Original text: CC0 1.0. Prerequisite proofs and component terms.

A coefficient domain for the dual GNS construction

OA-FLOW.DW.IMPORT.GNS — Exact weight-domain inputs

This is a refinement of IMP.MOD.WEIGHTS, owned by OA-MOD, not a proof of general weight theory. For every n.s.f. weight $\varphi$ on a von Neumann algebra $M$, we require the following results at arbitrary Hilbert dimension.

Write

The space $\mathfrak n_\varphi$ is a left ideal. The weight has its linear extension to $\mathfrak m_\varphi$. Its GNS map $\Lambda_\varphi:\mathfrak n_\varphi\to H_\varphi$ has dense range and satisfies

Inner products are linear in the first variable. The representation $\pi_\varphi$ is normal, unital and faithful. We identify $M$ with its image. In particular, for fixed $a,b\in\mathfrak n_\varphi$,

is a bounded normal functional on all of $M$, of norm at most $\|\Lambda_\varphi(a)\|\|\Lambda_\varphi(b)\|$. The expression on the left is defined because $xa\in\mathfrak n_\varphi$.

We also require a net $(e_i)$ of positive contractions in $\mathfrak n_\varphi\cap\mathfrak n_\varphi^*$ converging strongly to $1$. The net is not assumed increasing or sequential. This finite-domain approximation follows from the usual semifiniteness and density theory, but its complete general proof is an OA-MOD obligation. For the model below, we also use its precise criterion that a normal weight is semifinite if and only if its left ideal $\mathfrak n_\varphi$ is sigma-weakly dense. We use normal representations preserving bounded strong-star convergence. No faithful normal state or separable predual is being assumed.

OA-FLOW.DW.SETTING — Functions, measure and coefficient order

Let $G$ be a locally compact Hausdorff group, with a fixed left Haar measure $ds$, and let $\alpha:G\to\operatorname{Aut}(M)$ be point-ultraweakly continuous. Use the equivalent action topology and bounded strong-star continuity contract in OA-FLOW.INT.SETTING. No second countability, sigma compactness or unimodularity is imposed, and $\varphi$ need not be invariant under $\alpha$.

Put $\mathcal K=\mathcal K_\alpha$, the compactly supported strong-star continuous functions $G\to M$. Recall the formulas proved in OA-FLOW.INT.ALGEBRA:

The convention for the modular function is $\int h(rs)\,dr=\Delta_G(s)^{-1}\int h(r)\,dr$. Integration of a coefficient function means its ultraweak integral, equivalently its strong integral on each fixed Hilbert vector in the circumstances established in lesson 08. We do not assume operator-norm continuity of $f$.

The regular integrated operator on $L^2(G,H_\varphi)$ is

We use $L^2(G,H_\varphi)\simeq H_\varphi\otimes L^2(G)$; compactly supported continuous vector functions and the algebraic scalar-vector tensor products are dense. This is the Hilbert-space section construction, with no countable fundamental family attached to $H_\varphi$.

OA-FLOW.DW.GNSDOMAIN — An algebraic domain with a continuous GNS image

Define the finite-sum space

This is an algebraic span inside $\mathcal K$, not a completion and not the set of all pointwise weight-finite functions. Define

Proposition. The map in (D8) is well-defined and injective as a map to $L^2(G,H_\varphi)$. Its image lies in $C_c(G,H_\varphi)$. The space $\mathfrak b_\varphi$ is a left convolution ideal in $\mathcal K$ and is stable under the left coefficient action of $M$.

Proof. Write $g=\sum_{j=1}^m f_j\cdot a_j$. Since $\mathfrak n_\varphi$ is a left ideal, every value $g(s)$ belongs to $\mathfrak n_\varphi$. Formula (D2) gives

Each summand is continuous and compactly supported as a Hilbert-space function. Formula (D8) therefore has those properties. Its value is determined by $g(s)$ itself, so a different finite-sum presentation gives the same vector function. Compact support and boundedness imply square integrability; more explicitly,

If $\widetilde\Lambda_\varphi(g)=0$ in $L^2$, its continuous representative is zero everywhere. Indeed, a continuous vector function nonzero at a point has norm bounded below on a nonempty open set, and Haar measure is positive on that set. Thus $\varphi(g(s)^*g(s))=0$ for every $s$. Faithfulness implies $g(s)=0$ for every $s$.

For $h\in\mathcal K$, the coefficient identities from lesson 08 give

Summing proves the left convolution ideal assertion. Likewise

No multiplication of a weight-finite coefficient on the right by an arbitrary element of $M$ was used. $\square$

The ambient space $\mathcal K$ is an $M$-bimodule, but $\mathfrak b_\varphi$ need not be a right $M$-module. The relation $a\in\mathfrak n_\varphi\Rightarrow ac\in\mathfrak n_\varphi$ is generally false. A concrete failure appears below.

