Direct Integral Decomposition and Type I Groups — Examples

By [F1], the assumed AC supplies both DC and countable choice. Thus the DC hypotheses in [F9] and the countable-choice hypotheses in [F6] are met; the direct-integral assumptions are also covered by the given AC.

The metric and compact intervals in [F2] make $\mathbb{R}$ Hausdorff and locally compact, and rational intervals give second countability. With the group operations verified in [F3], $\mathbb{R}$ is a second-countable LCA group. Therefore [F5] applies to show that $\widehat{\mathbb{R}}$ is LCH abelian.

Let $J$ initially act on Borel representatives by $Jf(\chi )=f({\chi }^{-1})$. Inversion preserves the dual Haar measure by [F10], and it is Borel by [F5], so composition preserves measurability and null equivalence. For every nonnegative Borel function $g$, the map $s\mapsto s\circ \iota$ bijects simple minorants of $g$ and $g\circ \iota$; their simple integrals agree because inversion preserves the measures of their level sets. Taking suprema in the definition of the nonnegative integral [F10] gives $\int g\circ \iota d\mu =\int gd\mu$. Applying this to $g=\mathrm{\mid }f{\mathrm{\mid }}^2$ proves that $J$ is a linear isometry on $L^2$. Since inversion is involutive, $J^2=I$, so $J$ is unitary. Put ${\mathcal{F}}_{+}:=J{\mathcal{F}}_{-}$; it is unitary by [F9]. For every $f$ and $t$, $J{M}_{\overline{\chi (t)}}f(\chi )=\overline{{\chi }^{-1}(t)}f({\chi }^{-1})=\chi (t)Jf(\chi )\mathrm{.}$

By Step 1.2, the LCA hypotheses in [F9] hold. Let $s$ be a simple function with finite-measure support. It lies in $L^1\cap L^2$; [F9] says the $L^2$ Plancherel transform ${\mathcal{F}}_{-}$ agrees there with the integral Fourier transform. Since $\lambda (t)s=T_{t}s$, the Fourier translation formula in [F9] gives ${\mathcal{F}}_{-}\lambda (t)s=M_{\overline{\chi (t)}}{\mathcal{F}}_{-}s$. Such simple functions are dense in $L^2$ by [F9], while $\lambda (t)$, $M_{\overline{\chi (t)}}$ and ${\mathcal{F}}_{-}$ are bounded by [F4, F9, F11], so the equality extends to every $f\in L^2(\mathbb{R})$.

By [F4], $\mathrm{\Phi }:\mathbb{R}\to \widehat{\mathbb{R}}$, $\mathrm{\Phi }(\xi )={\chi }_{\xi }$, is a bijection and a group homomorphism. The evaluation formula is jointly continuous: near $({\xi }_0,t_0)$, $\mathrm{\mid }\xi t-{\xi }_0{t}_0\mathrm{\mid }\le \mathrm{\mid }\xi \mathrm{\mid }\mathrm{\mid }t-t_0\mathrm{\mid }+\mathrm{\mid }{t}_0\mathrm{\mid }\mathrm{\mid }\xi -{\xi }_0\mathrm{\mid }$, and continuity of the complex exponential in [F4] then gives continuity of $e^{2\pi i\xi t}$. For a subbasic compact-open neighborhood $S(K,V)=\{\chi :\chi [K]\subseteq V\}$ containing ${\chi }_{{\xi }_0}$, this joint continuity and compactness of $K$ give finitely many product neighborhoods covering $\{{\xi }_0\}\times K$ on which the exponential remains in $V$; intersecting their parameter neighborhoods gives an interval around ${\xi }_0$ mapped into $S(K,V)$. The inverse is continuous at the identity character: for $\delta >0$, put $K_{\delta }=[-1\mathrm{/}(2\delta ),1\mathrm{/}(2\delta )]$ and $V_0=\{z\in \mathbb{T}:\mathrm{\mid }z-1\mathrm{\mid }<1\}$. If ${\chi }_{\xi }[K_{\delta }]\subseteq V_0$ and $\mathrm{\mid }\xi \mathrm{\mid }\ge \delta$, then $t=1\mathrm{/}(2\mathrm{\mid }\xi \mathrm{\mid })\in K_{\delta }$ and ${\chi }_{\xi }(t)=e^{\pm \pi i}=-1\mathrm{\notin }{V}_0$, a contradiction. Continuity of translations in the dual group from [F5] gives continuity of ${\mathrm{\Phi }}^{-1}$ everywhere. Hence $\mathrm{\Phi }$ is a homeomorphism.

