The Predual, Normal States, and the σ-Weak Topology

Intuition Beginner

A state on an algebra of operators 39.03.01 is a way of averaging measurements: you hand it an operator and it returns a number, the expected value, in a way that respects adding and scaling and never reports a negative answer for a positive measurement. Quantum mechanics runs on exactly such averages. Among all the ways of averaging, some respect limits gracefully and some do not, and the well-behaved ones are called normal.

Normal means: if you build up a measurement out of infinitely many smaller pieces that fit together, the average of the whole equals the sum of the averages of the pieces. A state that lacks this property can assign weight to behavior that lives only "at infinity," in a way no honest limiting process ever reaches. The normal states are the ones a physicist or a measure theorist can actually use, because they cooperate with the limits that build complicated operators out of simple ones.

The collection of all normal averages, packaged together, forms a space called the predual. The striking fact is that the original algebra is recovered as the dual of this smaller space — the operators are exactly the bounded linear readings of the normal averages.

Visual Beginner

Two kinds of averaging functional sitting over the same algebra: the normal ones, which add up correctly across infinitely many fitted-together pieces, and the singular ones, which only see behavior at infinity. Every averaging functional splits cleanly into one piece of each kind.

The picture to hold: the predual is the "real" space, smaller than the full dual, and the algebra is its dual. The operators are reconstructed from how they pair against normal averages.

Worked example Beginner

Work in the smallest setting where the predual is visible: two-by-two complex matrices, the algebra $M_2(\mathbb{C})$. A state here is given by a density matrix — a positive matrix $\rho$ with diagonal entries summing to one — through the averaging rule "value of $A$ equals the sum of the diagonal of $\rho A$." Take

$$\rho =\left(\begin{array}{cc}3/4& 0\\ 0& 1/4\end{array}\right).$$

To average the operator $A=\mathrm{diag}(2,6)$, form $\rho A=\mathrm{diag}(3/4\cdot 2,1/4\cdot 6)=\mathrm{diag}(3/2,3/2)$, and add the diagonal: the average is $3/2+3/2=3$.

Every functional on $M_2(\mathbb{C})$ comes from such a matrix $\rho$, because the matrices are finite-dimensional and nothing lives "at infinity." So here every state is normal, and the predual is the whole dual space. The split into normal and singular parts is invisible until the algebra is infinite-dimensional.

What this tells us: a normal state is averaging-against-a-density, and in finite dimensions that is the only kind there is. The interesting separation between normal and singular only appears once the Hilbert space is infinite-dimensional.

Check your understanding Beginner

Formal definition Intermediate+

Let $M\subseteq B(H)$ be a von Neumann algebra 39.03.01. A linear functional $\phi :M\to \mathbb{C}$ is positive if $\phi (a^{\ast }a)\ge 0$ for all $a\in M$, and a state if it is positive with $\phi (1)=1$. A positive functional is automatically bounded, with $\Vert \phi \Vert =\phi (1)$.

A positive functional $\phi$ on $M$ is normal if for every bounded increasing net $(a_{\lambda })$ of self-adjoint elements with supremum $a={\sup }_{\lambda }{a}_{\lambda }$ (the supremum taken in the order of $M$, which exists because $M$ is monotone complete), one has $\phi (a)={\sup }_{\lambda }\phi (a_{\lambda })$. A general (not necessarily positive) functional is normal if it is a linear combination of normal positive functionals. The normal states are the normal positive functionals with $\phi (1)=1$.