OA-FLOW.DW.GNSCONVOLUTION — Convolution becomes bounded left multiplication

Proposition. If $f\in\mathcal K$ and $g\in\mathfrak b_\varphi$, then

For $x\in M$, there is also the coefficient identity

Proof. By finite-sum linearity, it suffices to put $g=h\cdot a$, $a\in\mathfrak n_\varphi$. The left-ideal identity already proved gives $f*g=(f*h)\cdot a$. Apply (D2) and the strong-on-vectors meaning of the integral:

The vector $\xi(s)=h(s)\Lambda_\varphi(a)$ is in $C_c(G,H_\varphi)$. In the regular-kernel formula of OA-FLOW.INT.KERNEL, make the left-translation substitution $t=rs$, for fixed $r$. Its right side becomes exactly the last line of (D17), with no Haar factor. This proves the equality of continuous representatives in (D15). The norm bound follows from $\|F_\alpha(f)\|\leq\|f\|_1$.

Finally, for each $s$, the left-ideal GNS relation gives

which is (D16). $\square$

Thus $\|g\|_{\varphi,2}:=\|\widetilde\Lambda_\varphi(g)\|_2$ is a norm on $\mathfrak b_\varphi$, and left convolution by $f$ is bounded for that norm. This assertion concerns the GNS norm; it is not a claim that $\mathfrak b_\varphi$ is complete or closed in any operator topology.

OA-FLOW.DW.COMMON — A domain on which the involution is defined twice

Set

Lemma. The space $B_\varphi$ is a star algebra for convolution and $\sharp$. It contains

Proof. If $f,g\in B_\varphi$, then $f*g\in\mathfrak b_\varphi$ because this is a left convolution ideal. Also

since $f^\sharp\in\mathfrak b_\varphi$ and $g^\sharp\in\mathcal K$. Thus $f*g\in B_\varphi$. The definition and involutivity of $\sharp$ show that $B_\varphi$ is invariant under $\sharp$.

For a term in (D19), its membership in $\mathfrak b_\varphi$ follows from $b^*\cdot f\in\mathcal K$ and the final coefficient $a\in\mathfrak n_\varphi$. The coefficient-involution identities of lesson 08 give

This proves membership in both parts of the intersection. $\square$

OA-FLOW.DW.DENSITY — Vector density, operator generation and products

Theorem. The space $\mathcal A_\varphi$ is dense in $L^2(G,H_\varphi)$. The strong operator closure of $F_\alpha(B_\varphi)$ is the regular crossed product $N=M\rtimes_\alpha G$. With multiplication

the linear span of $\mathcal A_\varphi\mathcal A_\varphi$ is dense as well.

Proof. Use the positive contraction net $(e_i)$ specified in the GNS import. If $h\in C_c(G)$ is scalar and $a\in\mathfrak n_\varphi$, (D19) and (D16) give

The regular representation is normal. Hence its images of the bounded strong-convergent net $e_i\to1$ converge strongly to $1$. Equation (D21) tends in $L^2$ to the displayed elementary vector. The span of such elementary vectors is dense, because $\Lambda_\varphi(\mathfrak n_\varphi)$ is dense in $H_\varphi$ and $C_c(G)$ is dense in $L^2(G)$. This proves vector density. In particular, this argument uses convergence in a normal representation, not dominated convergence for an arbitrary net of functions.

For operator generation, fix $f\in\mathcal K$. Both $e_i,e_j$ are in the finite weight domain, and (D19) gives $e_i\cdot f\cdot e_j\in B_\varphi$. The integrated module identities imply

strongly on the product directed set. To check the convergence, for a fixed vector $\xi$ bound the difference by

The contraction bound makes the first term independent of $i$. Both terms tend to zero.

The scalar approximate identities in OA-FLOW.INT.GENERATION also put $1$ in the strong closure of $F_\alpha(\mathcal K)$. Since $F_\alpha(B_\varphi)$ is a star algebra and its strong closure contains $F_\alpha(\mathcal K)$ by (D22), it contains $1$. The bicommutant theorem and the generation result of lesson 08 now identify that strong closure with $N$.

Finally suppose $\eta$ is orthogonal to every product in (D20). By (D15), for every $f,g\in B_\varphi$,

Vector density gives $F_\alpha(f)^*\eta=0$. Replacing $f$ by $f^\sharp$ shows $F_\alpha(f)\eta=0$ for every $f\in B_\varphi$. Because $1$ is in the strong closure of these operators, $\eta=0$. This proves density of products. $\square$

OA-FLOW.DW.IMPORT.TOMITA — Relative operators used for closability

This section states further results that these lessons use without proof. Realize $M$ in the standard form associated with $\varphi$, with conjugation $J$. For each n.s.f. weight $\psi$, use its canonical standard-form GNS map $\Lambda_\psi:\mathfrak n_\psi\to H_\varphi$. We require a closed antilinear relative Tomita operator $S_{\psi,\varphi}$ and a positive nonsingular self-adjoint $\Delta_{\psi,\varphi}$, satisfying

The second formula includes the assertion that its vector belongs to the operator domain. We do not assert that an arbitrary vector lies there merely because the formula is formally meaningful.

