Spectral measure of a unitary representation of an abelian group, covariance, and ergodicity

If $\mathcal{H}=0$, the zero PVM uniquely satisfies the statement, and the irreducibility premise does not hold. Assume $\mathcal{H}\mathrm{\ne }0$. Let $C_j$ be increasing compact sets covering $N$. The closed subspace of $L^2(N)$ supported in $C_j$ is separable by [F7], and its inclusion into $L^1(N)$ is continuous, with norm at most $\mu (C_j{)}^{1\mathrm{/}2}$. Choosing a countable dense family in each such subspace gives a countable $L^1$-dense family: for any $f\in L^1$, the truncations $f{\mathbf{1}}_{C_j}{\mathbf{1}}_{\mathrm{\mid }f\mathrm{\mid }\le r}$ lie in these $L^2$ subspaces and converge to $f$ in $L^1$ as $j,r\to \mathrm{\infty }$. Thus $L^1(N)$ is separable. On its dual unit ball, evaluation on a countable norm-dense family induces the pointwise-evaluation topology, because the norm bounds uniformly control the error of replacing any argument by a dense one. This embeds that ball into a countable product of complex lines. By [F3] the dual $\widehat{N}$ is homeomorphic to its character subspace and therefore second countable; since it is LCH, it is standard Borel by [F7].

For $f\in L^1(N)$ define $\pi (f)\xi ={\int }_{N}f(n)\pi (n)\xi dn$. The integrand is strongly measurable (a.e. limit of $C_c$-approximants times the continuous map $n\mapsto \pi (n)\xi$) and ${\int }_N\mathrm{\parallel }f(n)\pi (n)\xi \mathrm{\parallel }dn=\mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }\xi \mathrm{\parallel }<\mathrm{\infty }$, so $\pi (f)$ is a bounded operator with $\mathrm{\parallel }\pi (f)\mathrm{\parallel }\le \mathrm{\parallel }f{\mathrm{\parallel }}_1$; $f\mapsto \pi (f)$ is linear and multiplicative: for $f,g\in C_c(N)$ Fubini, applicable since the Haar measure is $\sigma$-finite by [F7], gives $\pi (f)\pi (g)={\int }_N{\int }_{N}f(m)g(n)\pi (mn)dmdn=\pi (f\ast g)$, and both sides extend by density; and $\pi (f{)}^{\ast }=\pi (f^{\ast })$ by the adjoint computation and the inversion formula of [F1]. Nondegeneracy: for the approximate identity $e_U$ one has $\pi (e_U)\xi \to \xi$ because $\mathrm{\parallel }\pi (e_U)\xi -\xi \mathrm{\parallel }\le \int e_U(n)\mathrm{\parallel }\pi (n)\xi -\xi \mathrm{\parallel }dn$ and strong continuity makes the integrand small on $\mathrm{supp}{e}_U$ eventually.

Let $A$ be the norm closure of $\pi (L^1(N))$ in $\mathcal{B}(\mathcal{H})$; it is a commutative C*-algebra (the image of the commutative $L^1(N)$ is a commutative $\ast$-algebra) and it is nonzero when $\mathcal{H}\mathrm{\ne }0$ by [step 2.1]. For $y\in \mathrm{\Delta }(A)$ the composite $y\circ \pi$ is a nonzero complex-linear multiplicative functional on $L^1(N)$: if it vanished on the dense subalgebra $\pi (L^1(N))$ then $y=0$ by continuity. Hence by [F3] there is $\chi \in \widehat{N}$ with $y(\pi (f))=\widehat{f}(\chi )$; therefore, using the isometry of [F4], $\mathrm{\parallel }\pi (f)\mathrm{\parallel }={\sup }_{y\in \mathrm{\Delta }(A)}\mathrm{\mid }y(\pi (f))\mathrm{\mid }\le {\sup }_{\chi \in \widehat{N}}\mathrm{\mid }\widehat{f}(\chi )\mathrm{\mid }=\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{\mathrm{\infty }}$.

The assignment $\widehat{f}\mapsto \pi (f)$ is well defined and linear because $\widehat{f}=0$ forces $\pi (f)=0$ by [step 3.1], and it is bounded for the uniform norm; since the Fourier transforms are uniformly dense in $C_0(\widehat{N})$ by [F3], it extends uniquely to a bounded linear map $T:C_0(\widehat{N})\to \mathcal{B}(\mathcal{H})$ with $T(\widehat{f})=\pi (f)$. Multiplicativity and $\ast$-preservation extend from the dense subalgebra of Fourier transforms, using continuity of the products, so $T$ is a nondegenerate star representation: $T(C_0(\widehat{N}))\mathcal{H}$ is dense because $T({\widehat{e}}_U)\xi =\pi (e_U)\xi \to \xi$ for the approximate identity.

By [F5] there is a unique regular PVM $P$ on $\widehat{N}$ with $T(g)={\int }_{\widehat{N}}gdP$ for all $g\in C_0(\widehat{N})$; in particular $\pi (f)={\int }_{\widehat{N}}\widehat{f}dP$ for every $f\in L^1(N)$, and $P(\widehat{N})=I$.