The predual of $M$ is the Banach space $M_{\ast }$ of all $\sigma$-weakly continuous functionals on $M$ (equivalently, by the theorem below, the normal functionals), with the norm inherited from $M^{\ast }$. The defining structural fact is the canonical isometric isomorphism $$ (M_*)^* \cong M, $$ so that $M$ is a dual Banach space with $M_{\ast }$ as a predual ^{[Takesaki Ch. II §2]}. The $\sigma$-weak (ultraweak) topology on $M$ is precisely the weak-$\ast$ topology $\sigma (M,M_{\ast })$ induced by this duality: a net $a_{\lambda }\to a$ $\sigma$-weakly iff $\phi (a_{\lambda })\to \phi (a)$ for every $\phi \in M_{\ast }$. Concretely, on $M\subseteq B(H)$, $$ \varphi(a) = \sum_{n} \langle a\xi_n, \eta_n\rangle, \qquad \sum_n |\xi_n|^2 < \infty,\ \sum_n |\eta_n|^2 < \infty, $$ runs over all $\sigma$-weakly continuous functionals; these are exactly the functionals of the form $a\mapsto \mathrm{Tr}(\rho a)$ for a trace-class $\rho$, restricted to $M$. The $\sigma$-strong topology is generated by the seminorms $a\mapsto \phi (a^{\ast }a{)}^{1/2}$ for normal positive $\phi$.

A unital $\ast$-homomorphism $\pi :M\to N$ between von Neumann algebras is normal if it is $\sigma$-weakly continuous, equivalently if it preserves suprema of bounded increasing nets, equivalently if $\phi \circ \pi \in M_{\ast }$ for every $\phi \in N_{\ast }$.

Counterexamples to common slips

• Not every state on $M$ is normal. On $M={\ell }^{\infty }(\mathbb{N})$ acting on ${\ell }^2(\mathbb{N})$ by multiplication, a Banach limit (a state extending the limit functional on convergent sequences, built from a free ultrafilter) is singular: it vanishes on every sequence supported on a finite set, yet assigns the value $1$ to the unit, so it cannot be additive across the increasing net of finite-support projections, whose supremum is $1$ but each of whose values is $0$.
• Normality is not norm-continuity plus something cosmetic — it is genuinely stronger. Every normal functional is norm-continuous, but the singular Banach-limit functional above is norm-continuous (indeed of norm $1$) and not normal. The gap is detected only by the order-continuity / complete-additivity condition, never by the norm alone.
• The predual is a property of $M$, not of the representation. Two von Neumann algebras that are spatially different but $\ast$-isomorphic have isometrically isomorphic preduals; the $\sigma$-weak topology and the normal states transport along any $\ast$-isomorphism, even one not implemented by a unitary between the Hilbert spaces.

Key theorem with proof Intermediate+

Theorem (characterisations of normality; existence and uniqueness of the predual). Let $M\subseteq B(H)$ be a von Neumann algebra and $\phi$ a positive functional on $M$. The following are equivalent: (i) $\phi$ is normal (order-continuous on bounded increasing nets); (ii) $\phi$ is completely additive on projections: for every family $(p_i{)}_{i\in I}$ of mutually orthogonal projections in $M$, $\phi \left({\sum }_i{p}_i\right)={\sum }_i\phi (p_i)$; (iii) $\phi$ is $\sigma$-weakly continuous; (iv) there is a positive trace-class operator $\rho$ on $H$ with $\phi (a)=\mathrm{Tr}(\rho a)$ for all $a\in M$. Consequently the space $M_$ofnormalfunctionalsisaclosedsubspaceof$M^$satisfying$(M_)^* \cong M$isometrically,andanypredualof$M$iscanonicallyisometricallyisomorphicto$M_*$.* ^{[Takesaki Ch. II §2–3; Dixmier Ch. I §4]}

Proof. $(iv)\Rightarrow (iii)$: for fixed trace-class $\rho$, the map $a\mapsto \mathrm{Tr}(\rho a)$ is by construction one of the generating functionals of the $\sigma$-weak topology, hence $\sigma$-weakly continuous.

$(iii)\Rightarrow (ii)$: let $(p_i)$ be orthogonal projections with sum $p={\sum }_i{p}_i$, the supremum of the finite partial sums $p_F={\sum }_{i\in F}{p}_i$ over finite $F\subseteq I$. The net $(p_F)$ is increasing and bounded by $1$, and $p_F\to p$ $\sigma$-weakly (indeed $\sigma$-strongly: $\Vert (p-p_F)\xi {\Vert }^2=⟨(p-p_F)\xi ,\xi ⟩\to 0$ as $(p-p_F)$ decreases to $0$ strongly). A $\sigma$-weakly continuous $\phi$ therefore satisfies $\phi (p)={\lim }_F\phi (p_F)={\sum }_i\phi (p_i)$.

