Group C Star Algebras and the Fell Unitary Dual — Examples

$G$ is a second-countable locally compact Hausdorff group: the product ${\mathbb{R}}_{>0}\times \mathbb{R}$ is second-countable and locally compact, the multiplication $(a,b)(a^{\mathrm{\prime }},b^{\mathrm{\prime }})=(a{a}^{\mathrm{\prime }},b+a{b}^{\mathrm{\prime }})$ and the inverse $(a,b{)}^{-1}=(a^{-1},-b\mathrm{/}a)$ are continuous, so $G$ is a topological group.

$\pi$ is a unitary representation. Each $\pi (a,b)$ is the composition of the translation $\eta \mapsto \eta (\cdot +\log a)$ and the multiplication by the modulus-one function $u\mapsto e^{ib{e}^u}$, hence unitary by [F2]; and $\pi (a,b)\pi (a^{\mathrm{\prime }},b^{\mathrm{\prime }})\eta (u)=e^{ib{e}^u}{e}^{i{b}^{\mathrm{\prime }}{e}^{u+\log a}}\eta (u+\log a+\log a^{\mathrm{\prime }})=e^{i(b+a{b}^{\mathrm{\prime }})e^u}\eta (u+\log (a{a}^{\mathrm{\prime }}))=\pi (a{a}^{\mathrm{\prime }},b+a{b}^{\mathrm{\prime }})\eta (u)$ by $e^{u+\log a}=a{e}^u$.

$\pi$ is strongly continuous. Write ${\tau }_h\eta (u)=\eta (u+h)$ and $M_b\eta (u)=e^{ib{e}^u}\eta (u)$. Since $M_b$ is unitary, $\mathrm{\parallel }\pi (a,b)\eta -\pi (a_0,b_0)\eta \mathrm{\parallel }\le \mathrm{\parallel }{\tau }_{\log a}\eta -{\tau }_{\log a_0}\eta \mathrm{\parallel }+\mathrm{\parallel }(M_b-M_{b_0}){\tau }_{\log a_0}\eta \mathrm{\parallel }$. The first term tends to zero by continuity of translations. In the second, the vector is fixed, the multiplier converges pointwise as $b\to b_0$, and its squared modulus is bounded by $4\mathrm{\mid }{\tau }_{\log a_0}\eta {\mathrm{\mid }}^2\in L^1$. Dominated convergence gives the second limit; its sequential form suffices because the parameter space is metrizable.

Every bounded operator commuting with $\pi$ is a multiplication $M_m$. For $n\ge 1$, $\pi (1,\pm 1\mathrm{/}n)$ is multiplication by $e^{\pm i{e}^u\mathrm{/}n}$, so a commuting $T$ commutes with their real and imaginary parts, the multiplications $M_{{\phi }_n}$, $M_{{\psi }_n}$ for ${\phi }_n(u)=\cos (e^u\mathrm{/}n)$ and ${\psi }_n(u)=\sin (e^u\mathrm{/}n)$. The family $\mathcal{F}=\{{\phi }_n,{\psi }_n:n\ge 1\}$ consists of bounded real Borel functions with $\sigma (\mathcal{F})=$ Borel: indeed $n\sin (e^u\mathrm{/}n)\to e^u$ pointwise, so $e^u$, hence $u=\log e^u$, is $\sigma (\mathcal{F})$-measurable, and the sets $\{u\le c\}$ generate the Borel $\sigma$-algebra. By [F4], $T=M_m$ for some bounded measurable $m$.

