# Unitary representations of abelian groups: the spectral theorem

**Lesson HA-LCA-13.** Self-checked by the writing AI.

A strongly continuous unitary representation of an abelian group can be resolved into characters, with orthogonal projections replacing scalar weights. We prove this for an arbitrary locally compact Hausdorff abelian group \(G\) and an arbitrary complex Hilbert space \(\mathcal H\). Neither space is assumed separable or \(\sigma\)-compact.

Write \(\Gamma=\widehat G\), \(\gamma^{-1}=\overline\gamma\), and take inner products linear in the first variable. Our conventions are
\[
 \widehat f(\gamma)=\int_G f(x)\overline{\gamma(x)}\,dx,\qquad
 \pi(f)=\int_G f(x)\pi(x)\,dx,\qquad
 \lambda_x f(y)=f(y-x).                                      \tag{1}
\]
The integrated operator is defined by its matrix coefficients; its existence, star law, nondegeneracy, translate formula and commutant equality are proved in [nonabelian Haar Theorem 7.2](../prerequisites/src/nonabelian-haar-integration.md#ha-lca-pre-nonabelian-haar-theorem-7-2). We use the complete earlier proofs of [the \(L^1\) spectrum](the-dual-group-as-the-gelfand-spectrum-of-l1.md#ha-lca-03-theorem-2-1), [Plancherel](the-plancherel-theorem.md#ha-lca-08-theorem-1-1) and [Pontryagin duality](the-pontryagin-duality-theorem.md#ha-lca-09-theorem-2-1).

The free readings are Karl-Hermann Neeb, [*An Introduction to Unitary Representations of Lie Groups*](https://www.math.fau.de/wp-content/uploads/2024/01/rep.pdf), 2 June 2010, §§6.2–6.4; Jacob Lurie, [*Math 261y, Lecture 5*](https://people.math.harvard.edu/~lurie/261ynotes/lecture5.pdf), 9 September 2011, pp. 1–2; and Gerald Teschl, [*Mathematical Methods in Quantum Mechanics*, free author online edition](https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe2.pdf), second edition, §§2.2, 2.6, 3.1 and 5.1. Every required argument appears here or in an earlier reading linked at its use.

Written and checked by GPT-6 Astra (OpenAI), Ultra, October 2026. The new exposition in this lesson is released under CC0. Separately linked prerequisite readings and their source packages retain their own licences.

## 1. Operator closure

For \(S\subseteq B(\mathcal H)\), put
\[
 S'=\{T:Ts=sT\text{ for every }s\in S\}.
\]
Strong operator convergence means convergence on each vector in norm; weak operator convergence means convergence of every matrix coefficient. These topologies use nets. The von Neumann algebra generated by a self-adjoint set \(S\) is the weakly closed unital star algebra it generates.

<a id="ha-lca-13-lemma-1-1"></a>
**Lemma 1.1 — The double commutant theorem.** If \(A\subseteq B(\mathcal H)\) is a unital star algebra, its strong and weak operator closures both equal \(A''\).

**Proof.** Multiplication on either side by a fixed bounded operator is weakly continuous: for example,
\(\langle Txu,v\rangle=\langle xu,T^*v\rangle\).
Thus commutants are weakly closed. Consequently
\[
 A\subseteq\overline A^{\,s}\subseteq\overline A^{\,w}\subseteq A''.
\]
Let \(T\in A''\) and \(u\in\mathcal H\). The subspace \(M=\overline{Au}\) is invariant under \(A\), as is \(M^\perp\), because \(A=A^*\). Its orthogonal projection \(E\), whose existence is proved in [Hilbert Corollary 2.2](../prerequisites/src/hilbert-spaces-and-unitary-representations.md#ha-lca-pre-hilbert-corollary-2-2), therefore commutes with \(A\). Since \(I\in A\), \(Eu=u\); hence \(ETu=TEu=Tu\). Thus \(Tu\in\overline{Au}\).

To approximate on \(u_1,\ldots,u_n\) simultaneously, act diagonally on \(\mathcal H^n\). A bounded operator on this finite sum has entries \(R_{ij}\in B(\mathcal H)\), obtained by coordinate inclusion and projection. It commutes with every \(\operatorname{diag}(a,\ldots,a)\) precisely when every \(R_{ij}\) commutes with every \(a\in A\). Consequently \(\operatorname{diag}(T,\ldots,T)\) belongs to the double commutant of the diagonal algebra. Apply the preceding one-vector argument to \((u_1,\ldots,u_n)\): for every \(\varepsilon>0\) there is \(a\in A\) with
\[
 \sum_{j=1}^n\|(a-T)u_j\|^2<\varepsilon^2.                  \tag{2}
\]
These are a basis of strong neighbourhoods, so \(T\in\overline A^{\,s}\). The inclusions above are equalities. In particular the weak closure is a unital star algebra, as is \(A''\). \(\square\)

## 2. Completing the group algebra

The full group norm on \(L^1(G)\) is
\(\|f\|_u=\sup_\pi\|\pi(f)\|\), over all strongly continuous unitary representations. The bound \(\|\pi(f)\|\leq\|f\|_1\) makes this supremum finite. The reduced norm is \(\|\lambda(f)\|\). Their completions, after quotienting any zero seminorm, are \(C^*(G)\) and \(C_r^*(G)\).

<a id="ha-lca-13-theorem-2-1"></a>
**Theorem 2.1 — Full and reduced group algebras.** Fourier transformation gives isometric star isomorphisms
\[
 C^*(G)\cong C_0(\Gamma)\cong C_r^*(G),\qquad
 \|f\|_u=\|\lambda(f)\|=\|\widehat f\|_\infty.                \tag{3}
\]

**Proof.** Set \(a=f^**f\). In every integrated representation, \(A=\pi(a)=\pi(f)^*\pi(f)\) is self-adjoint and
\(\|A\|=\|\pi(f)\|^2\). The [operator \(C^*\)-identity](../prerequisites/src/hilbert-spaces-and-unitary-representations.md#ha-lca-pre-hilbert-proposition-3-1), iterated on the self-adjoint powers of \(A\), gives
\[
 \|\pi(f)\|^2
 =\|A^{2^n}\|^{1/2^n}
 \leq\|a^{*\,2^n}\|_1^{1/2^n}
 \longrightarrow r_{L^1(G)^+}(a)
 =\|\widehat f\|_\infty^2.                                  \tag{4}
\]
Here \(a^{*\,k}\) means the \(k\)-fold convolution power, not an iterated involution. The limit is the [Banach spectral-radius formula](../prerequisites/src/banach-spectrum.md#ha-lca-pre-banach-theorem-2-2). The final equality follows from the [character description of the commutative spectrum and its unitization](../prerequisites/src/banach-spectrum.md#ha-lca-pre-banach-theorem-3-4), the earlier \(L^1(G)\) spectrum theorem, and
\(\widehat {f^**f}=|\widehat f|^2\). The additional unitization character, when present, takes value zero on \(a\).