$(ii)\Rightarrow (i)$: this is the substantive implication. Let $(a_{\lambda })$ increase to $a=\sup a_{\lambda }$ with all $a_{\lambda }$ self-adjoint and the net bounded; by translating and scaling assume $0\le a_{\lambda }\le a\le 1$. Set $c={\sup }_{\lambda }\phi (a_{\lambda })\le \phi (a)$; the goal is $c=\phi (a)$. Suppose not, so $\delta :=\phi (a)-c>0$. Using the spectral resolution of $a-a_{\lambda }\ge 0$ inside $M$, choose for each $\lambda$ a spectral projection $q_{\lambda }=1_{[ϵ,1]}(a-a_{\lambda })\in M$ carving off where $a-a_{\lambda }$ is bounded below by a fixed $ϵ>0$. Complete additivity controls the functional on the lattice of projections, and a maximality argument (Zorn, over orthogonal families of subprojections on which $\phi$ already accounts for the deficit $\delta$) produces a single projection $q\le 1$ with $\phi (q)>0$ on which $a-a_{\lambda }$ stays bounded below for cofinally many $\lambda$, contradicting $a=\sup a_{\lambda }$. Hence $\delta =0$ and $\phi$ is normal.

$(i)\Rightarrow (iv)$: given normality, the GNS representation ${\pi }_{\phi }$ of $\phi$ is itself normal, so the cyclic vector ${\Omega }_{\phi }$ yields, after transporting back to $H$ via the spatial theory of normal representations of a von Neumann algebra, a square-summable family $({\xi }_n)$ with $\phi (a)={\sum }_n⟨a{\xi }_n,{\xi }_n⟩=\mathrm{Tr}(\rho a)$, where $\rho ={\sum }_n|{\xi }_n⟩⟨{\xi }_n|$ is positive trace-class with $\mathrm{Tr}\rho =\Vert \phi \Vert$. This closes the cycle.

For the predual: $M_{\ast }$ is the norm-closed linear span of the normal positive functionals, a closed subspace of $M^{\ast }$. Banach–Alaoglu and the bipolar theorem identify $(M_{\ast }{)}^{\ast }$ with the $\sigma (M^{\ast \ast },M^{\ast })$-closure of $M$ in $M^{\ast \ast }$ intersected with the normal functionals' annihilator; the equivalence $(i)\iff (iii)$ shows the evaluation map $M\to (M_{\ast }{)}^{\ast }$ is an isometric isomorphism onto. Uniqueness: if $X$ is any Banach space with $X^{\ast }\cong M$ isometrically, the unit ball of $M$ is $\sigma (M,X)$-compact, and a positive functional in $X\subseteq M^{\ast }$ is order-continuous on bounded monotone nets because suprema are attained as $\sigma (M,X)$-limits; hence $X\subseteq M_{\ast }$, and a dimension/norming count forces $X=M_{\ast }$. $\square$

Bridge. The predual builds toward the abstract theory of $W^{\ast }$-algebras, and it appears again in 39.04.01 where Tomita–Takesaki modular theory is built from a normal faithful state and its $\sigma$-strong analytic structure. The foundational reason normality matters is exactly the equivalence $(i)\iff (iii)$: order-continuity on the projection lattice is the same datum as $\sigma$-weak continuity, which is the central insight that turns the measure-theoretic "complete additivity" into the topological "weak-$\ast$ continuity" and back. This is exactly the noncommutative replacement for countable additivity of a measure, so the von Neumann algebra plays the role of bounded measurable functions and the predual plays the role of the $L^1$ of integrable densities. Putting these together, the duality $M=(M_{\ast }{)}^{\ast }$ generalises the duality $L^{\infty }=(L^1{)}^{\ast }$, and it is dual to the bicommutant identity $M=M^{\prime \prime }$ of 39.03.01 in that one fixes $M$ by its weak closure and the other fixes it by its predual.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib carries Banach spaces, dual spaces, the weak-$\ast$ topology with Banach–Alaoglu, and the Schatten classes that host the trace-class operators, but neither the predual of a von Neumann algebra, the normalState predicate, nor Sakai's dual-space characterisation is packaged.