Coefficient estimate. For $t\in \mathbb{R}$ and $R>0$ put $I_R=[-R,-R\mathrm{/}2]$ and ${\eta }_R:=(R\mathrm{/}2{)}^{-1\mathrm{/}2}{e}^{itu}{\mathbf{1}}_{I_R}$, a unit vector of $L^2(\mathbb{R})$. For $(a,b)\in G$ with $h:=\log a$, the change of variable $u\mapsto u+h$ and $\mathrm{\mid }{e}^{it(u+h)}\mathrm{\mid }=1$ give $⟨\pi (a,b){\eta }_R,{\eta }_R⟩=(2\mathrm{/}R){\int }_{I_R\cap (I_R-h)}{e}^{ib{e}^u}{e}^{ith}du=a^{it}(2\mathrm{/}R){\int }_{I_R\cap (I_R-h)}{e}^{ib{e}^u}du$. If $\mathrm{\mid }h\mathrm{\mid }\le R\mathrm{/}2$, the intersection has measure $R\mathrm{/}2-\mathrm{\mid }h\mathrm{\mid }$ by [F8], and $\mathrm{\mid }{e}^{ib{e}^u}-1\mathrm{\mid }\le \min (2,\mathrm{\mid }b\mathrm{\mid }{e}^{-R\mathrm{/}2})$ on $I_R$ because $u\le -R\mathrm{/}2$ there; therefore $\mid ⟨\pi (a,b){\eta }_R,{\eta }_R⟩-{\chi }_t(a,b)\mid \le (2\mathrm{/}R){\int }_{I_R\cap (I_R-h)}\mathrm{\mid }{e}^{ib{e}^u}-1\mathrm{\mid }du+2\mathrm{\mid }h\mathrm{\mid }\mathrm{/}R\le 2\mathrm{\mid }h\mathrm{\mid }\mathrm{/}R+\mathrm{\mid }b\mathrm{\mid }{e}^{-R\mathrm{/}2}$.

The operator $M_m$ commutes with all translations $U_h\eta (u)=\eta (u+h)$, $h\in \mathbb{R}$, because $\pi (e^h,0)=U_h$. The family of translations is ergodic for Lebesgue measure: if a Lebesgue measurable set $E$ is invariant under every translation up to null sets, replace it by a Borel set equal to it a.e. using [F5]. Translation preserves null sets, so the new indicator is still invariant a.e. under each translation, and its composition with $(u,t)\mapsto u+t$ is Borel, hence product-measurable. Then ${\int }_{\mathbb{R}}{\int }_{\mathbb{R}}\mathrm{\mid }{1}_E(u+t)-1_E(u){\mathrm{\mid }}^{2}dudt=0$ by Tonelli [F5], since every $t$-slice vanishes; hence $(u,t)\mapsto \mathrm{\mid }{1}_E(u+t)-1_E(u)\mathrm{\mid }$ is $0$ a.e., so for some $u_0$ one has $1_E(u_0+t)=1_E(u_0)$ for a.e. $t$, and $1_E$ is a.e. equal to the constant $1_E(u_0)$; being an indicator, $E$ is null or conull. Applying the moreover clause of [F4] to the commuting translation unitaries gives that $m$ is constant a.e., so $T=\lambda I$.

Hence ${\chi }_t\prec \pi$. Let $K\subseteq G$ be compact and $ϵ>0$, and let $z\in \mathbb{C}$; since $\log a$ and $b$ stay in bounded ranges $\mathrm{\mid }h\mathrm{\mid }\le L$, $\mathrm{\mid }b\mathrm{\mid }\le B$ on $K$, choose $R>2L$ so large that $2L\mathrm{/}R+B{e}^{-R\mathrm{/}2}<ϵ\mathrm{/}(1+\mathrm{\mid }z{\mathrm{\mid }}^2)$. Then step 1.5 gives ${\sup }_{(a,b)\in K}\mid \mathrm{\mid }z{\mathrm{\mid }}^2{\chi }_t(a,b)-⟨\pi (a,b)(z{\eta }_R),z{\eta }_R⟩\mid \le \mathrm{\mid }z{\mathrm{\mid }}^2(2L\mathrm{/}R+B{e}^{-R\mathrm{/}2})<ϵ$, so every diagonal coefficient of the one-dimensional representation ${\chi }_t$, hence every function of positive type associated to ${\chi }_t$, is a compact-uniform limit of coefficients of $\pi$; by [F9], ${\chi }_t\prec \pi$.