The homeomorphism in Step 2.2 makes $\widehat{\mathbb{R}}$ second countable; it is LCH by Step 1.2. Thus [F5] gives its standard-Borel structure. The compact sets $K_n=\mathrm{\Phi }([-n,n])$ cover the dual, and [F5] gives $\mu (K_n)<\mathrm{\infty }$; hence $\mu$ is sigma-finite by [F5].

Set $e_0(\chi )=1$ and $e_n(\chi )=0$ for $n>0$. Its Gram coefficients are constant and its values span $\mathbb{C}$, so [F7] makes $(H_{\chi })$ the constant measurable Hilbert field. Every ${\pi }_{\chi }$ is a unitary homomorphism and is strongly continuous by [F4, F8]. For fixed $t$, the scalar field $\chi \mapsto \chi (t)$ is continuous by [F8], hence weakly measurable. The standard-Borel sigma-finite base was established in Step 3.1, so the in-run definition [F8] applies and forms $\mathrm{\Pi }={\int }^{\oplus }{\pi }_{\chi }d\mu (\chi )$.

The map $\mathcal{V}:L^2(\widehat{\mathbb{R}},\mu )\to {\int }_{\widehat{\mathbb{R}}}^{\oplus }\mathbb{C}d\mu$, $\mathcal{V}f=[\chi \mapsto f(\chi )]$, is a unitary by the quotient definition in [F7]. From the pointwise definition of the direct-integral representation in [F8], ${\mathcal{V}}^{-1}\mathrm{\Pi }(t)\mathcal{V}$ is multiplication by $\chi \mapsto \chi (t)$.

By Steps 1.3 and 2.1, ${\mathcal{F}}_{+}=J{\mathcal{F}}_{-}$ intertwines $\lambda (t)$ with multiplication by $\chi (t)$. By Step 5.1 this is $\mathrm{\Pi }(t)$ under the canonical direct-integral identification, so $\mathcal{V}{\mathcal{F}}_{+}$ intertwines the left regular representation with ${\int }^{\oplus }{\pi }_{\chi }d\mu$. Each fibre is exactly $\mathbb{C}$, and [F4] parametrizes each character exactly once. The vector ${\mathbf{1}}_{(0,1]}$ is nonzero in $L^2(\mathbb{R})$ because $m_{\mathbb{R}}((0,1])=1$ by [F6], so the decomposition is not the zero Hilbert space. The homeomorphic group isomorphism $\mathrm{\Phi }$ pulls $\mu$ back to a nonzero regular Borel measure finite on compact sets and invariant under translations, hence to a left Haar measure on $\mathbb{R}$ by [F6]. By [F12] it is a positive multiple of Lebesgue measure; its countable subsets are null, so the parameter measure has no point masses. This is the stated multiplicity-one continuous-spectrum model.

The vector functional $\tau$ is positive and unital, since $\tau (T^{\ast }T)=\mathrm{\parallel }T{\delta }_e{\mathrm{\parallel }}^2$ and $\mathrm{\parallel }{\delta }_e\mathrm{\parallel }=1$, and is WOT-continuous, hence normal. On generators, $\tau (\lambda (g)\lambda (h))={\mathbf{1}}_{gh=e}={\mathbf{1}}_{hg=e}=\tau (\lambda (h)\lambda (g))$. Bilinearity proves the trace identity on their linear span. For fixed $A$ in that span, approximate $B\in M$ in WOT and use separate multiplication continuity to obtain $\tau (AB)=\tau (BA)$; then fix this $B$ and approximate arbitrary $A\in M$, proving traciality on $M$. Every $T\in M$ commutes with $\rho (\mathrm{\Gamma })$ by [F1,F2]. If $\tau (T^{\ast }T)=0$, then $T{\delta }_e=0$ and $T{\delta }_g=T\rho (g^{-1}){\delta }_e=\rho (g^{-1})T{\delta }_e=0$ for all $g$, so $T=0$ on a dense basis. Thus $\tau$ is faithful. The operators $\lambda (g)$ are linearly independent: applying a finite linear relation to ${\delta }_e$ gives the corresponding relation among the distinct basis vectors ${\delta }_g$. Hence $M$ is infinite dimensional.