Put $R(n)={\int }_{\widehat{N}}\chi (n)dP(\chi )$, using the PVM just constructed. The PVM calculus gives $R(n)R(m)=R(nm)$ and $R(n{)}^{\ast }R(n)=R(n)R(n{)}^{\ast }=I$. For every sequence $n_j\to n_0$, $\mathrm{\parallel }(R(n_j)-R(n_0))\xi {\mathrm{\parallel }}^2=\int \mathrm{\mid }\chi (n_j)-\chi (n_0){\mathrm{\mid }}^{2}d{P}_{\xi }\to 0$ by dominated convergence; since $N$ is metrizable, this proves strong continuity. The zero Hilbert space has the zero PVM throughout, so the same conclusions hold there.

For $f\in L^1(N)$ one has ${\int }_{N}f(n)R(n)dn=\pi (f)$: the evaluation $(n,\chi )\mapsto \chi (n)$ is jointly continuous: restrict $n$ to a compact neighbourhood of $n_0$ and use uniform convergence of characters there together with continuity of the limiting character. Thus the kernel below is jointly Borel. Pairing with $\eta$ and commuting the bounded functional through the Bochner integral, $⟨{\int }_{N}f(n)R(n)dn\xi ,\eta ⟩={\int }_{N}f(n){\int }_{\widehat{N}}\chi (n)d{P}_{\xi ,\eta }(\chi )dn$, and Fubini, applied to the product of the $\sigma$-finite Haar measure and the finite measure $P_{\xi ,\eta }$ by [F7], identifies this with ${\int }_{\widehat{N}}\widehat{f}d{P}_{\xi ,\eta }=⟨\pi (f)\xi ,\eta ⟩$ by [step 5.1]. Hence the continuous function $h(n)=⟨\pi (n)\xi ,\eta ⟩-⟨R(n)\xi ,\eta ⟩$ satisfies ${\int }_{N}fhdn=0$ for every $f\in L^1(N)$; if $h(n_0)\mathrm{\ne }0$, rotate $h$ by a scalar of modulus one so its value at $n_0$ has positive real part; a nonnegative compactly supported cutoff supported where that real part remains positive has a nonzero integral against $h$, a contradiction, so $h=0$. As $\xi ,\eta$ were arbitrary, $\pi (n)=R(n)$ for every $n$, i.e. $\pi (n)={\int }_{\widehat{N}}\chi (n)dP(\chi )$.

Uniqueness of $P$: if $P^{\mathrm{\prime }}$ is another regular PVM on $\widehat{N}$ with $\pi (n)=\int \chi (n)d{P}^{\mathrm{\prime }}(\chi )$ for all $n$, then for every $f\in L^1(N)$, $\int \widehat{f}d{P}^{\mathrm{\prime }}=\int f(n)\pi (n)dn$ as above, so $\int gd{P}^{\mathrm{\prime }}=\int gdP$ for all $g$ in the uniformly dense algebra of Fourier transforms; both sides are bounded linear in $g\in C_0(\widehat{N})$, so the equality holds on all of $C_0(\widehat{N})$, and [F5] applied to the common representation gives $P^{\mathrm{\prime }}=P$.

Covariance: fix $k\in K$. The map $\chi \mapsto k\cdot \chi$ is a homeomorphism of $\widehat{N}$, so $Q(E):=P(k\cdot E)$ is a regular PVM, and $P_k(E):=\tau (k{)}^{-1}Q(E)\tau (k)$ is again a regular PVM. Its integrated representation is $\int \chi (n)d{P}_k(\chi )=\tau (k{)}^{-1}\int \chi (n)dP(k\cdot \chi )\tau (k)=\tau (k{)}^{-1}\pi ({\alpha }_k(n))\tau (k)=\pi (n)$ for every $n$, where the change of variables in the dual and the covariance relation were used. By [step 7.1] $P_k=P$, that is $\tau (k)P(E)\tau (k{)}^{-1}=P(k\cdot E)$.

Ergodicity: suppose $E$ is Borel and $P(E)$ is invariant under $\tau$, $\tau (k)P(E)\tau (k{)}^{-1}=P(E)$ for all $k$. For every $n$, $\pi (n)P(E)=P(E)\pi (n)$, since $P(E)$ is a spectral projection of $P$ and $\pi (n)=\int \chi (n)dP$. Hence the range of $P(E)$ is a closed subspace invariant under both $\pi (N)$ and $\tau (K)$; if the semidirect-product representation is irreducible, $P(E)=0$ or $P(E)=I$. This is precisely the ergodicity of the measure class of $P$ for the dual action.

Steps 5.1, 7.1 and 8.1 give existence, uniqueness and covariance of the regular PVM $P$, and [step 9.1] gives ergodicity under irreducibility; the zero-dimensional case $\mathcal{H}=\{0\}$ is the zero PVM and is immediate.