$\pi$ is irreducible. If $M\subseteq L^2(\mathbb{R})$ were a closed invariant subspace different from $0$ and $H$, its orthogonal projection $P_M$ would commute with every $\pi (g)$ by [F6] and would satisfy $P_M\mathrm{\ne }0,I$, contradicting step 2.1. Hence $H$ has no nontrivial closed invariant subspace, and $\pi$ is irreducible; its carrier $L^2(\mathbb{R})$ is infinite dimensional, since the indicators of $[n,n+1)$, $n\in \mathbb{Z}$, are an infinite orthonormal family.

Each ${\chi }_t(a,b)=a^{it}$ is a continuous unitary character, i.e. a one-dimensional strongly continuous unitary representation; its diagonal coefficients at $z\in \mathbb{C}$ are $\mathrm{\mid }z{\mathrm{\mid }}^2{\chi }_t$. No ${\chi }_t$ is unitarily equivalent to $\pi$, since unitarily equivalent representations have isomorphic carriers and $\dim L^2(\mathbb{R})=\mathrm{\infty }$ while $\dim \mathbb{C}=1$; in particular $\pi \simeq ̸1_G={\chi }_0$, and $[\pi ]\mathrm{\ne }[1_G]$.

Fell closure and failure of Hausdorffness. Since ${\chi }_t$ and $\pi$ are irreducible, [F10] gives ${\chi }_t\in \overline{\{[\pi ]\}}$ in the Fell topology; in particular $[1_G]=[{\chi }_0]\in \overline{\{[\pi ]\}}$ with $[1_G]\mathrm{\ne }[\pi ]$ by step 4.1, so $\{[\pi ]\}$ is not closed. If the Fell topology were Hausdorff, the distinct points $[1_G]$ and $[\pi ]$ would have disjoint open neighbourhoods $U\ni [1_G]$, $V\ni [\pi ]$; since $[1_G]$ lies in the closure of $\{[\pi ]\}$, the neighbourhood $U$ meets $\{[\pi ]\}$ and therefore $[1_G]\in U$ and $[\pi ]\in U\cap V$, contradicting disjointness. Hence $\widehat{G}$ is not Hausdorff.

The Axiom of Choice is inherited from the multiplication lemma and the Fell suppliers; the explicit representation, the coefficient estimates and the ergodicity computation use no further choice (The Axiom of Choice).

Every irreducible strongly continuous unitary representation of the abelian group $\mathbb{R}$ is one-dimensional. Indeed, for fixed $g$ the operator $\pi (g)$ commutes with every $\pi (h)$, hence is a bounded self-intertwiner of $\pi$; [F1] makes it a scalar $\chi (g)I$. Every linear subspace is then invariant, so irreducibility forces $\dim H=1$; the resulting map $\chi :\mathbb{R}\to \mathbb{T}$ is a continuous unitary character.

By [F2] every continuous unitary character of $\mathbb{R}$ is ${\chi }_t$ for a unique $t\in \mathbb{R}$, and each ${\chi }_t$ is a continuous unitary character of $\mathbb{R}$.

Hence $\widehat{\mathbb{R}}=\{{\chi }_t:t\in \mathbb{R}\}$, with $t\mapsto {\chi }_t$ bijective: step 1.1 exhibits every irreducible class as a character, step 1.2 identifies the characters, and distinct $t$ give distinct characters (evaluate at a suitable $x$).

