The standard crossed-product commutant

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

The standard crossed-product commutant

Let $G$ be a locally compact group, $M$ a von Neumann algebra, and $\alpha:G\to\operatorname{Aut}(M)$ a point-ultraweakly continuous action. Represent $M$ faithfully in a standard form on $K$ , with modular conjugation $J$ . Its canonical unitary implementers $U_g$ are strongly continuous, satisfy $U_gxU_g^*=\alpha_g(x)$ , and commute with $J$ , as in lesson 10. On $L^2(G,K)$ let

[\pi_\alpha(x)\xi](s)=\alpha_{s^{-1}}(x)\xi(s),\qquad [\lambda_g\xi](s)=\xi(g^{-1}s),\qquad [R_g\xi](s)=\Delta_G(g)^{1/2}\xi(sg). \tag{I1}

Write $P=M\rtimes_\alpha G=(\pi_\alpha(M)\cup\lambda(G))''$ . Choose an n.s.f. weight. The general standard-form equivalence theorem OA-MOD-SE-10 transports its canonical GNS standard form onto the prescribed form, preserving J and the canonical automorphism implementers by SE-11; its arbitrary directed support-corner patching does not require a faithful state on the whole algebra. For the standard model chosen initially in BF77, one may simply start with this weight-constructed form. The coefficient construction in OA-FLOW.DW.LEFTHILBERT supplies a left Hilbert algebra $\mathcal A_\varphi\subset L^2(G,K)$ whose generated left von Neumann algebra is $P$ . The closed-operator theorem OA-FLOW.DW.INVARIANTCORE, equation M25, identifies the closure of its coefficient involution and its polar conjugation, including equality of domains. In the conventions of (I1) the conjugation is

[\mathcal J\xi](s)=\Delta_G(s)^{-1/2}U_s^*J\xi(s^{-1}).

The general Hilbert-algebra commutant theorem OA-MOD-MF-05 applies directly to this left Hilbert algebra and its closed polar decomposition. Its full-completion step WH-03–04 preserves both the generated algebra and the closed involution. The theorem therefore gives

\mathcal J P\mathcal J=P'. \tag{I2}

This argument retains arbitrary Hilbert dimension and the full locally compact group scope. The coefficient-domain construction requires the exact general GNS input and the relative Tomita domain and standard transport identities D23–D24. Equation M25 additionally uses the real-power domain theorem, M23 imaginary-power invariance and OA-FLOW.GRAPH.POWERS. The displayed antiunitary formula alone proves neither the closed polar identification nor (I2). Reconstructing the associated dual weight in lesson 14 is needed for its weight and modular-action identifications; that second reconstruction is not an additional premise of this commutant calculation.

The two conjugations can be checked on continuous compactly supported $K$ -valued sections. Since $U_sJ=JU_s$ and $U_{s^{-1}}^*=U_s$ , direct substitution gives

\mathcal J\pi_\alpha(x)\mathcal J=JxJ\otimes1. \tag{I3}

For a group generator, the argument arriving at the innermost section is $sg$ , while its operator coefficient is $U_s^*U_{sg}=U_g$ . The Haar scalars reduce to $\Delta_G(g)^{1/2}$ , so

[\mathcal J\lambda_g\mathcal J\xi](s) =\Delta_G(g)^{1/2}U_g\xi(sg) =[(U_g\otimes R_g)\xi](s). \tag{I4}

These identities hold on all vectors by boundedness. Standard form gives $JMJ=M'$ . Since antiunitary conjugation preserves generated von Neumann algebras, (I2)–(I4) prove

P'=\bigl(M'\otimes1\ \cup\ \{U_g\otimes R_g:g\in G\}\bigr)''. \tag{I5}

There is no abelian, discrete, compact, unimodular, or separability assumption in this standard-representation calculation. The right regular factor in (I4) is essential for a nonunimodular group. Formula (I5) proves the standard-form case of the source's crossed-product commutant theorem X.1.21. Its additional concrete assertion for every possibly nonstandard covariant representation is a separate representation-extension obligation; the intrinsic consequences below need only (I5).

Editable source · Proof dependencies and component terms