The one-dimensional representation \(x\mapsto\overline{\gamma(x)}\) has integrated value \(\widehat f(\gamma)\), so the reverse universal bound holds. Fourier transformation is a star homomorphism, is injective by [Fourier uniqueness](the-pontryagin-duality-theorem.md#ha-lca-09-corollary-4-2), and has uniformly dense image in \(C_0(\Gamma)\) by [HA-LCA-03, Theorem 3.2](the-dual-group-as-the-gelfand-spectrum-of-l1.md#ha-lca-03-theorem-3-2). This proves the full completion assertion.

The [\(L^1\)-\(L^2\) convolution formula](the-plancherel-theorem.md#ha-lca-08-proposition-2-1) gives
\[
 \mathcal F\lambda(f)\mathcal F^{-1}=M_{\widehat f},
\]
where \(M_hv=hv\). Its norm is \(\|\widehat f\|_\infty\). Indeed the upper bound is immediate. If \(b<\|h\|_\infty\) for continuous \(h\), the open set \(\{|h|>b\}\) is nonempty and has positive Haar measure. Inner regularity supplies a compact subset \(K\) of positive finite measure; testing on \(1_K\) gives \(\|M_h\|\geq b\). These Haar properties are proved in [nonabelian Haar Theorem 3.1](../prerequisites/src/nonabelian-haar-integration.md#ha-lca-pre-nonabelian-haar-theorem-3-1). Let \(b\) increase to the supremum. Thus the reduced and full norms agree, proving (3). \(\square\)

## 3. From continuous functions to spectral projections

A **normalized projection-valued measure**, or PVM, on a locally compact Hausdorff space \(X\) assigns an orthogonal projection \(P(B)\) to each Borel set, with \(P(X)=I\), \(P(\varnothing)=0\), and
\(P(\bigcup_n B_n)u=\sum_nP(B_n)u\) in norm for disjoint Borel sets. It is **regular** when every finite positive measure
\(\mu_u(B)=\langle P(B)u,u\rangle\) is Radon. This is scalar regularity; no countable compact exhaustion of \(X\) is asserted.

<a id="ha-lca-13-lemma-3-0"></a>
**Lemma 3.0 — Bounded spectral integration and regularity.** A PVM determines a unital star homomorphism \(h\mapsto P(h)\) on the bounded Borel functions, satisfying
\[
 \langle P(h)u,v\rangle=\int_X h\,d\mu_{u,v},\qquad
 \|P(h)u\|^2=\int_X|h|^2\,d\mu_u,\qquad
 \|P(h)\|\leq\|h\|_\infty,                                  \tag{5}
\]
where \(\mu_{u,v}(B)=\langle P(B)u,v\rangle\). For a regular PVM these complex measures are finite Radon measures, and a uniformly bounded net converging uniformly on compact subsets yields strongly convergent spectral integrals. For any PVM, the same conclusion holds for a uniformly bounded pointwise convergent sequence.

If \(\mu\) is finite Radon and \(w\geq0\) is integrable, \(w\mu\) is finite Radon. A sum of positive finite Radon measures whose total masses have finite sum is finite Radon.

**Proof.** For disjoint \(B,C\), write \(E=P(B)\), \(F=P(C)\). Since \(E+F\) is a projection, \(EF+FE=0\). Multiplication on the right by \(F\) shows \(EF=-FEF\), which is self-adjoint; therefore \(EF=FE\), and the displayed identity gives \(EF=0\). Decomposing two general sets into their intersection and disjoint differences proves
\(P(B)P(C)=P(B\cap C)\).

For a simple Borel function \(h=\sum_j c_j1_{B_j}\) on a disjoint finite partition, set \(P(h)=\sum_jc_jP(B_j)\). Orthogonality proves (5), multiplication and the adjoint law. Subdividing the bounded range of any Borel function into squares of arbitrarily small diameter gives uniform simple approximations. The operator norm bound extends the definition and all these identities.

Polarization expresses \(\mu_{u,v}\) as a finite linear combination of positive scalar measures. Its variation is finite; more precisely for disjoint \(B_j\),
\[
 \sum_j|\langle P(B_j)u,v\rangle|
 \leq\Big(\sum_j\|P(B_j)u\|^2\Big)^{1/2}
      \Big(\sum_j\|P(B_j)v\|^2\Big)^{1/2}
 \leq\|u\|\|v\|.
\]
Thus its total variation is at most \(\|u\|\|v\|\). Polarization also gives Radon regularity from that of the positive scalar measures; finite variation and linear combinations are covered by [finite-Radon Lemma 4.1 and Theorem 4.3](../prerequisites/src/finite-radon-representation.md#ha-lca-pre-radon-lemma-4-1).

If \(|h_\alpha|,|h|\leq M\), choose compact \(K\) with \(\mu_u(X\setminus K)<\delta\). Then
\[
 \|(P(h_\alpha)-P(h))u\|^2
 \leq \mu_u(K)\sup_K|h_\alpha-h|^2+4M^2\delta.               \tag{6}
\]
This proves the net assertion. For sequences, (5) and the proved [dominated convergence theorem](../prerequisites/src/integration-and-l1.md#ha-lca-pre-integral-theorem-2-2) prove the pointwise assertion.

For weighted regularity, first choose \(M\) with
\(\int(w-\min(w,M))\,d\mu<\varepsilon\). For each Borel \(B\), choose compact \(K\subseteq B\) and open \(O\supseteq B\) with
\(\mu(B\setminus K),\mu(O\setminus B)<\varepsilon/(M+1)\).
Then both weighted losses are less than \(2\varepsilon\). This proves inner and outer regularity. For a summable family of finite measures, only countably many are nonzero: for each positive threshold, only finitely many masses exceed it. Discard a tail of total mass less than \(\varepsilon\), approximate the finitely many remaining measures, and take the union of their inner compact approximations and the intersection of their outer open approximations. The total losses tend to zero. Countable additivity of the sum follows from monotone convergence for nonnegative sums, proved in [integration Theorem 1.2](../prerequisites/src/integration-and-l1.md#ha-lca-pre-integral-theorem-1-2). \(\square\)

<a id="ha-lca-13-theorem-3-1"></a>
**Theorem 3.1 — The spectral measure of a \(C_0\) representation.** Let \(X\) be locally compact Hausdorff and let
\(\rho:C_0(X)\to B(\mathcal H)\) be a nondegenerate star representation, meaning
\(\overline{\operatorname{span}\rho(C_0(X))\mathcal H}=\mathcal H\).
There is a unique normalized regular PVM \(P\) with
\[
 \rho(f)=P(f)\quad(f\in C_0(X)).                             \tag{7}
\]
Moreover
\[
 \rho(C_0(X))'=\{P(B):B\text{ Borel}\}',                     \tag{8}
\]
and there is a unique unital star representation
\(\overline\rho:C_b(X)\to B(\mathcal H)\) extending \(\rho\). It is given by \(\overline\rho(h)=P(h)\).

**Proof.** We first prove contractivity, without assuming it in the definition. If \(M=\|f\|_\infty>0\), put
\(g=(M^2-|f|^2)^{1/2}-M\in C_0(X)\). It is real and
\(g^2+2Mg=-|f|^2\). Applying \(\rho\) gives
\[
 (MI+\rho(g))^*(MI+\rho(g))=M^2I-\rho(f)^*\rho(f)\geq0.
\]
Hence \(\|\rho(f)\|\leq M\). The case \(M=0\) is immediate.