The intended formalisation reads schematically:

import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.Topology.Algebra.Module.WeakDual

variable {M : Type*} [VonNeumannAlgebra M]

/-- A normal positive functional: σ-weakly continuous, equivalently
completely additive on orthogonal projections. -/
structure NormalState (M : Type*) [VonNeumannAlgebra M] where
  toFun : M → ℂ
  positive : ∀ a, 0 ≤ (toFun (star a * a)).re
  unital : toFun 1 = 1
  normal : ∀ {ι} (p : ι → M), OrthogonalProjections p →
    toFun (⨆ i, p i) = ∑' i, toFun (p i)

/-- Sakai's theorem: a C*-algebra is a W*-algebra iff it is (isometric to)
a dual Banach space; the predual is then unique. -/
theorem sakai_dual_iff (A : Type*) [CStarAlgebra A] :
    IsWStarAlgebra A ↔ ∃ (X : Type*) (_ : NormedSpace ℂ X), Nonempty (NormedSpace.Dual ℂ X ≃ₗᵢ⋆[ℂ] A) :=
  sorry  -- predual existence ⟺ σ-weak compactness of the unit ball

Advanced results Master

The duality $M=(M_{\ast }{)}^{\ast }$ makes von Neumann algebras the noncommutative measure spaces, and the structural theorems below organise the consequences.

Sakai's theorem. A C*-algebra $A$ is $\ast$-isomorphic to a von Neumann algebra (a $W^{\ast }$-algebra) if and only if $A$ is isometrically isomorphic to the dual of some Banach space, and in that case the predual is unique up to isometric isomorphism ^{[Sakai §1.16]}. This is the abstract characterisation parallel to Gelfand–Naimark for C*-algebras: where a C*-algebra is the abstract counterpart of a concrete norm-closed operator algebra, a $W^{\ast }$-algebra is the abstract counterpart of a concrete $\sigma$-weakly closed one, and the extra datum that promotes the first to the second is precisely the existence of a predual. The uniqueness of the predual is what makes the $\sigma$-weak topology, the normal states, and normal homomorphisms intrinsic — independent of any representation on a Hilbert space.

The Takesaki decomposition. Every positive functional $\phi$ on a von Neumann algebra $M$ decomposes uniquely as $\phi ={\phi }_n+{\phi }_s$ with ${\phi }_n$ normal and ${\phi }_s$ singular, where singular means ${\phi }_s$ majorises no nonzero normal functional. The normal part is the largest normal functional dominated by $\phi$, obtained as the supremum of the upward-directed set of normal positive functionals $\psi \le \phi$. This is the noncommutative Lebesgue decomposition: ${\phi }_n$ is the "absolutely continuous" part, given by a density in $M_{\ast }$, and ${\phi }_s$ is the "purely atomic at infinity" part that the GNS construction sends to a representation with no normal subrepresentation.

$\sigma$-weak continuity of multiplication. Multiplication $M\times M\to M$ is jointly $\sigma$-strongly continuous on bounded sets and separately $\sigma$-weakly continuous, and on the unit ball the $\sigma$-weak and weak operator topologies coincide, as do the $\sigma$-strong and strong operator topologies. The adjoint is $\sigma$-weakly continuous (unlike the strong operator topology, where it fails). These continuity facts are what let one manipulate normal functionals and normal homomorphisms by density arguments, the Kaplansky density theorem of 39.03.01 supplying the bounded approximants.

Normal homomorphisms and the category of $W^{\ast }$-algebras. With morphisms taken to be normal unital $\ast$-homomorphisms, von Neumann algebras form a category in which kernels are $\sigma$-weakly closed ideals (each of the form $Mz$ for a central projection $z$), images are von Neumann subalgebras, and every normal $\ast$-homomorphism factors as a central cut-down followed by a spatial isomorphism. The functor $M\mapsto M_{\ast }$ is a contravariant equivalence onto a category of $L^1$-type Banach spaces, the noncommutative-measure-theory mirror of the contravariant Gelfand equivalence $A\mapsto \hat{A}$ for commutative C*-algebras: C*-algebras are noncommutative topology, von Neumann algebras are noncommutative measure theory.

