Group C Star Algebras and the Fell Unitary Dual

Assume the Axiom of Choice. Let $G$ be an LCH group with a fixed left Haar measure, let $(\pi ,H)$ be a strongly continuous unitary representation of $G$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Hilbert space), and let $f\in L^1(G)$ with its norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1$ (Complex Haar L^p spaces and compactly supported functions). The integrated form of $\pi$ at $f$ is the operator $\pi (f)\in \mathcal{B}(H)$ (A bounded linear operator between normed spaces) characterised by the weak integral identity $⟨\pi (f)\xi ,\eta ⟩={\int }_{G}f(g)⟨\pi (g)\xi ,\eta ⟩dg(\xi ,\eta \in H)\mathrm{.}$ The integral is a Haar integral over the fixed measure, and the right-hand side is the pairing convention of the Hilbert space, linear in the first argument and conjugate-linear in the second.

• Well-definedness (existence). Fix $\xi \in H$. Since $\pi$ is unitary, $\mathrm{\mid }⟨\pi (g)\xi ,\eta ⟩\mathrm{\mid }\le \mathrm{\parallel }\xi \mathrm{\parallel }\mathrm{\parallel }\eta \mathrm{\parallel }$ for all $g\in G$ and $\eta \in H$, so $g\mapsto f(g)⟨\pi (g)\xi ,\eta ⟩$ is measurable with $\mathrm{\mid }f(g)⟨\pi (g)\xi ,\eta ⟩\mathrm{\mid }\le \mathrm{\mid }f(g)\mathrm{\mid }\mathrm{\parallel }\xi \mathrm{\parallel }\mathrm{\parallel }\eta \mathrm{\parallel }$; this majorant lies in $L^1(G)$ when $\xi$ and $\eta$ are fixed. The assignment $\eta \mapsto {\int }_{G}f(g)⟨\pi (g)\xi ,\eta ⟩dg$ is therefore a well-defined conjugate-linear functional, bounded by $\mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }\xi \mathrm{\parallel }\mathrm{\parallel }\eta \mathrm{\parallel }$; by the Hilbert Riesz representation theorem (Riesz representation for Hilbert spaces) there is a unique vector, written $\pi (f)\xi$, with $⟨\pi (f)\xi ,\eta ⟩={\int }_{G}f(g)⟨\pi (g)\xi ,\eta ⟩dg$ for every $\eta \in H$, and $\mathrm{\parallel }\pi (f)\xi \mathrm{\parallel }\le \mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }\xi \mathrm{\parallel }$.
• Linearity. For scalars $a,b$ and $\xi ,{\xi }^{\mathrm{\prime }}\in H$ the defining functionals satisfy the identity for $a\xi +b{\xi }^{\mathrm{\prime }}$ by linearity of the integral and of the inner product in the first argument, so $\pi (f)(a\xi +b{\xi }^{\mathrm{\prime }})=a\pi (f)\xi +b\pi (f){\xi }^{\mathrm{\prime }}$; thus $\xi \mapsto \pi (f)\xi$ is a linear map $H\to H$ with $\mathrm{\parallel }\pi (f)\mathrm{\parallel }\le \mathrm{\parallel }f{\mathrm{\parallel }}_1$. Likewise, for scalars $a,b$ and $f,h\in L^1(G)$ the integral identity gives $\pi (af+bh)=a\pi (f)+b\pi (h)$, because both sides have the same pairing with every $\eta \in H$.
• No further hypotheses. The construction applies to every strongly continuous unitary representation of every LCH group: no irreducibility, separability, unimodularity or compactness is assumed. The modular function enters this page only through the involution of $L^1(G)$ (Involution on L1 of a locally compact group), not through the definition of $\pi (f)$.
• Use of choice. The Axiom of Choice is declared as a standing hypothesis of the completion chain of this page. In this definition it is needed only through the Hilbert Riesz representation step, which is established under Countable Choice (Riesz representation for Hilbert spaces) and therefore under AC (The Axiom of Choice).

