = L^{\infty}(X)$ of bounded measurable functions of the analysis layer (the $L^{p}$ spaces are Part III), with its integral-defined form, the completion is $L^{2}(X)$ and the generated von Neumann algebra is $A$ itself acting by multiplication; the standard form is the abelian one, with $\mathcal{M}' = \mathcal{M}$ and the modular conjugation the complex conjugation.

Summary

A Hilbert algebra has a completion whose left multiplications generate a von Neumann algebra $\mathcal{M}$ on the completed Hilbert space, and this is the precise sense of the completeness of a Hilbert algebra: the operator algebra obtained is weakly closed, not merely a $\ast$-algebra of operators. Conversely, a von Neumann algebra $\mathcal{M}$ with a cyclic and separating vector $\xi$ gives a Hilbert algebra $A = \mathcal{M}\xi$ with product $x\xi\cdot y\xi = xy\xi$, involution $(x\xi)^{\dagger} = x^{*}\xi$ and the Hilbert-space form, whose completion is $H$ and whose generated algebra is $\mathcal{M}$; the two constructions are inverse, so the Hilbert algebras and the von Neumann algebras with a faithful normal state are the same objects. The operator-theoretic form of the correspondence is the standard form $(\mathcal{M},H,J,P)$ with $J\mathcal{M}J = \mathcal{M}'$ and a self-dual cone $P$, and it is the avatar of the completion of the Hilbert algebra. The algebra is Hilbert Algebras, the completion and the regular representations are The Completion of a Hilbert Algebra and The Left and the Right Regular Representation, the modular objects are The Modular Operator and Tomita-Takesaki Theory, and the positivity is Self-Adjoint Elements and the Positive Cone.

Summary of Notation

Symbol	Meaning
$A$, Cauchy–Schwarz algebra	Hilbert algebra
$\bar L_x$, $\mathcal{M}$	Extended left multiplications and generated algebra
$A = \mathcal{M}\xi$	Reconstruction of the algebra from a cyclic and separating vector
$x\xi\cdot y\xi = xy\xi$	Product in the reconstructed algebra
$(x\xi)^{\dagger} = x^{*}\xi$	Involution in the reconstructed algebra
$(\mathcal{M},H,J,P)$	Standard form, $J$ modular conjugation, $P$ self-dual cone
$J\mathcal{M}J = \mathcal{M}'$	The conjugation identifies the commutant
Completeness of $A$	The generated operator algebra is closed