The commutative model. For $M=L^{\infty }(X,\mu )$ acting on $L^2(X,\mu )$ by multiplication, the predual is $M_{\ast }=L^1(X,\mu )$, the $\sigma$-weak topology is the weak-$\ast$ topology $\sigma (L^{\infty },L^1)$, the normal states are integration against probability densities $g\ge 0$, $\int gd\mu =1$, and a singular state is one concentrated on the "ideal points" of the Stone space not seen by $\mu$. Sakai's theorem specialises to the classical fact that $L^{\infty }$ is a dual space with predual $L^1$, and the Takesaki decomposition specialises to the Lebesgue decomposition of a finitely additive measure into its countably additive and purely finitely additive parts.

Synthesis. The predual is the foundational reason von Neumann algebras are the measurable world: $M=(M_{\ast }{)}^{\ast }$ is exactly the noncommutative duality $L^{\infty }=(L^1{)}^{\ast }$, and this is exactly what makes the $\sigma$-weak topology, the normal states, and normal homomorphisms intrinsic invariants of $M$. Sakai's theorem is dual to Gelfand–Naimark — one characterises norm-closed algebras as functions on a space, the other characterises predual-having algebras as densities on a measure space — and putting these together gives the central insight that "C*-algebra is to commutative C*-algebra as topological space" while "von Neumann algebra is to commutative von Neumann algebra as measure space." The Takesaki decomposition generalises the Lebesgue decomposition, and the bicommutant identity $M=M^{\prime \prime }$ of 39.03.01 is dual to the predual identity $M=(M_{\ast }{)}^{\ast }$: weak-operator closure and predual-existence are two faces of the same $\sigma$-weak compactness of the unit ball, which is the bridge from concrete operator topology to abstract Banach-space duality, and the standard form that modular theory 39.04.01 then animates with the modular automorphism group is built on a chosen faithful normal state in $M_{\ast }$.

Full proof set Master

Proposition (the normal part is the largest dominated normal functional). Let $\phi \ge 0$ on $M$ and let $S=\{\psi :0\le \psi \le \phi ,\psi \text{ normal}\}$. Then $S$ is upward-directed and norm-bounded by $\Vert \phi \Vert$, its supremum ${\phi }_n=\sup S$ is normal, and ${\phi }_s=\phi -{\phi }_n\ge 0$ is singular. Proof. For ${\psi }_1,{\psi }_2\in S$ the functional ${\psi }_1\vee {\psi }_2$ (least upper bound in the order of $M^{\ast }$) is again normal and $\le \phi$, so $S$ is directed. The net of $\psi \in S$ increases and is bounded, so converges in norm to ${\phi }_n$, which is normal since $M_{\ast }$ is norm-closed in $M^{\ast }$ and a norm limit of normal functionals is normal. Maximality gives ${\phi }_n=\sup S$. If some nonzero normal $\chi \le {\phi }_s$ existed, then ${\phi }_n+\chi \le \phi$ is normal and strictly larger than ${\phi }_n$, contradicting maximality; hence ${\phi }_s$ dominates no nonzero normal functional, i.e. is singular. Uniqueness follows because any decomposition $\phi ={\phi }_n^{\prime }+{\phi }_s^{\prime }$ with ${\phi }_n^{\prime }$ normal forces ${\phi }_n^{\prime }\le \phi$, so ${\phi }_n^{\prime }\le {\phi }_n$, and symmetry of the singular condition gives equality. $\square$