Let $T\in Z(M)$ and write $T{\delta }_e={\sum }_g{c}_g{\delta }_g$. For $s\in \mathrm{\Gamma }$ the unitary $C_s=\lambda (s)\rho (s)$ fixes ${\delta }_e$ and sends ${\delta }_g$ to ${\delta }_{sg{s}^{-1}}$. Centrality and [F1,F2] imply that $T$ commutes with both factors of $C_s$, so $C_{s}T{\delta }_e=T{\delta }_e$. Thus $c_g$ is constant on each conjugacy class. An ${\mathrm{\ell }}^2$ sequence cannot have a nonzero constant value on an infinite set, so ICC gives $T{\delta }_e=c_e{\delta }_e$. Commutation with $\rho$ then gives $T{\delta }_g=c_e{\delta }_g$ for every $g$. Therefore $T=c_{e}I$, proving the factor assertion.

If $M$ were type I, [F4] would give the spatial algebra $\mathcal{B}(E)\otimes I_L$. Finite-dimensional $E$ would make $M$ finite dimensional. If $E$ is infinite dimensional and separable, choose a countable orthonormal basis and the isometries $V_1,V_2$ onto its even and odd basis subspaces. Their range projections $p,q$ are orthogonal. Transferring $V_i\otimes I_L$ to $M$, traciality gives $\tau (p)=\tau (V_1^{\ast }{V}_1)=1$ and $\tau (q)=1$, while positivity and $p+q\le I$ give $2=\tau (p+q)\le 1$, a contradiction. Thus $M$ is not type I and [F4] excludes an irreducible multiple. In particular $\lambda$ is not irreducible. Its one-point integral is a central factor decomposition, since diagonal operators on that point are $\mathbb{C}I=Z(M)$; any central diagonal model must have a one-atom measure algebra on its effective support, because its diagonal algebra is scalar. The fibre therefore remains this non-irreducible factor, rather than an irreducible.

For completeness, $F_2$ is countable by its finite reduced words and infinite by the powers of $a$. If $w\mathrm{\notin }A$, all conjugates $a^{n}w{a}^{-n}$ are distinct: equality at two different integers would make $w$ commute with a nonzero power $a^k$. Then $A\cap w^{-1}Aw$ contains $⟨a^k⟩$, of finite index in both infinite cyclic groups, contradicting ${\mathrm{Comm}}_{F_2}(A)=A$ from [F5]. If $w\in A\setminus \{e\}$, then $w\mathrm{\notin }B$ by [F5], and the same argument with $b$ gives infinitely many conjugates. Thus $F_2$ is ICC. The finite-support permutation group is a countable union of finite permutation groups and is infinite. For a nonidentity permutation with finite moved support $S$, move $S$ to infinitely many pairwise disjoint blocks of the same size by finite permutations. Its conjugates then have distinct moved supports and are distinct, proving ICC. Both examples therefore satisfy all conclusions above.

A countable open base generates the Borel sigma-algebra of $K$, and normalized Haar measure is finite. By [F3] real $L^2(K)$ has a countable dense subset $D$; the set $D+iD$ is countable and dense in complex $L^2(K)$, since real and imaginary parts can be approximated separately. Choose a unit vector in each nonzero Peter--Weyl coefficient block [F2]. These vectors are orthogonal, so pairwise disjoint balls of radius $1\mathrm{/}3$ each meet a countable dense set. Assigning the first dense point in each ball proves that $\widehat{K}$ is at most countable.

Use [F1] to choose the representatives and multiplicity spaces stated above. The countable dual with discrete metric is complete and separable, hence standard Borel, and its counting measure is sigma-finite. Choose an orthonormal basis in each nonzero separable fibre and enumerate all pairs consisting of an atom and a basis vector. The section associated with a pair equals that vector at its atom and zero elsewhere. Their Gram coefficients are measurable and their values span densely at each atom, so they form a countable fundamental family in [F4]; if all fibres are zero, use a sequence of zero sections. Every section is measurable because every scalar function on a countable discrete space is measurable. For fixed $k\in K$, all matrix coefficients of the fibre action are likewise measurable.

Counting integration gives $\mathrm{\parallel }\xi {\mathrm{\parallel }}^2={\sum }_{\alpha \in \widehat{K}}\mathrm{\parallel }\xi (\alpha ){\mathrm{\parallel }}^2$, and its only null subset is empty. Thus the direct integral is exactly the completed Hilbert direct sum, with precisely the isotypic action of [F1]. This action is strongly continuous: approximate a vector by its finitely many nonzero coordinates, use continuity on those coordinates, and bound the remaining displacement by twice the tail norm. The regular multiplicity assertion follows from [F1], including its conjugate-class reindexing convention.