For each compact \(K\subseteq X\), choose \(e_K\in C_c(X)\) with
\(0\leq e_K\leq1\) and \(e_K=1\) on \(K\), using [finite-Radon Lemma 1.3](../prerequisites/src/finite-radon-representation.md#ha-lca-pre-radon-lemma-1-3). Ordered by inclusion of compact sets, these cutoffs satisfy \(e_Kf\to f\) uniformly for every \(f\in C_0(X)\): outside a sufficiently large compact set \(f\) is uniformly small, and on that set the product is exact. Nondegeneracy and the uniform operator bound imply
\[
 \rho(e_K)u\longrightarrow u\quad(u\in\mathcal H).           \tag{9}
\]

For \(v\in\mathcal H\), the cyclic subspace
\(\mathcal H_v=\overline{\rho(C_0(X))v}\) contains \(v\) by (9). It and its orthogonal complement are invariant under \(\rho\), since the algebra is self-adjoint. The functional
\(\ell_v(f)=\langle\rho(f)v,v\rangle\) is bounded and positive: for \(f\geq0\), its square root lies in \(C_0(X)\), and
\(\ell_v(f)=\|\rho(\sqrt f)v\|^2\). The proved [finite positive Riesz theorem](../prerequisites/src/finite-radon-representation.md#ha-lca-pre-radon-theorem-3-1) supplies a finite Radon measure \(\nu_v\) with \(\ell_v(f)=\int f\,d\nu_v\). Thus
\[
 \|\rho(f)v\|^2=\int_X|f|^2\,d\nu_v.                         \tag{10}
\]
By [Hilbert Proposition 6.2](../prerequisites/src/hilbert-spaces-and-unitary-representations.md#ha-lca-pre-hilbert-proposition-6-2), \(C_c(X)\), hence \(C_0(X)\), is dense in \(L^2(\nu_v)\). Formula (10) extends \(f\mapsto\rho(f)v\) to a unitary \(W_v:L^2(\nu_v)\to\mathcal H_v\). On its dense continuous subspace, multiplication by \(a\in C_0(X)\) corresponds to \(\rho(a)\); boundedness extends this to all vectors.

Choose a maximal family of mutually orthogonal nonzero cyclic subspaces, using the maximal principle and the [Hilbert direct-sum construction](../prerequisites/src/hilbert-spaces-and-unitary-representations.md#ha-lca-pre-hilbert-theorem-4-3). Its orthogonal complement is invariant. If nonzero, a vector there produces another nonzero cyclic subspace by (9), contradicting maximality. Hence
\[
 \mathcal H=\bigoplus_{j\in J}\mathcal H_{v_j}
 \cong\bigoplus_{j\in J}L^2(\nu_{v_j}).                       \tag{11}
\]
On this sum define \(P(B)=\bigoplus_j M_{1_B}\). These are projections, normalized, and countably additive strongly: for \(u=(u_j)\), the squared norm of a disjoint-union tail is the tail of the finite measure
\[
 \mu_u(B)=\sum_j\int_B|u_j|^2\,d\nu_{v_j},\qquad
 \mu_u(X)=\sum_j\|u_j\|_2^2=\|u\|^2.
\]
Each vector has only countably many nonzero coordinates, by the finite-sum definition of its squared norm. Lemma 3.0 shows that each weighted measure and their sum are Radon. Thus \(P\) is regular and (7) holds.

If another regular PVM satisfies (7), its scalar measures have the same integrals on \(C_0(X)\). Uniqueness in the finite complex Riesz theorem makes all matrix-coefficient measures equal, so all projections agree.

Suppose \(T\rho(f)=\rho(f)T\) for all \(f\). Then
\[
 \int f\,d\mu_{Tu,v}
 =\langle\rho(f)Tu,v\rangle
 =\langle\rho(f)u,T^*v\rangle
 =\int f\,d\mu_{u,T^*v}.
\]
Complex Riesz uniqueness gives \(P(B)T=TP(B)\). Conversely, commuting with every projection implies commuting with simple spectral integrals and their uniform limits. This proves (8).

Lemma 3.0 supplies \(\overline\rho(h)=P(h)\), with
\(\overline\rho(h)\rho(f)=\rho(hf)\).
Any other star representation extending \(\rho\) satisfies the same identity by multiplication. The vectors \(\rho(f)u\) span a dense subspace, so this identity forces uniqueness. Its value at \(1\) is \(I\) by nondegeneracy. Finally, (9) and Lemma 1.1 show that the von Neumann algebra generated by \(\rho(C_0(X))\) is \(\rho(C_0(X))''\): adjoining \(I\) does not change its strong closure or commutant. \(\square\)

## 4. The spectral theorem for an abelian group

<a id="ha-lca-13-lemma-4-1"></a>
**Lemma 4.1 — An approximate identity on compact dual sets.** Let \(\psi_U\geq0\) be a mass-one continuous compactly supported function with support in a neighbourhood \(U\) of zero. As \(U\) shrinks, \(\widehat{\psi_U}\to1\) uniformly on each compact subset of \(\Gamma\).

**Proof.** Existence of these functions and their approximate-identity property is proved in [nonabelian Haar Corollary 6.2](../prerequisites/src/nonabelian-haar-integration.md#ha-lca-pre-nonabelian-haar-corollary-6-2). For compact \(K\subseteq\Gamma\),
\[
 \sup_{\gamma\in K}|\widehat{\psi_U}(\gamma)-1|
 \leq\sup_{\substack{x\in U\\\gamma\in K}}|\gamma(x)-1|
 \longrightarrow0.                                       \tag{12}
\]
For the last limit, joint continuity of evaluation, proved in [HA-LCA-09, Lemma 1.1](the-pontryagin-duality-theorem.md#ha-lca-09-lemma-1-1), gives for each point of \(K\) a product neighbourhood of \((0,\gamma)\) on which the difference is small. A finite cover of \(K\) and intersection of the corresponding neighbourhoods of zero work uniformly. This is a net argument and requires no first countability. \(\square\)

<a id="ha-lca-13-theorem-4-2"></a>
**Theorem 4.2 — Stone–Naimark–Ambrose–Godement.** Every strongly continuous unitary representation \(\pi\) of \(G\) on \(\mathcal H\) has a unique normalized regular PVM \(P\) on \(\Gamma\) such that
\[
 \pi(x)=P(\gamma\mapsto\gamma(x)),\qquad
 \pi(f)=P(\gamma\mapsto\widehat f(\overline\gamma)).
                                                                  \tag{13}
\]
Conversely every normalized regular PVM on \(\Gamma\) gives a strongly continuous unitary representation by the first formula. In either case
\[
 \pi(G)'=\pi(L^1(G))'=\{P(B):B\text{ Borel}\}'.              \tag{14}
\]

**Proof.** By Theorem 2.1, the assignment
\[
 \rho(\widehat f\circ\mathrm{inv})=\pi(f),
 \qquad \mathrm{inv}(\gamma)=\overline\gamma,
\]
extends to a star representation of \(C_0(\Gamma)\). It is well-defined and bounded by (3); the transformed \(L^1\) functions are uniformly dense. It is nondegenerate because the integrated representation is. Theorem 3.1 supplies \(P\) and the second formula in (13).