Let $A$ be a complex Banach $\ast$-algebra without a required unit (Banach star-algebra without a required unit) and let $K$ be a complex Hilbert space (Hilbert space). A star-representation of $A$ on $K$ is a bounded complex-linear map $\sigma :A\to \mathcal{B}(K)$ (A bounded linear operator between normed spaces, Bounded Hilbert operators form a C star algebra) satisfying $\sigma (ab)=\sigma (a)\sigma (b),\sigma (a^{\ast })=\sigma (a{)}^{\ast }(a,b\in A),$ where the adjoint is the Hilbert adjoint on $\mathcal{B}(K)$. It is nondegenerate if the closed linear span of $\{\sigma (a)\xi :a\in A,\xi \in K\}$ (Linear combination of a finite list, and the span $\mathrm{span}(S)$ as the smallest linear subspace containing $S$) is all of $K$. When $A$ is a C*-algebra, a bounded star-representation of $A$ is exactly a bounded star-homomorphism $A\to \mathcal{B}(K)$ in the sense of C star algebra.

• Equivalence with the common-kernel condition. Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the Hilbert-space decomposition and adjoint suppliers in this paragraph. Nondegeneracy is equivalent to: every $\xi \in K$ with $\sigma (a)\xi =0$ for all $a\in A$ satisfies $\xi =0$. Indeed, let $S$ be the closed linear span of $\{\sigma (a)\xi \}$. By Orthogonal decomposition by a closed subspace one has $K=S\oplus S^{\perp }$, so $S=K$ exactly when $S^{\perp }=\{0\}$; and a vector $\eta$ lies in $S^{\perp }$ exactly when $⟨\sigma (a)\xi ,\eta ⟩=0$ for all $a\in A$ and $\xi \in K$. Since $⟨\sigma (a)\xi ,\eta ⟩=⟨\xi ,\sigma (a{)}^{\ast }\eta ⟩=⟨\xi ,\sigma (a^{\ast })\eta ⟩$ (Orthogonality and the orthogonal complement, Bounded Hilbert operators form a C star algebra), and since $\eta \perp K$ forces $\eta =0$, this holds exactly when $\sigma (a^{\ast })\eta =0$ for all $a$, i.e. when $\sigma (a)\eta =0$ for all $a$.
• Continuity is part of the definition. A star-representation is required to be bounded; no automatic-continuity statement is asserted here. For $A=L^1(G)$ the contractive bound is proved, not assumed, by the recovery lemma used on this page.

Let $A$ be a complex C*-algebra (C star algebra) and recall that positivity in $A$ is the algebraic condition $a=b^{\ast }b$, without spectral hypotheses (Self-adjoint positive unitary and normal elements). A linear functional $\omega :A\to \mathbb{C}$ is positive if $\omega (a^{\ast }a)\ge 0\text{for every }a\in A,$ and a state if in addition $\mathrm{\parallel }\omega \mathrm{\parallel }=1$, the norm being the operator norm of the bounded linear functional on $A$.