A character of $A$ is uniquely determined by $z=\chi (a)\in \{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$, with $\chi (a^n)=z^n$. This identifies $\widehat{A}$ with the torus: compact subsets of the discrete cyclic group are finite, so its compact-open character topology is exactly the topology of this evaluation. Transport normalized torus Haar measure to $\widehat{A}$; do the same for $B$ and $\psi (b)$. These are standard-Borel probability parameter spaces. A transversal $T$ of $A\mathrm{\backslash }F$ is countable. Realize all ${\pi }_z={\mathrm{Ind}}_A^F{\chi }_z$ on the fixed space ${\mathrm{\ell }}^2(T)$ as in [F2]. For fixed $g$ every basis matrix coefficient is zero or a monomial $z^n$, hence continuous in $z$. Thus [F4] defines ${\mathrm{\Pi }}_A={\int }_{\widehat{A}}^{\oplus }{\pi }_{z}d\mu (z)$ on $L^2(\widehat{A};{\mathrm{\ell }}^2(T))$. The pointwise group law is exact for every parameter; since $F$ is discrete, the integrated unitary representation is automatically strongly continuous.

The constant section $\xi (z)={\delta }_e$ is a unit vector. Its diagonal coefficient at $g$ is the integral of the coefficient in [F2]: it is zero if $g\mathrm{\notin }A$, and if $g=a^n$ it is $\int z^{n}d\mu (z)={\mathbf{1}}_{n=0}$. Hence it equals ${\mathbf{1}}_{g=e}$. Crucially, $\xi$ is cyclic for the entire integral. Indeed, for every $t\in T$ and $n\in \mathbb{Z}$, the group law and [F2] give ${\mathrm{\Pi }}_A(t^{-1}{a}^n)\xi (z)=z^n{\delta }_t$. The span of these orbit vectors contains every finite sum of Fourier polynomials times coset basis vectors. Such sums are dense: the squared norm is the sum of the scalar-coordinate squared $L^2$ norms, so truncating the countable coset coordinates makes the tail arbitrarily small, and [F5] approximates each of the finitely many remaining coordinates by Fourier polynomials. This proves global cyclicity, without inferring it from fibrewise cyclicity.

The regular representation and $({\mathrm{\Pi }}_A,\xi )$ now have the same diagonal coefficient and cyclic unit vectors, so [F6] gives ${\lambda }_F\cong {\mathrm{\Pi }}_A$. Repeating steps 1.1 and 2.1 with $B$ gives ${\lambda }_F\cong {\int }_{\widehat{A}}^{\oplus }{\mathrm{Ind}}_A^F\chi d\mu (\chi )\cong {\int }_{\widehat{B}}^{\oplus }{\mathrm{Ind}}_B^F\psi d\nu (\psi )$. Both measures are normalized Haar probabilities.

By self-commensuration in [F1], [F3] makes every fibre in both integrals irreducible. Every intersection $g^{-1}Bg\cap A$ is trivial, hence has infinite index in the infinite group $A$. The cross-family inequivalence hypothesis is therefore vacuous, and [F3] gives ${\mathrm{Ind}}_A^F\chi \cong ̸{\mathrm{Ind}}_B^F\psi$ for every pair of parameters. Within one family, distinct characters also give inequivalent fibres: if the intersection $g^{-1}Ag\cap A$ has finite index in both groups, self-commensuration forces $g\in A$; conjugation then fixes every character of the abelian $A$, and different characters differ on $A$. The same inequivalence criterion applies, and similarly for $B$. Thus each model has multiplicity one on its own irreducible classes, and the two sets of classes are disjoint.

No removal of null parameter sets, change of measure within its class, or parameter identification can match the irreducible fibres of these two probability models: every cross pair is inequivalent by step 4.1. This refutes canonical irreducible measure-class/multiplicity data. No standard-Borel structure on the full dual $\widehat{F}$ is being assumed; the two models already use standard-Borel circles and disjoint images among irreducible classes. Moreover [F7] proves that $F$ is not type I. In this example the central factor datum remains the one-point non-irreducible regular factor of [F7]; the two irreducible disintegrations are not central diagonalizations. Hence the uniqueness appropriate to central factor decompositions cannot be transferred to irreducible multiplicity data outside type I.