Put \(e_x(\gamma)=\gamma(x)\). Fourier translation gives
\[
 \widehat{L_x\psi_U}(\overline\gamma)
 =\gamma(x)\widehat{\psi_U}(\overline\gamma).
\]
The integrated translate identity and multiplicativity of spectral integration therefore give
\[
 \pi(x)\pi(\psi_U)
 =\pi(L_x\psi_U)
 =P(e_x)\pi(\psi_U).
\]
The integrated approximate identity converges strongly to \(I\), so \(\pi(x)=P(e_x)\). Equivalently, Lemma 4.1 and the compact-tail estimate (6) prove that same strong limit directly from \(P\). This proves existence without applying pointwise dominated convergence to a net.

Conversely, \(P(e_x)\) is unitary and multiplicative because \(e_{x+y}=e_xe_y\) and \(|e_x|=1\). For fixed \(u\),
\[
 \|(P(e_x)-I)u\|^2=\int_\Gamma|\gamma(x)-1|^2\,d\mu_u(\gamma)
 \longrightarrow0
\]
by uniform convergence on compact subsets, proved exactly as in Lemma 4.1, and (6). Continuity at all group elements follows by unitarity.

Its integrated formula follows by interchanging the \(x\) and \(\gamma\) integrals:
\[
 \int_G f(x)\langle P(e_x)u,v\rangle\,dx
 =\int_\Gamma\left(\int_G f(x)\gamma(x)\,dx\right)d\mu_{u,v}(\gamma).
                                                                  \tag{15}
\]
Here are the needed measure hypotheses. Choose a Borel representative of \(f\) vanishing outside an open \(\sigma\)-compact subgroup, as in [nonabelian Haar Lemma 5.1](../prerequisites/src/nonabelian-haar-integration.md#ha-lca-pre-nonabelian-haar-lemma-5-1). Haar measure on that subgroup is \(\sigma\)-finite Radon, while \(|\mu_{u,v}|\) is finite Radon. Joint evaluation is continuous, and the integrand is bounded in absolute integral by \(\|f\|_1\|\mu_{u,v}\|\). Thus the proved [full-Borel \(\sigma\)-finite Fubini theorem](../prerequisites/src/integration-and-l1.md#ha-lca-pre-integral-theorem-4-4), applied also to the real and imaginary signed parts of \(\mu_{u,v}\), justifies (15). The inner integral is \(\widehat f(\overline\gamma)\).

If two regular PVMs give the same group representation, (15) makes their integrals agree on the dense transformed \(L^1\) subalgebra of \(C_0(\Gamma)\). Theorem 3.1 gives equality. Finally, the first commutant equality in (14) is the earlier integrated-representation theorem; the second follows from uniform density and (8). \(\square\)

<a id="ha-lca-13-corollary-4-3"></a>
**Corollary 4.3 — Recovering a group representation from \(L^1(G)\).** Integration is a bijection between strongly continuous unitary representations of \(G\) and bounded nondegenerate star representations of \(L^1(G)\), on a fixed Hilbert space.

**Proof.** The forward construction and its properties are the earlier nonabelian Haar Theorem 7.2. For the converse let \(\theta:L^1(G)\to B(\mathcal H)\) be bounded, nondegenerate and star preserving. If \(\mathcal H=\{0\}\), both representations are uniquely determined. Otherwise its norm \(C=\|\theta\|\) is positive. The proof of (4), now with
\(\|\theta(a^{*\,2^n})\|\leq C\|a^{*\,2^n}\|_1\), gives
\(\|\theta(f)\|\leq\|\widehat f\|_\infty\), since \(C^{1/2^n}\to1\). Hence
\(\rho(\widehat f\circ\mathrm{inv})=\theta(f)\) extends to a nondegenerate star representation of \(C_0(\Gamma)\). Its regular PVM from Theorem 3.1 gives, by the converse part of Theorem 4.2, a strongly continuous group representation whose integrated operators are exactly \(\theta(f)\). If two group representations have these operators, then for each \(x\),
\(\pi(x)\pi(\psi_U)=\pi(L_x\psi_U)\); the common right side and the strong limit \(\pi(\psi_U)\to I\) make their values at \(x\) equal. Thus integration is injective as well. \(\square\)

## 5. Multiplication operators and the regular representation

All \(L^p\) spaces below use the full Haar measure constructed earlier, including its locally completed measurable sets. This matters when the group is not \(\sigma\)-compact.

<a id="ha-lca-13-lemma-5-1"></a>
**Lemma 5.1 — Maximal abelianness on full Haar measure.** For an LCA group \(X\), let
\(\mathcal M=\{M_h:h\in L^\infty(X)\}\subseteq B(L^2(X))\). Then
\[
 \|M_h\|=\|h\|_\infty,\qquad
 \mathcal M'=\mathcal M,\qquad
 \{M_f:f\in C_0(X)\}''=\mathcal M.                           \tag{16}
\]
Thus \(\mathcal M\) is a maximal abelian von Neumann algebra.

**Proof.** Use the disjoint measurable partition \(X=\coprod_{i\in I}E_i\) into sets of finite measure constructed in [dual-convex Lemma 1.1](../prerequisites/src/dual-balls-and-compact-convexity.md#ha-lca-pre-dual-convex-lemma-1-1). That lemma proves arbitrary measurable gluing, gluing of null sets, and
\[
 \int_X w\,d\mu=\sum_i\int_{E_i}w\,d\mu\quad(w\geq0).
                                                                  \tag{17}
\]
In particular finite-piece truncations are dense in \(L^2(X)\), by applying (17) to \(|u|^2\). If \(b<\|h\|_\infty\), the positive-measure set \(\{|h|>b\}\) meets some \(E_i\) in positive measure. Its indicator tests \(\|M_h\|\geq b\). The reverse inequality follows by integration, proving the norm assertion.

Suppose \(T\in\mathcal M'\), and put \(h_i=T1_{E_i}\). Commutation with \(M_{1_{E_i}}\) shows that \(h_i\) vanishes off \(E_i\). For every measurable \(F\subseteq E_i\),
\[
 T1_F=1_Fh_i,\qquad
 \int_F|h_i|^2\,d\mu\leq\|T\|^2\mu(F).
\]
Taking \(F=\{x\in E_i:|h_i(x)|>\|T\|+\varepsilon\}\) shows that it is null, and then a countable union over positive rational \(\varepsilon\) gives \(|h_i|\leq\|T\|\) almost everywhere. Choose bounded representatives and glue them to a measurable bounded \(h\) by the cited lemma. Equality \(T=M_h\) holds on simple functions in each finite piece, hence on all of \(L^2(E_i)\), and finally on \(L^2(X)\) by finite-piece density. To justify the simple-function step explicitly, truncate \(u\in L^2(E_i)\) on \(\{|u|\leq n\}\); dominated convergence makes the omitted \(L^2\) tail tend to zero. Subdivide the bounded range of the truncation into squares of diameter \(\delta\). The resulting measurable simple function has \(L^2\) error at most \(\delta\mu(E_i)^{1/2}\). Thus simple functions are dense. We have \(\mathcal M'\subseteq\mathcal M\); the reverse inclusion follows from pointwise commutativity. It follows also that \(\mathcal M=\mathcal M'\) is weakly closed, by the commutant argument in Lemma 1.1.