Fell convergence is compact-uniform convergence of the parameters. If $t_i\to t$, then for a compact $Q\subseteq \mathbb{R}$ and $R:={\sup }_{x\in Q}\mathrm{\mid }x\mathrm{\mid }$ one has ${\sup }_{x\in Q}\mathrm{\mid }{\chi }_{t_i}(x)-{\chi }_t(x)\mathrm{\mid }\le R\mathrm{\mid }{t}_i-t\mathrm{\mid }\to 0$; hence for every finite family $c_j{\chi }_t$ of tests, every compact $Q$ and every $ϵ>0$, the test $c_j{\chi }_t$ is within $ϵ$ on $Q$ of the coefficient $c_j{\chi }_{t_i}$ once $i$ is large, so ${\chi }_{t_i}\in W({\chi }_t;\cdot ,Q,ϵ)$ eventually and ${\chi }_{t_i}\to {\chi }_t$ in the Fell topology. Conversely, suppose ${\chi }_{t_i}\to {\chi }_t$, let $0<ϵ<1$ and fix $\delta >0$; put $Q:=[-\pi \mathrm{/}\delta ,\pi \mathrm{/}\delta ]$. By [F3] the neighbourhood determined by the coefficient ${\chi }_t$, the set $Q$ and $ϵ$ is met eventually: there are $c_i^{\mathrm{\prime }}\ge 0$ with ${\sup }_Q\mathrm{\mid }{\chi }_t-c_i^{\mathrm{\prime }}{\chi }_{t_i}\mathrm{\mid }<ϵ$. Evaluating at $x=0$ gives $\mathrm{\mid }1-c_i^{\mathrm{\prime }}\mathrm{\mid }<ϵ$, hence $c_i^{\mathrm{\prime }}>1-ϵ$. If $\mathrm{\mid }{t}_i-t\mathrm{\mid }\ge \delta$, then $x_i:=\pi \mathrm{/}\mathrm{\mid }{t}_i-t\mathrm{\mid }$ lies in $Q$ and ${\chi }_{t_i}(x_i)=-{\chi }_t(x_i)$, so $\mathrm{\mid }{\chi }_t(x_i)-c_i^{\mathrm{\prime }}{\chi }_{t_i}(x_i)\mathrm{\mid }=1+c_i^{\mathrm{\prime }}>2-ϵ>1>ϵ$, a contradiction. Hence eventually $\mathrm{\mid }{t}_i-t\mathrm{\mid }<\delta$, and since $\delta >0$ was arbitrary, $t_i\to t$.

Weak containment of characters: if ${\chi }_s\prec {\chi }_t$, then applying the defining approximation to the coefficient ${\chi }_s$ (the diagonal coefficient at a unit vector), the compact set $Q=[-\pi \mathrm{/}\delta ,\pi \mathrm{/}\delta ]$ and some $0<ϵ<1$, and using [F3], we find $c^{\mathrm{\prime }}\ge 0$ with ${\sup }_Q\mathrm{\mid }{\chi }_s-c^{\mathrm{\prime }}{\chi }_t\mathrm{\mid }<ϵ$; evaluating at $0$ gives $c^{\mathrm{\prime }}>1-ϵ$, and if $\mathrm{\mid }s-t\mathrm{\mid }\ge \delta$, the point $x=\pi \mathrm{/}\mathrm{\mid }s-t\mathrm{\mid }\in Q$ gives $\mathrm{\mid }{\chi }_s(x)-c^{\mathrm{\prime }}{\chi }_t(x)\mathrm{\mid }=1+c^{\mathrm{\prime }}>1>ϵ$, a contradiction. Hence $\mathrm{\mid }s-t\mathrm{\mid }<\delta$ for every $\delta >0$, so $s=t$; the converse is immediate by taking the identical coefficient. This agrees with The Fell closure of a single representation is its weak containment closure, since by step 3.1 the point ${\chi }_s$ lies in the Fell closure of $\{{\chi }_t\}$ exactly when $s=t$.

The Axiom of Choice is inherited from Schur's lemma and the Fell topology suppliers; the character and parameter computations use no further choice (The Axiom of Choice).

The evaluation map $\mathrm{\Phi }:L^1(G)\to {⨁}_{\pi \in \widehat{G}}\mathrm{End}(H_{\pi })$, $f\mapsto (\pi (f){)}_{\pi \in \widehat{G}}$, is injective and $\ast$-multiplicative. It is $\ast$-multiplicative by [F3]; for injectivity suppose $\pi (f)=0$ for every $\pi \in \widehat{G}$. Then every matrix element $⟨\pi (f)e_j,e_i⟩$ vanishes, and these are $(1\mathrm{/}\sqrt{d_{\pi }}){\int }_{G}f(g)u_{ji}^{\pi }(g)d\mu (g)$, the inner products of $f$ with $\overline{u_{ji}^{\pi }}$ up to nonzero constants. Conjugation sends the complete orthonormal family of [F2] to another complete orthonormal family: it preserves norms and turns each inner product into its conjugate. Hence vanishing of all these inner products forces $f=0$.