Proposition (the predual is unique). If $X$ and $Y$ are Banach spaces with isometric isomorphisms $X^{\ast }\cong M\cong Y^{\ast }$, then $X\cong Y$ isometrically, and both equal $M_{\ast }$. Proof. The unit ball of $M$ is compact in both $\sigma (M,X)$ and $\sigma (M,Y)$ by Banach–Alaoglu. A positive functional $f\in X\subseteq M^{\ast }$ is order-continuous on bounded increasing nets: such a net has its supremum as its $\sigma (M,X)$-limit (the limit exists by compactness and equals the supremum because $f^{\prime }\ge 0$ in $X$ separate points and respect order), so $f(\sup a_{\lambda })=\lim f(a_{\lambda })=\sup f(a_{\lambda })$. Hence $X\subseteq M_{\ast }$ as sets of functionals, and likewise $Y\subseteq M_{\ast }$. But $M_{\ast }\subseteq X$ as well: a normal functional is $\sigma (M,X)$-continuous on bounded sets, and by the Krein–Smulian theorem a functional continuous on the unit ball for the weak-$\ast$ topology lies in the predual $X$. Thus $X=M_{\ast }=Y$. $\square$

Proposition ($\sigma$-weak and weak-operator topologies agree on bounded sets). On the unit ball $M_1=\{a:\Vert a\Vert \le 1\}$ of a von Neumann algebra $M\subseteq B(H)$, the $\sigma$-weak topology coincides with the weak operator topology. Proof. The $\sigma$-weak topology is finer than the WOT (it has more seminorms: WOT uses single $⟨a\xi ,\eta ⟩$, $\sigma$-weak uses summable families). On a bounded set the two have the same convergent nets: if $a_{\lambda }\to a$ WOT with $\Vert a_{\lambda }\Vert \le 1$, then for a $\sigma$-weak generating functional $\phi (a)={\sum }_n⟨a{\xi }_n,{\eta }_n⟩$ with $\sum \Vert {\xi }_n{\Vert }^2,\sum \Vert {\eta }_n{\Vert }^2<\infty$, split the sum at $N$: the tail is bounded by $({\sum }_{n>N}\Vert {\xi }_n{\Vert }^2{)}^{1/2}({\sum }_{n>N}\Vert {\eta }_n{\Vert }^2{)}^{1/2}$ uniformly in $\lambda$ (using $\Vert a_{\lambda }\Vert \le 1$), and the finite head converges by WOT. An $ϵ/3$ argument gives $\phi (a_{\lambda })\to \phi (a)$. So WOT-convergence implies $\sigma$-weak convergence on $M_1$, and the reverse always holds; the topologies agree on $M_1$. $\square$

Proposition (normality $⟺$ $\sigma$-weak continuity for a positive functional). A positive $\phi$ on $M$ is normal iff it is $\sigma$-weakly continuous. Proof. $\sigma$-weak continuity implies complete additivity on projections hence normality, as in the key theorem. Conversely, a normal $\phi$ is $\mathrm{Tr}(\rho \cdot )$ for trace-class $\rho \ge 0$; on the unit ball this is WOT-continuous (finite-rank truncations of $\rho$ converge in trace norm, and each gives a WOT-continuous functional, with uniform tail control), hence $\sigma$-weakly continuous on $M_1$. A linear functional that is $\sigma$-weakly continuous on the unit ball is $\sigma$-weakly continuous by the Krein–Smulian theorem, since $\sigma$-weak continuity on every ball $r{M}_1$ assembles to global weak-$\ast$ continuity. $\square$

Proposition (kernel of a normal homomorphism is $Mz$). Let $\pi :M\to N$ be a normal $\ast$-homomorphism. Then $\ker \pi =Mz$ for a central projection $z\in Z(M)$, and $\pi$ restricts to a $\ast$-isomorphism of $M(1-z)$ onto $\pi (M)$. Proof. $\ker \pi$ is a $\sigma$-weakly closed two-sided $\ast$-ideal (closed because $\pi$ is $\sigma$-weakly continuous). A $\sigma$-weakly closed two-sided ideal of a von Neumann algebra equals $Mz$ for a central projection $z$: the ideal is generated by its projections, which have a supremum $z$ lying in the ideal (by $\sigma$-weak closure and monotone completeness), and centrality follows because the ideal is two-sided so $z$ commutes with all of $M$. Then $\pi (z)=0$, $\pi$ kills $Mz$, and on $M(1-z)$ the kernel reduces to zero, so $\pi {|}_{M(1-z)}$ is an injective normal $\ast$-homomorphism, hence (injective $\ast$-homomorphisms of C*-algebras are isometric) a $\ast$-isomorphism onto its image. $\square$

Connections Master

• Von Neumann algebras and the bicommutant theorem 39.03.01 — the $\sigma$-weak topology defined there as a topology on $B(H)$ is identified here intrinsically as the weak-$\ast$ topology of the predual; the bicommutant identity $M=M^{\prime \prime }$ and the predual identity $M=(M_{\ast }{)}^{\ast }$ are the concrete and abstract faces of the same $\sigma$-weak closedness.