It remains to show that continuous multipliers generate all of \(\mathcal M\). Fix \(h\in L^\infty(X)\), with a representative bounded by \(M\), and finitely many \(u_1,\ldots,u_n\in L^2(X)\). Put
\(w=\sum_j|u_j|^2\in L^1(X)\). The positive finite measure \(d\nu=w\,d\mu\) on Borel sets is Radon by [HA-LCA-03, Lemma 4.1](the-dual-group-as-the-gelfand-spectrum-of-l1.md#ha-lca-03-lemma-4-1). Both \(w\) and \(hw\) have Borel representatives by the earlier Haar support lemma. Dividing these representatives where \(w>0\), and setting the quotient to zero elsewhere, gives a Borel version of \(h\) for \(\nu\); modify it on its \(\nu\)-null exceptional set by clipping to the disk of radius \(M\). This justifies using \(h\) as an element of \(L^2(\nu)\), even though the original full-Haar representative need not be globally Borel.

By Hilbert Proposition 6.2, choose \(a\in C_c(X)\) approximating \(h\) in \(L^2(\nu)\). Clip \(a\) radially to the closed disk of radius \(M\), obtaining \(b\in C_c(X)\) with \(\|b\|_\infty\leq M\). For \(M=0\) take \(b=0\). Since \(|h|\leq M\),
\[
 |b-h|\leq|b-a|+|a-h|\leq2|a-h|.
\]
Consequently the approximation can achieve
\[
 \sum_{j=1}^n\|(M_b-M_h)u_j\|_2^2
 =\int_X|b-h|^2\,d\nu<\varepsilon^2.                        \tag{18}
\]
Directing by finite vector sets and tolerances proves strong approximation of \(M_h\) by continuous compactly supported multipliers, with the common norm bound \(M\).

An operator commuting with all continuous multipliers therefore commutes with every \(M_h\), by passing to the strong limit on \(u\) and on \(Tu\). The continuous-multiplier commutant is exactly \(\mathcal M'=\mathcal M\). Taking commutants proves the last identity in (16). Any abelian algebra containing \(\mathcal M\) lies in \(\mathcal M'=\mathcal M\), proving maximality. \(\square\)

<a id="ha-lca-13-theorem-5-2"></a>
**Theorem 5.2 — The abelian group von Neumann algebra.** Under Plancherel,
\[
 \mathcal F\lambda_x\mathcal F^{-1}
   =M_{\gamma\mapsto\overline{\gamma(x)}},\qquad
 VN(G):=\lambda(G)''
   =\mathcal F^{-1}\mathcal M_\Gamma\mathcal F.               \tag{19}
\]
In particular \(VN(G)\) is maximal abelian on \(L^2(G)\). The regular representation's SNAG measure is
\[
 P_\lambda(B)=\mathcal F^{-1}M_{1_{\mathrm{inv}(B)}}\mathcal F,
 \qquad B\subseteq\Gamma\text{ Borel}.                       \tag{20}
\]

**Proof.** For \(f\in L^1(G)\cap L^2(G)\), translation in the Fourier integral gives
\(\widehat{\lambda_xf}(\gamma)=\overline{\gamma(x)}\widehat f(\gamma)\), as proved in [HA-LCA-03, Theorem 3.1](the-dual-group-as-the-gelfand-spectrum-of-l1.md#ha-lca-03-theorem-3-1). Density in \(L^2\), from Hilbert Proposition 6.2, extends the identity to all vectors.

The integrated commutant theorem and the norm-dense Fourier image give
\[
 \mathcal F\lambda(G)'\mathcal F^{-1}
 =\{M_{\widehat f}:f\in L^1(G)\}'
 =\{M_a:a\in C_0(\Gamma)\}'
 =\mathcal M_\Gamma.
\]
Lemma 5.1 and taking commutants prove (19). Lemma 1.1 identifies this double commutant with the generated von Neumann algebra.

The projections in (20) are normalized and countably additive by multiplication on \(L^2(\Gamma)\). Their scalar measures are the images under inversion of the finite Radon measures
\(|\widehat u|^2\,d\gamma\). These are Radon by the earlier \(L^1\)-density measure lemma; inversion is a homeomorphism, so compact and open approximations remain such. Thus (20) is regular. Integrating \(\eta\mapsto\eta(x)\) against (20) yields multiplication by \(\overline{\gamma(x)}\). Uniqueness in Theorem 4.2 proves the assertion. The inversion in (20) is required by the sign convention (1). \(\square\)

## 6. The precise input for Weyl systems

<a id="ha-lca-13-theorem-6-1"></a>
**Theorem 6.1 — The conjugated-character convention.** Let
\(V:\widehat G\to\mathcal U(\mathcal H)\) be strongly continuous, with no separability assumption on \(\mathcal H\). There is a unique nondegenerate star representation
\(\rho:C_0(G)\to B(\mathcal H)\) whose extension to \(C_b(G)\) satisfies
\[
 \overline\rho(\overline\chi)=V_\chi
 \quad(\chi\in\widehat G),\qquad
 \overline\chi(x)=\overline{\chi(x)}.                        \tag{21}
\]
Moreover
\[
 \rho(C_0(G))'=V(\widehat G)',\qquad
 \rho(C_0(G))''=V(\widehat G)''.                             \tag{22}
\]
The last equality is equality of their generated von Neumann algebras.

**Proof.** Apply Theorem 4.2 to \(\widehat G\), and identify its dual with \(G\) by the proved Pontryagin theorem. The resulting regular PVM \(Q\) on \(G\) satisfies
\(V_\chi=Q(x\mapsto\chi(x))\).
Define
\[
 \rho(f)=Q(x\mapsto f(-x)).
\]
This is a star representation. It is nondegenerate: the cutoffs of Theorem 3.1, composed with negation, converge uniformly to one on compact sets and are bounded by one, so their \(Q\)-integrals converge strongly to \(I\). Its extension is
\(\overline\rho(h)=Q(h\circ(-\mathrm{id}))\). Since
\(\overline{\chi(-x)}=\chi(x)\), (21) follows.

Conversely, given any representation satisfying (21), Theorem 3.1 supplies its unique regular PVM \(R\). The reflected measure \(Q_1(B)=R(-B)\) satisfies
\[
 Q_1(x\mapsto\chi(x))=R(x\mapsto\chi(-x))
 =\overline\rho(\overline\chi)=V_\chi.
\]
SNAG uniqueness makes \(Q_1=Q\), so \(\rho\) is unique.