$\mathrm{\Phi }$ is bijective. It is injective by step 1.1, $\dim L^1(G)=\mathrm{\mid }G\mathrm{\mid }$ for the finite group, and the orthonormal basis of [F2] is indexed by the triples $(\pi ,i,j)$, so ${\sum }_{\pi \in \widehat{G}}{d}_{\pi }^2=\mathrm{\mid }G\mathrm{\mid }=\dim L^2(G)$ [F2]; hence the two finite-dimensional spaces have equal dimension and $\mathrm{\Phi }$ is a linear isomorphism. Consequently $\mathrm{\Phi }$ is a $\ast$-isomorphism of $L^1(G)$ onto the finite-dimensional C*-algebra ${⨁}_{\pi \in \widehat{G}}\mathrm{End}(H_{\pi })$, and it is isometric for the transported norm $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\Phi }}:=\mathrm{\parallel }\mathrm{\Phi }(f)\mathrm{\parallel }={\max }_{\pi \in \widehat{G}}\mathrm{\parallel }\pi (f)\mathrm{\parallel }$.

The full and reduced norms coincide on $L^1(G)$. By [F4] $\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}={\sup }_{\rho }\mathrm{\parallel }\rho (f)\mathrm{\parallel }$ over all unitary representations $\rho$; by [F1] each $\rho$ is a discrete direct sum ${\widehat{⨁}}_i{\sigma }_i$ of irreducibles, so $\mathrm{\parallel }\rho (f)\mathrm{\parallel }={\sup }_i\mathrm{\parallel }{\sigma }_i(f)\mathrm{\parallel }$ and therefore ${\sup }_{\rho }\mathrm{\parallel }\rho (f)\mathrm{\parallel }={\max }_{\pi \in \widehat{G}}\mathrm{\parallel }\pi (f)\mathrm{\parallel }=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\Phi }}$. By [F2] $\lambda \cong {\widehat{⨁}}_{\pi }{d}_{\pi }\pi$, so also $\mathrm{\parallel }\lambda (f)\mathrm{\parallel }={\max }_{\pi }\mathrm{\parallel }\pi (f)\mathrm{\parallel }=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\Phi }}$; as $C_r^{\ast }(G)$ is the completion of $\lambda (L^1(G))$, its norm on $L^1(G)$ is $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\Phi }}$ as well.

The canonical surjection $C^{\ast }(G)\twoheadrightarrow C_r^{\ast }(G)$ of [F4] is isometric on the dense image of $L^1(G)$ by step 3.1, hence is injective on a dense subspace and therefore an isometric isomorphism of C*-algebras; both algebras are the completion of $L^1(G)$ in the common norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{\Phi }}$. That completion is $L^1(G)$ itself, because step 2.1 exhibits it as isometric to the finite-dimensional, hence complete, algebra ${⨁}_{\pi }\mathrm{End}(H_{\pi })$. Therefore $C^{\ast }(G)=C_r^{\ast }(G)\cong {⨁}_{\pi \in \widehat{G}}\mathrm{End}(H_{\pi })\cong {⨁}_{\pi \in \widehat{G}}{M}_{d_{\pi }}(\mathbb{C})$ by [F5], a finite-dimensional algebra with one matrix block per irreducible class. It is semisimple by [F6]: the regular module of $M_d(\mathbb{C})$ is the direct sum of its column left ideals, each simple because matrix units send any nonzero column vector to every coordinate vector. The central block projections give the corresponding direct sum for the finite product of matrix algebras.

The Axiom of Choice is inherited from the Peter-Weyl decomposition, the representation correspondence and the group C*-algebra constructions; the dimension count and the norm comparison add no further choice (The Axiom of Choice).