The standard implementation $U_s$ of $\alpha_s$ is a strongly continuous unitary representation commuting with $J$. Its GNS transport identity is

These statements concern general weights, standard forms and relative derivatives, so they belong to general modular theory. The argument that follows uses only (D23)–(D24) and closedness of the indicated operators. It does not need a formula for their joint imaginary powers yet.

OA-FLOW.DW.CLOSED-FIELDS — A pointwise closedness lemma

Lemma. On any measure space $(X,\mu)$, let $A_x$ be a closed linear operator on a fixed Hilbert space $H$, for each $x$. Define the maximal operator $T$ on $L^2(X,H)$ by

Then $T$ is closed. No measurability assertion about the whole field $x\mapsto A_x$ is needed for this conclusion; measurability is part of the domain condition. This lemma alone does not assert that the domain is dense.

Proof. Suppose $\xi_n\to\xi$ and $T\xi_n\to\eta$ in $L^2$. Choose a subsequence $n_k$ for which

Monotone convergence applied to the sum of the nonnegative pointwise squared norms shows that the sum is finite almost everywhere. Thus both vector differences tend to zero pointwise outside one null set. Remove also the countable union of the null sets where $\xi_{n_k}(x)$ fails to lie in $\operatorname{Dom}(A_x)$. At every remaining point, closedness of $A_x$ implies

The right side is already a strongly measurable $L^2$ section. Thus $\xi\in\operatorname{Dom}(T)$ and $T\xi=\eta$. This proves closedness. Neither sigma-finiteness of $\mu$ nor separability of $H$ was used. $\square$

OA-FLOW.DW.RELATIVETOMITA — Closing the coefficient involution

The injectivity of $\widetilde\Lambda_\varphi$ makes the following operator unambiguous:

Theorem. Assuming the exact relative Tomita import, $S_0$ is closable on the dense domain $\mathcal A_\varphi$.

Proof. On $C_c(G,H_\varphi)$ define

The output is continuous and compactly supported. Inversion of Haar measure gives

Since $U_sJ=JU_s$, direct substitution gives $\mathcal J^2\xi=\xi$. It follows that $\mathcal J$ extends to an antiunitary involution on $L^2(G,H_\varphi)$. Formula (D27) remains valid almost everywhere for the extension: approximate by continuous compact-support sections in $L^2$, take pointwise-convergent subsequences as in the preceding lemma, and use inversion preserving Haar null sets. This also supplies the needed strong measurability without a global separability assumption.

For $s\in G$, write $\psi_s=\varphi\circ\alpha_s$. Take the closed positive operator

on $H_\varphi$, and let $T$ be its maximal pointwise operator (D25). The preceding lemma makes $T$ closed.

Fix $f\in B_\varphi$. Its value $f(s)$ lies in $\mathfrak n_\varphi$. Also

By the definition of $\psi_s$, this says $f(s)^*\in\mathfrak n_{\psi_s}$. Therefore the domain assertion in (D23) applies to $f(s)$. Using (D24), then (D23), gives

The left side is an $L^2$ section, since $f^\sharp\in\mathfrak b_\varphi$ and $\mathcal J$ is antiunitary. Thus (D28) verifies every condition in the domain definition of $T$, and proves

Consequently $S_0\subset\mathcal JT$. The operator $\mathcal JT$ is closed: convergence of its second graph coordinate is equivalent, after applying the bounded involution $\mathcal J$, to convergence of the second coordinate for $T$. Therefore $S_0$ has a closed extension and is closable. $\square$

This proof gives an inclusion of graph closures. Equality with $\mathcal JT$ has not been proved. For the later modular identification one must show that $T=\widetilde\Delta_\varphi^{1/2}$ for the positive operator defined through the proposed modular unitary group, and establish invariance of $\mathcal A_\varphi$ under its imaginary powers. Once those facts hold, OA-FLOW.GRAPH.POWERS supplies the graph-core step and upgrades the inclusion to equality.

OA-FLOW.DW.LEFTHILBERT — What the domain construction now provides

A left Hilbert algebra is a dense involutive algebra in a Hilbert space such that left multiplication by each algebra vector extends to a bounded operator, the inner product obeys the left adjoint identity, its involution is closable, and the linear span of products is dense. These are the conventions used in the general OA-MOD correspondence with weights.

Corollary. Under the displayed GNS and relative Tomita imports, $\mathcal A_\varphi$, with (D20) and the involution (D26), is a left Hilbert algebra. Its left multiplication representation is

and its generated von Neumann algebra is $M\rtimes_\alpha G$.

Proof. Injectivity of $\widetilde\Lambda_\varphi$ and the star algebra properties of $B_\varphi$ make both operations well-defined. Equation (D15) supplies bounded left multiplication. For $f,g,h\in B_\varphi$, the integrated adjoint identity gives

This is the required left adjoint identity. Density, density of products and the generated algebra were proved above, and closability is the preceding theorem. $\square$

Editable source · Proof dependencies and component terms