The PVM of \(\rho\) is \(B\mapsto Q(-B)\). Reflection permutes all Borel sets, so its set of projections has the same commutant as that of \(Q\). Theorems 3.1 and 4.2 therefore give the first equality in (22). Taking commutants gives the second. By (9), \(I\) lies in the strong closure of \(\rho(C_0(G))\); Lemma 1.1 identifies its double commutant with its generated algebra. The group side already contains \(I\). The proof uses arbitrary vector families and finite simultaneous approximations, and imposes no countability condition. \(\square\)

## 7. Unbounded generators and familiar cases

For clarity, a densely defined operator \(A\) is **symmetric** when
\(\langle Au,v\rangle=\langle u,Av\rangle\) on its domain. Its adjoint is defined on the vectors \(v\) for which
\(\langle Au,v\rangle=\langle u,w\rangle\) for all \(u\in D(A)\), for some \(w\); density makes \(w\) unique, and \(A^*v=w\). It is **self-adjoint** when \(A=A^*\), including equality of domains.

<a id="ha-lca-13-lemma-7-0"></a>
**Lemma 7.0 — Real measurable spectral integrals.** Let \(P\) be a PVM and \(h:X\to\mathbb R\) a finite Borel function. Put
\[
 D(A)=\left\{u:\int_Xh^2\,d\mu_u<\infty\right\},\qquad
 Au=\lim_{n\to\infty}P(h1_{\{|h|\leq n\}})u.                \tag{23}
\]
Then \(A\) is densely defined and self-adjoint. The operators
\(U_t=P(e^{ith})\) form a strongly continuous unitary group, and
\[
 D(A)=\left\{u:\lim_{t\to0}\frac{U_tu-u}{t}
                    \text{ exists in norm}\right\},\qquad
 \left.\frac{d}{dt}U_tu\right|_{t=0}=iAu.                    \tag{24}
\]

**Proof.** Write \(E_n=\{|h|\leq n\}\), \(A_n=P(h1_{E_n})\). The measure inequalities
\(\mu_{\alpha u}=|\alpha|^2\mu_u\) and
\(\mu_{u+v}\leq2\mu_u+2\mu_v\), obtained from norms of projections, show that \(D(A)\) is linear. Formula (5) shows that \(A_nu\) is Cauchy exactly when the corresponding \(h^2\)-tails tend to zero, and gives
\[
 \|Au\|^2=\int h^2\,d\mu_u.
\]
The identity
\(d\mu_{P(g)u}=|g|^2\,d\mu_u\) follows by multiplying the bounded spectral integrals. Hence \(P(E_n)u\in D(A)\), and these vectors tend to \(u\). The domain is dense. Since each \(A_n\) is bounded self-adjoint, passing to limits in its matrix coefficients proves symmetry of \(A\).

The bounded operators
\[
 R_\pm=P((h\pm i)^{-1})
\]
map \(\mathcal H\) into \(D(A)\), because
\(\int h^2\,d\mu_{R_\pm u}
 =\int h^2/(h^2+1)\,d\mu_u\leq\|u\|^2\).
Multiplying the bounded truncations and passing to the limit gives
\[
 (A\pm i)R_\pm=I,\qquad R_\pm(A\pm i)u=u\quad(u\in D(A)).
                                                                  \tag{25}
\]
For example \(AR_\pm=P(h/(h\pm i))\), by dominated convergence, which proves the first identity; commuting \(R_\pm\) with \(A_n\) proves the second.

For any densely defined \(B\), its adjoint definition directly gives
\((\operatorname{ran}B)^\perp=\ker B^*\). In particular
\(\ker(A^*-i)=(\operatorname{ran}(A+i))^\perp=\{0\}\).
If \(v\in D(A^*)\), let \(w=R_-(A^*-i)v\in D(A)\).
Then \((A^*-i)(v-w)=0\), so \(v=w\in D(A)\). Symmetry gives the reverse domain inclusion, proving self-adjointness. This argument supplies the onto-resolvent criterion rather than citing it.

Multiplicativity of bounded spectral integration gives the group and unitary laws for \(U_t\). For strong continuity, split the finite measure \(\mu_u\) into \(E_n\) and its complement. The latter has mass tending to zero; on \(E_n\), \(e^{ith}\to1\) uniformly as \(t\to0\). Formula (5) proves continuity.

For \(u\in D(A)\), elementary integration of the scalar exponential, proved in [Banach calculus Lemma 1.2](../prerequisites/src/banach-spectrum.md#ha-lca-pre-banach-lemma-1-2), gives
\[
 \left|\frac{e^{ith}-1}{t}\right|\leq|h|,
 \qquad \frac{e^{ith}-1}{t}\longrightarrow ih.
\]
Formula (5), the defining truncation limit and dominated convergence with bound \(4h^2\) prove (24) in this direction. More explicitly, subtract \(iA_nu\), apply (5), and pass \(n\to\infty\) in the \(L^2(\mu_u)\) expressions. Conversely, if the norm derivative exists, its difference quotients have bounded norms. Along any nonzero real sequence \(t_k\to0\), [Fatou's lemma](../prerequisites/src/integration-and-l1.md#ha-lca-pre-integral-corollary-1-3) gives
\[
 \int h^2\,d\mu_u
 \leq\liminf_k\int\left|\frac{e^{it_kh}-1}{t_k}\right|^2d\mu_u
 <\infty.
\]
Thus \(u\in D(A)\), and the derivative has the already computed value. \(\square\)

<a id="ha-lca-13-corollary-7-1"></a>
**Corollary 7.1 — Stone, unitary operators and compact groups.**

1. Every strongly continuous unitary group \(U_t\), \(t\in\mathbb R\), has a unique self-adjoint generator \(H\) with \(U_t=e^{itH}\). If \(P\) is its SNAG measure in the coordinate \(\gamma_\xi(t)=e^{2\pi it\xi}\), then
   \[
    H=P(2\pi\xi),\qquad
    D(H)=\left\{u:\int_{\mathbb R}(2\pi\xi)^2\,d\mu_u(\xi)
                       <\infty\right\}.                    \tag{26}
   \]
   Every self-adjoint \(H\) gives such a group, whose norm derivative domain is exactly \(D(H)\).

2. For every unitary operator \(W\), there is a unique normalized regular PVM \(P_W\) on the complex unit circle with
   \[
    W=P_W(z),\qquad W^n=P_W(z^n)\quad(n\in\mathbb Z).
                                                                  \tag{27}
   \]

3. If \(G\) is compact, every representation has the orthogonal decomposition
   \[
    \mathcal H=\bigoplus_{\gamma\in\widehat G}\mathcal H_\gamma,
    \qquad
    \mathcal H_\gamma
     =\{u:\pi(x)u=\gamma(x)u\text{ for all }x\in G\}
     =P(\{\gamma\})\mathcal H.                               \tag{28}
   \]
   If \(G\) is discrete, its spectral PVM is on the compact space \(\widehat G\).

**Proof.** The characters of \(\mathbb R\) and \(\mathbb Z\), including their dual topologies, were proved in [HA-LCA-02, Theorem 3.1 and Corollary 3.2](characters-and-the-dual-group.md#ha-lca-02-theorem-3-1). SNAG for \(\mathbb R\), followed by Lemma 7.0 with \(h(\xi)=2\pi\xi\), gives (26), self-adjointness and the exact derivative domain. SNAG for the automatically continuous representation \(n\mapsto W^n\) of discrete \(\mathbb Z\) gives (27). A PVM with \(W=P_W(z)\) also gives every integral power by multiplicativity and adjoints, so SNAG uniqueness proves its uniqueness.