By [F1], $\widehat{\mathbb{Z}}=\{{\chi }_z:z\in \mathbb{T}\}$ with ${\chi }_z(n)=z^n$. By [F2] the Gelfand transform is an isometric $\ast$-isomorphism $C^{\ast }(\mathbb{Z})\to C_0(\widehat{\mathbb{Z}})=C(\mathbb{T})$, and under the identification $\widehat{\mathbb{Z}}\cong \mathbb{T}$ of [F1] the class of ${\delta }_1$ is sent to the function $\gamma \mapsto \gamma (1)$, that is to $z\mapsto z$; this proves the full-algebra statement.

The operator $U$ is unitary and $(U\xi )(k)=\xi (k-1)$, and $\sigma (U)\subseteq \mathbb{T}$. Unitarity gives $\mathrm{\parallel }U\mathrm{\parallel }=\mathrm{\parallel }{U}^{-1}\mathrm{\parallel }=1$. If $\mathrm{\mid }\lambda \mathrm{\mid }>1$, then $U-\lambda I=-\lambda (I-{\lambda }^{-1}U)$ with $\mathrm{\parallel }{\lambda }^{-1}U\mathrm{\parallel }=\mathrm{\mid }\lambda {\mathrm{\mid }}^{-1}<1$, so $U-\lambda I$ is invertible by [F4]; if $0<\mathrm{\mid }\lambda \mathrm{\mid }<1$, then $U-\lambda I=U(I-\lambda U^{-1})$ with $\mathrm{\parallel }\lambda U^{-1}\mathrm{\parallel }=\mathrm{\mid }\lambda \mathrm{\mid }<1$, so it is invertible by [F4]; and $\lambda =0$ gives the invertible operator $U$. Hence $\sigma (U)\subseteq \mathbb{T}$.

Conversely $\mathbb{T}\subseteq \sigma (U)$. For $z\in \mathbb{T}$ and $N\ge 0$ put ${\eta }_N(k)=z^{-k}{\mathbf{1}}_{[-N,N]}(k)\mathrm{/}\sqrt{2N+1}$, a unit vector in ${\mathrm{\ell }}^2(\mathbb{Z})$; then $(U{\eta }_N)(k)=z{z}^{-k}{\mathbf{1}}_{[-N,N]}(k-1)\mathrm{/}\sqrt{2N+1}$, so $(U-zI){\eta }_N$ has support in the two endpoints $\{-N,N+1\}$ and $\mathrm{\parallel }(U-zI){\eta }_N{\mathrm{\parallel }}^2=2\mathrm{/}(2N+1)\to 0$. If $U-zI$ were invertible with inverse $S$, then $1=\mathrm{\parallel }{\eta }_N\mathrm{\parallel }\le \mathrm{\parallel }S\mathrm{\parallel }\mathrm{\parallel }(U-zI){\eta }_N\mathrm{\parallel }\to 0$, a contradiction; hence $z\in \sigma (U)$ and $\sigma (U)=\mathbb{T}$ by step 1.2.

The continuous functional calculus of the normal operator $U$ with $\sigma (U)=\mathbb{T}$ gives an isometric unital $\ast$-isomorphism $C(\mathbb{T})\to C^{\ast }(I,U)$ sending the coordinate function $z\mapsto z$ to $U$; here $C^{\ast }(I,U)$ is the closed span of the powers $U^n$, $n\in \mathbb{Z}$, because $U^{-1}=U^{\ast }$. For $f\in L^1(\mathbb{Z})$ the integrated form is $\lambda (f)={\sum }_{n}f(n)U^n$; finitely supported $f$ give finite Laurent polynomials in $U$ and are dense in $L^1(\mathbb{Z})={\mathrm{\ell }}^1(\mathbb{Z})$, so by continuity of the integrated form the reduced group C*-algebra $C_r^{\ast }(\mathbb{Z})=\overline{\{\lambda (f)\}}$ equals $C^{\ast }(I,U)\cong C(\mathbb{T})$. Under this isomorphism the image of the group element ${\delta }_1$ is $\lambda ({\delta }_1)=\lambda (1)=U$ and hence to the coordinate function $z\mapsto z$.

The Axiom of Choice is inherited from the abelian duality corollary, the representation correspondence and the functional calculus; the shift computation and the spectral argument add no further choice (The Axiom of Choice).