• Tomita–Takesaki modular theory 39.04.01 — modular theory begins from a faithful normal state in $M_{\ast }$ and its GNS construction; the normality and $\sigma$-weak continuity established here are exactly the hypotheses that make the modular automorphism group $\sigma$-weakly continuous and the standard form intrinsic.

• Banach space fundamentals 02.11.04 — the predual is a Banach space and the duality $M=(M_{\ast }{)}^{\ast }$, the uniqueness of the predual, and the agreement of topologies on bounded sets all rest on Banach–Alaoglu, the bipolar theorem, and Krein–Smulian from the general theory.

• Hilbert space 02.11.08 — the concrete normal functionals $a\mapsto {\sum }_n⟨a{\xi }_n,{\eta }_n⟩$ are built from square-summable families in $H$, and the predual of $B(H)$ is the trace-class operators, a Hilbert–Schmidt-completed space over $H$.

• C-algebras: axioms, spectrum, functional calculus 39.01.01* — Sakai's dual-space characterisation of $W^{\ast }$-algebras is the measure-theoretic analogue of the Gelfand–Naimark characterisation of C*-algebras; both isolate an abstract class by an intrinsic property and recover the concrete model.

Historical & philosophical context Master

The predual was isolated by Dixmier, who in the early 1950s proved that a von Neumann algebra has a unique predual and that the normal functionals are exactly the $\sigma$-weakly continuous ones; the normal–singular decomposition appears in his treatment of forms on rings of operators and was sharpened by Takesaki ^{[Takesaki Ch. III]}. The abstract characterisation — that a C*-algebra is a $W^{\ast }$-algebra precisely when it is a dual Banach space — is due to Sakai (1956), who thereby freed the theory from any particular Hilbert-space representation and put $W^{\ast }$-algebras on the same abstract footing that Gelfand and Naimark had given C*-algebras ^{[Sakai §1.16]}.

The conceptual content is the identification of von Neumann algebras with noncommutative measure theory: a commutative von Neumann algebra is $L^{\infty }$ of a measure space, its predual is $L^1$, and the $\sigma$-weak topology is the $\sigma (L^{\infty },L^1)$ topology, so the passage from C*-algebras to von Neumann algebras is the passage from continuous functions on a topological space to bounded measurable functions on a measure space. Murray and von Neumann's original "rings of operators" already carried this measure-theoretic instinct in the type II trace; Sakai and Takesaki made the duality with $L^1$ exact, and the normal–singular dichotomy is the operator-algebraic form of the Lebesgue decomposition.

Bibliography Master

• Takesaki, M., Theory of Operator Algebras I, Springer, 1979. Ch. II §§2–3, Ch. III.
• Sakai, S., C-Algebras and W*-Algebras*, Springer, 1971. §1.13–1.16.
• Dixmier, J., Von Neumann Algebras, North-Holland, 1981. Ch. I §4.
• Sakai, S., "A characterization of W*-algebras", Pacific Journal of Mathematics 6 (1956), 763–773.
• Takesaki, M., "On the conjugate space of operator algebra", Tôhoku Mathematical Journal 10 (1958), 194–203.
• Kadison, R. V. and Ringrose, J. R., Fundamentals of the Theory of Operator Algebras II: Advanced Theory, Academic Press, 1986. Ch. 7.

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Operator-algebras spine, von Neumann-algebra duality unit. The predual $M_$andtheabstractSakaicharacterisationasthemeasure-theoreticstrengtheningofthebicommutanttheory(\mathrm{39.03.01});suppliesthenormalstatesand$\sigma$-weak topology that modular theory (39.04.01) is built on.*