To include the converse in the first assertion, we construct the calculus of an arbitrary self-adjoint \(H\). An adjoint is closed: if \(v_n\to v\) and \(A^*v_n\to w\), then
\(\langle Au,v\rangle=\langle u,w\rangle\) for all \(u\in D(A)\), so \(v\in D(A^*)\) and \(A^*v=w\). Thus \(H\) is closed. Symmetry gives
\[
 \|(H\pm i)u\|^2=\|Hu\|^2+\|u\|^2.                         \tag{29}
\]
The ranges of \(H\pm i\) are closed: convergence of their values implies, by (29), convergence both of \(u\) and of \(Hu\), and closedness of \(H\) supplies a limit in the domain. The orthogonal complements of those ranges are
\(\ker(H\mp i)=\{0\}\), again by (29). Hence both ranges are all of \(\mathcal H\), and their inverses have norm at most one.

The Cayley operator
\[
 W=(H-i)(H+i)^{-1}=I-2i(H+i)^{-1}
\]
is an isometry by (29), and is onto because \(H-i\) is onto. It is unitary. Also \(\ker(W-I)=0\), since \((H+i)^{-1}\) is injective. Apply (27) to \(W\). Formula (5) shows that
\(\ker(W-I)=P_W(\{1\})\mathcal H\): the integral of \(|z-1|^2\) vanishes exactly when the scalar measure is concentrated at \(1\). Thus \(P_W(\{1\})=0\).

Set \(h(z)=i(1+z)/(1-z)\) for \(z\ne1\), and \(h(1)=0\). This is Borel and real: for \(|z|=1\), replacing \(\overline z\) by \(z^{-1}\) in its conjugate gives the same expression. Lemma 7.0 defines a self-adjoint \(A=P_W(h)\). Outside the null spectral set \(\{1\}\),
\[
 (h(z)+i)^{-1}=\frac{1-z}{2i}.
\]
Consequently
\[
 (A+i)^{-1}=P_W((1-z)/(2i))=(H+i)^{-1}.
\]
Equality of these inverses gives equality of their ranges \(D(A)=D(H)\), and then \(A=H\). Define \(e^{itH}=P_W(e^{ith})\). Lemma 7.0 proves that this is a strongly continuous group with precisely generator \(iH\).

For completeness, a group is determined by this generator. If \(U_t\) has derivative \(iH\) on exactly \(D(H)\), then its group law implies \(U_tD(H)=D(H)\) and \(HU_tu=U_tHu\), by taking difference quotients and then using \(U_{-t}\). The constructed group \(V_t=e^{itH}\) has the same properties. For \(u\in D(H)\), the norm derivative of \(U_{-t}V_tu\) is zero: the two product-rule terms are \(-iU_{-t}HV_tu\) and \(iU_{-t}HV_tu\). The product rule here follows by adding and subtracting \(U_{-(t+s)}V_tu\), using unitary norm bounds and the two strong derivatives. Scalar fundamental calculus applied to every matrix coefficient makes the vector constant. Hence \(U_{-t}V_tu=u\), and density of \(D(H)\) gives \(U_t=V_t\). Conversely the domain and value of \(H\) are uniquely recovered from the derivative (24). This completes both directions of Stone's theorem.

If \(G\) is compact, its dual is discrete by [HA-LCA-02, Theorem 2.1](characters-and-the-dual-group.md#ha-lca-02-theorem-2-1). A compact subset of a discrete space is finite, since its singleton cover has a finite subcover. Regularity of \(\mu_u\) therefore gives
\[
 \mu_u(\Gamma)=\sup_{F\subseteq\Gamma,\ F\text{ finite}}\mu_u(F).
\]
Thus the finite sums \(\sum_{\gamma\in F}P(\{\gamma\})u\) converge to \(u\). The summands are orthogonal, proving the direct sum in (28); each lies in the stated eigenspace by (13). Conversely, if \(u\) is a \(\gamma\)-eigenvector, for each \(\eta\ne\gamma\) choose \(x\) with \(\eta(x)\ne\gamma(x)\). Commuting \(P(\{\eta\})\) with \(\pi(x)\) gives
\[
 (\eta(x)-\gamma(x))P(\{\eta\})u=0.
\]
Every other component vanishes, and the finite-sum limit then gives \(u=P(\{\gamma\})u\). The discrete-group assertion is the other half of the same earlier compact/discrete duality theorem. \(\square\)

<a id="ha-lca-13-example-7-2"></a>
**Example 7.2 — Translation, rotation and signs.** For \(G=\mathbb R\), with
\(\widehat f(\xi)=\int f(s)e^{-2\pi is\xi}\,ds\), left translation has
\[
 P_\lambda(B)=\mathcal F^{-1}M_{1_{-B}}\mathcal F,\qquad
 \mathcal F H\mathcal F^{-1}=-2\pi M_\xi,\qquad
 D(H)=\{f:\xi\widehat f(\xi)\in L^2(\mathbb R)\}.             \tag{30}
\]
On \(C_c^1(\mathbb R)\), \(Hf=if'\). For left rotation on \(L^2(\mathbb T)\), the character \(e_n(s)=e^{2\pi ins}\) spans the eigenspace with representation character \(t\mapsto e^{-2\pi int}\).

**Proof.** Theorem 5.2 gives (30)'s projection formula, because inversion on the dual coordinate is \(\xi\mapsto-\xi\). Integrating \(2\pi\eta\) against this reflected measure, with the truncations of Lemma 7.0, gives multiplication by \(-2\pi\xi\) and the displayed domain.

For \(f\in C_c^1(\mathbb R)\), both \(f\) and \(f'\) belong to \(L^1\cap L^2\). Integration by parts, as proved in [real calculus Lemma 1.3](../prerequisites/src/real-variable-calculations.md#ha-lca-pre-real-lemma-1-3), gives
\(\widehat{f'}(\xi)=2\pi i\xi\widehat f(\xi)\), with zero boundary terms. Plancherel then puts \(f\) in the domain in (30), and
\(\widehat{if'}=-2\pi\xi\widehat f\). Fourier injectivity in \(L^2\) gives \(Hf=if'\), consistent with
\(\left.\frac{d}{dt}f(s-t)\right|_{t=0}=-f'(s)=iHf(s)\).

