Mackey's imprimitivity theorem

Zero case: if $H_0=\{0\}$, take $K=\{0\}$, the zero representation of $H$ and the zero unitary; the induced space is $\{0\}$, the canonical system has $P(G\mathrm{/}H)=I=0$, and the displayed identities hold; the same data give the zero system from the zero representation. Hence assume $H_0\mathrm{\ne }\{0\}$.

Apply [F1] to obtain $W,\mu ,K$ and put $T_g=W{U}_g{W}^{-1}$; [F2] gives the Borel source-variable cocycle fields. Choose $\nu$ and $a=d\mu \mathrm{/}d\nu$ from [F8] and set $Jf=\sqrt{a}f$. Then $J:L^2(\mu ;K)\to L^2(\nu ;K)$ is unitary. Pushing the equality $d\mu =ad\nu$ through the base translation gives $D_g^{\mu }(x)=a(g^{-1}x)a(x{)}^{-1}{D}_g^{\nu }(x)$ almost everywhere, so $J{V}_g^{\mu }{J}^{-1}=V_g^{\nu }$. The scalar $V_g^{\nu }$ is induction of the trivial representation of $H$ and is strongly continuous by [F8]; this extends to $K$-valued sections first on finite sums $f(x)\xi$ and then by their density and unitarity. Hence $V_g^{\mu }$ is strongly continuous. Since $T_g$ is strongly continuous, so is $W_g=(V_g^{\mu }{)}^{-1}{T}_g$, by the triangle inequality and unitary norm bounds.

Reverse direction: given $\sigma :H\to U(K)$ on separable $K$, [F5] endows the induced representation on its covariant completion with the multiplication PVM $P(E)=M_{{\mathbf{1}}_E}$, which is a projection-valued measure with $P(G\mathrm{/}H)=I$ and the covariance identity, hence a transitive system of imprimitivity on $G\mathrm{/}H$; this is the converse construction.

Fix $g_0\in G$, a Borel $E\subseteq G\mathrm{/}H$ with $\mu (E)<\mathrm{\infty }$, and $\xi \in K$. Since ${\mathbf{1}}_E\xi \in L^2(\mu ;K)$ and $W_g=M_{{\phi }_g}$, step 1.2 gives ${\int }_E\mathrm{\parallel }({\phi }_g(x)-{\phi }_{g_0}(x))\xi {\mathrm{\parallel }}^{2}d\mu (x)=\mathrm{\parallel }(W_g-W_{g_0})({\mathbf{1}}_E\xi ){\mathrm{\parallel }}_2^2\to 0$. Therefore $\mu \{x\in E:\mathrm{\parallel }({\phi }_g(x)-{\phi }_{g_0}(x))\xi \mathrm{\parallel }>\epsilon \}\le {\epsilon }^{-2}\mathrm{\parallel }(W_g-W_{g_0})({\mathbf{1}}_E\xi ){\mathrm{\parallel }}_2^2\to 0$. On the unitary group, the strong topology is determined by a countable dense set of vectors in separable $K$: finite-vector tests pass to every vector using $\mathrm{\parallel }(u-v)(\xi -\eta )\mathrm{\parallel }\le 2\mathrm{\parallel }\xi -\eta \mathrm{\parallel }$. Finite unions of the displayed exceptional sets thus prove local convergence in measure in the strong topology. This is the continuity hypothesis required in [F3].

Now [F3] applies with its continuity hypothesis verified by step 2.1 and supplies $B,\sigma$. In the Haar proof the lifted coboundary $b$ has $b(th)=b(t)\sigma (h)$ for each $h$ and almost every $t$. Thus $\sigma (hk)=\sigma (h)\sigma (k)$; integrating the Borel matrix coefficients of $b(t{)}^{-1}b(th)$ against a fixed Haar probability density makes every coefficient of $\sigma$ Borel. Since $K$ is separable, [F8] applies to this Borel homomorphism and gives strong continuity. Invariant Borel descent of $b(s(x)h)\sigma (h{)}^{-1}$ gives $B(x)$, and the resulting section-cocycle formula is strict on all pairs after replacing the fields by their equal a.e. representatives.

Multiplication by the Borel unitaries $B(x)$ is unitary and commutes with indicator multiplications. Put $\mathrm{\Phi }=M_B^{-1}W$. Substituting the factorization from step 3.1 at the source point $g^{-1}x$ gives $(\mathrm{\Phi }{U}_g{\mathrm{\Phi }}^{-1}f)(x)=D_g^{\mu }(x{)}^{1\mathrm{/}2}\sigma (s(x{)}^{-1}gs(g^{-1}x))f(g^{-1}x)$ and $\mathrm{\Phi }P(E){\mathrm{\Phi }}^{-1}=M_{{\mathbf{1}}_E}$. This is the section-coordinate form of the canonical induced action, as in [F4], so $\mathrm{\Phi }$ is the required unitary and may be denoted $W$ in the statement.

Start with the canonical system of $\sigma$. Its source-variable cocycle is ${\phi }_g(x)=\sigma (s(gx{)}^{-1}gs(x))$, so $(B,\sigma )=(I,\sigma )$ is already a factorization. Any recovered representation ${\sigma }^{\mathrm{\prime }}$ is unitarily equivalent to $\sigma$ by the uniqueness clause of [F3]. Conversely step 4.1 reconstructs a system unitarily equivalent to the starting $(U,P)$. Thus the constructions are inverse up to unitary equivalence.

Steps 1.2, 2.1, 3.1 and 4.1 prove the forward direction, including the local-measure continuity and strongly continuous stabilizer action; steps 1.3 and 5.1 give the converse and inverse character up to unitary equivalence. The zero case is covered by step 1.1.