On the circle, direct substitution gives
\(\lambda_t e_n=e^{-2\pi int}e_n\). The [compact-group Fourier basis theorem](the-plancherel-theorem.md#ha-lca-08-corollary-4-1) makes the \(e_n\) a complete orthonormal family for probability Haar measure; therefore each eigenspace is exactly its stated line. Right rotation \(f(s)\mapsto f(s+t)\) instead has eigencharacter \(e^{2\pi int}\). \(\square\)

The convention translations, all proved above, are:

| Object | Convention or resulting formula |
|---|---|
| Fourier transform | \(\widehat f(\gamma)=\int f(x)\overline{\gamma(x)}\,dx\) |
| Integrated representation | \(\pi(f)=\int f(x)\pi(x)\,dx=P(\widehat f\circ\mathrm{inv})\) |
| SNAG coordinate | \(\pi(x)=P(\gamma\mapsto\gamma(x))\) |
| Left translation after Plancherel | \(\mathcal F\lambda_x\mathcal F^{-1}=M_{\overline{\gamma(x)}}\) |
| Weyl-system convention | \(\overline\rho(\overline\chi)=V_\chi\), using reflection of the SNAG measure on \(G\) |
| Stone frequency | \(\gamma_\xi(t)=e^{2\pi it\xi}\), \(H=P(2\pi\xi)\), \(U_t=e^{itH}\) |
| Norm derivative generator | \(D= iH\) on \(D(H)\) |
| An \(e^{-itA}\) convention | \(A=-H\), with the same domain |
| Left translation on \(\mathbb R\) | \(H=if'\) on \(C_c^1\); the full domain is (30) |

## 8. Exercises with complete solutions

<a id="ha-lca-13-exercise-8-1"></a>
**Exercise 8.1 — An uncountable discrete spectrum.** Prove the compact-group decomposition without assuming that \(\widehat G\) is countable. Determine how many components a single vector can have.

**Solution.** Fix \(u\). Its scalar measure is finite Radon on the discrete dual. For every \(\varepsilon>0\), inner regularity gives a finite \(F\) with
\[
 \left\|u-\sum_{\gamma\in F}P(\{\gamma\})u\right\|^2
 =\mu_u(\Gamma\setminus F)<\varepsilon.
\]
This proves convergence along the directed set of all finite subsets. Orthogonality gives
\(\sum_\gamma\|P(\{\gamma\})u\|^2=\|u\|^2\).
For each positive integer \(k\), only finitely many terms can exceed \(1/k\), so only countably many terms are nonzero. The family of possible nonzero spaces over all vectors can still be uncountable. The eigenvector and converse projection calculations in Corollary 7.1 identify each summand with \(\mathcal H_\gamma\), proving the assertion. \(\square\)

<a id="ha-lca-13-exercise-8-2"></a>
**Exercise 8.2 — Stone's domain and normalization.** Starting with SNAG for \(\mathbb R\), identify the self-adjoint operator, its domain, the infinitesimal generator and the change for an \(e^{-itA}\) convention.

**Solution.** SNAG writes \(U_t=P(e^{2\pi it\xi})\). Apply Lemma 7.0 with \(h=2\pi\xi\). This gives \(H=P(2\pi\xi)\) on exactly the vectors with finite \(\int(2\pi\xi)^2\,d\mu_u\). The bounded inverses \(P((2\pi\xi\pm i)^{-1})\), and the adjoint-domain argument (25), prove self-adjointness. The difference quotient converges to \(iHu\) precisely on this domain; the reverse implication follows from Fatou, not merely a formal differentiation under the integral.

Thus the character-frequency operator is \(H/(2\pi)\), while the norm derivative generator is \(iH\). Writing the same group as \(e^{-itA}\) gives \(A=-H\), since equality of derivatives implies \(-iA=iH\) on the same derivative domain. For left translation, the Plancherel multiplier is \(e^{-2\pi it\xi}\), so \(H=-2\pi\mathcal F^{-1}M_\xi\mathcal F\), in agreement with \(Hf=if'\) on \(C_c^1\). \(\square\)

<a id="ha-lca-13-exercise-8-3"></a>
**Exercise 8.3 — Recover the group \(C^*\)-norm from SNAG.** Give a second proof of (3) using the spectral theorem.

**Solution.** By (13) and (5),
\[
 \|\pi(f)\|=\|P(\widehat f\circ\mathrm{inv})\|
 \leq\|\widehat f\|_\infty
\]
in every representation. For each \(\gamma\), the one-dimensional representation \(x\mapsto\overline{\gamma(x)}\) has integrated value \(\widehat f(\gamma)\). Taking suprema gives \(\|f\|_u=\|\widehat f\|_\infty\). The earlier Fourier laws give preservation of multiplication and involution, Fourier uniqueness gives injectivity, and HA-LCA-03, Theorem 3.2 gives uniform density in \(C_0(\Gamma)\). Extending the isometry to the completion gives the full algebra identification. Under Plancherel, the reduced integrated operator is \(M_{\widehat f}\); its norm is the same by Lemma 5.1 and the continuous full-support argument in Theorem 2.1. Thus the reduced completion is identical as well. No additional representation theorem for abstract \(C^*\)-algebras is needed for this calculation. \(\square\)

<a id="ha-lca-13-exercise-8-4"></a>
**Exercise 8.4 — Equivalence of spectral measures.** Show that two group representations are unitarily equivalent if and only if their SNAG PVMs are unitarily equivalent.

**Solution.** Suppose \(W:\mathcal H_1\to\mathcal H_2\) is unitary and
\(W\pi_1(x)W^*=\pi_2(x)\). Set \(Q(B)=WP_1(B)W^*\). It is a normalized PVM. Its scalar measure at \(u\in\mathcal H_2\) is the scalar measure of \(P_1\) at \(W^*u\), hence is finite Radon. For simple Borel functions, and then uniform limits, \(Q(h)=WP_1(h)W^*\). Consequently \(Q(\gamma\mapsto\gamma(x))=\pi_2(x)\). SNAG uniqueness gives \(Q=P_2\).

Conversely, if \(WP_1(B)W^*=P_2(B)\) for every Borel set, the same simple-function approximation gives equality of all bounded spectral integrals. Apply it to \(\gamma\mapsto\gamma(x)\) to obtain \(W\pi_1(x)W^*=\pi_2(x)\) for every \(x\). \(\square\)

<a id="ha-lca-13-exercise-8-5"></a>
**Exercise 8.5 — Generated algebras on arbitrary Hilbert spaces.** Prove the von Neumann algebra assertion in Theorem 6.1 by commutants, without a separable-space spectral representation.

**Solution.** Let \(Q\) be the SNAG measure of \(V\) on \(G\), and \(R(B)=Q(-B)\) the PVM of \(\rho\). For any bounded \(T\), the complex Riesz uniqueness argument in Theorem 3.1 gives
\[
 T\in\rho(C_0(G))'
 \ \Longleftrightarrow\
 TR(B)=R(B)T\text{ for every Borel }B.
\]
Negation permutes Borel sets, so the last condition is equivalent to commuting with every \(Q(B)\). By Theorem 4.2 this is equivalent to \(T\in V(\widehat G)'\). Thus the commutants are equal, and taking their commutants proves (22). Nondegeneracy supplies a strongly convergent approximate identity on the \(C_0\) side; the group side contains \(I\). Lemma 1.1 therefore identifies both double commutants with the generated algebras. Every step applies to arbitrary vectors on an arbitrary Hilbert space. We have not asserted that every operator in the generated algebra is a single bounded Borel spectral integral; that assertion is unnecessary here. \(\square\)

## Proof dependencies

The new operator inputs are proved in Lemma 1.1, Lemma 3.0, Theorem 3.1, Lemma 5.1 and Lemma 7.0, including the unbounded domain arguments in Corollary 7.1. Earlier results enter only through the exact programme proofs linked at their use. No external source citation replaces a proof. General direct-integral multiplicity theory and a characterization of all measurable functional calculi on nonseparable spaces are outside this lesson's assertions.