• The positive functionals form a convex cone. If $\omega ,\phi$ are positive, $s,t\ge 0$ and $a\in A$, then $(s\omega +t\phi )(a^{\ast }a)=s\omega (a^{\ast }a)+t\phi (a^{\ast }a)\ge 0$. In particular the positive functionals of norm at most one form a convex set, since $\mathrm{\parallel }s\omega +t\phi \mathrm{\parallel }\le s+t$ whenever $s,t\ge 0$. Being bounded with norm one is part of the definition of a state, not a consequence claimed here.
• Cauchy–Schwarz. Every positive functional satisfies $\mathrm{\mid }\omega (b^{\ast }a){\mathrm{\mid }}^2\le \omega (a^{\ast }a)\omega (b^{\ast }b)(a,b\in A)\mathrm{.}$ Indeed, for all $z\in \mathbb{C}$ linearity gives $\omega ((a+zb{)}^{\ast }(a+zb))=\omega (a^{\ast }a)+z\omega (a^{\ast }b)+\bar{z}\omega (b^{\ast }a)+\mathrm{\mid }z{\mathrm{\mid }}^2\omega (b^{\ast }b)\ge 0.$ The left side is a nonnegative real number for every $z$. Taking $z=1$ and $z=i$ and using that both resulting values are real shows $\omega (a^{\ast }b)+\omega (b^{\ast }a)\in \mathbb{R}$ and $i(\omega (a^{\ast }b)-\omega (b^{\ast }a))\in \mathbb{R}$; hence $\omega (b^{\ast }a)=\overline{\omega (a^{\ast }b)}$. Writing $p=\omega (a^{\ast }a)\ge 0$, $q=\omega (b^{\ast }b)\ge 0$ and $u=\omega (a^{\ast }b)$, the displayed inequality reads $p+2\mathrm{Re}(zu)+\mathrm{\mid }z{\mathrm{\mid }}^{2}q\ge 0$ for all $z$. If $q>0$, insert $z=-\bar{u}\mathrm{/}q$ to obtain $p-\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{/}q\ge 0$, that is $\mathrm{\mid }u{\mathrm{\mid }}^2\le pq$; if $q=0$, the same inequality forces $\mathrm{Re}(zu)\ge -p\mathrm{/}2$ for all $z$, which is impossible unless $u=0$, and then $\mathrm{\mid }u{\mathrm{\mid }}^2=0\le pq$. Since $\mathrm{\mid }\omega (b^{\ast }a)\mathrm{\mid }=\mathrm{\mid }\overline{\omega (a^{\ast }b)}\mathrm{\mid }=\mathrm{\mid }u\mathrm{\mid }$, this is the stated inequality.
• Continuity and the norm formula. Assume AC for the calculus and approximate-unit suppliers (The Axiom of Choice, Positive calculus and order estimates in a C star algebra, Positive contractive approximate units for C star algebras and ideals). Write $A^{+}$ here for the positive cone, not the unitization, and put $M=\sup \{\omega (b):b\in A^{+},\mathrm{\parallel }b\mathrm{\parallel }\le 1\}$. This supremum is finite: otherwise choose positive contractions $b_n$ with $\omega (b_n)>n{2}^n$. The norm-convergent series $b={\sum }_{n\ge 1}{2}^{-n}{b}_n$ is positive, and $b-2^{-n}{b}_n$ is positive because the positive cone is closed. Positivity would give $\omega (b)\ge 2^{-n}\omega (b_n)>n$ for every $n$, a contradiction. Every self-adjoint $h$ is $h_{+}-h_{-}$ with $h_{\pm }\ge 0$ and $\mathrm{\parallel }{h}_{\pm }\mathrm{\parallel }\le \mathrm{\parallel }h\mathrm{\parallel }$ by the calculus. Thus $\omega (h)$ is real and $\mathrm{\mid }\omega (h)\mathrm{\mid }\le M\mathrm{\parallel }h\mathrm{\parallel }$. Decomposing $x=h+ik$, with $\mathrm{\parallel }h\mathrm{\parallel },\mathrm{\parallel }k\mathrm{\parallel }\le \mathrm{\parallel }x\mathrm{\parallel }$, gives $\mathrm{\mid }\omega (x)\mathrm{\mid }\le 2M\mathrm{\parallel }x\mathrm{\parallel }$, proving continuity. For a positive contractive approximate unit $u_{\lambda }$, Cauchy–Schwarz gives $\mathrm{\mid }\omega (u_{\lambda }x){\mathrm{\mid }}^2\le \omega (x^{\ast }x)\omega (u_{\lambda }^2)\le M^2\mathrm{\parallel }x{\mathrm{\parallel }}^2$. Passing to $u_{\lambda }x\to x$ gives $\mathrm{\parallel }\omega \mathrm{\parallel }\le M$; the reverse inequality follows from the definition of the operator norm. Hence $\mathrm{\parallel }\omega \mathrm{\parallel }=M$. A nonzero positive functional therefore becomes a state upon division by its norm. The zero algebra has no state.

Assume the Axiom of Choice. Let $G$ be a topological group. Two strongly continuous unitary representations $\pi$ on $H$ and $\rho$ on $K$ are unitarily equivalent when there is a unitary intertwiner $U:H\to K$ with $U\pi (g)=\rho (g)U$ for every $g\in G$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners). Irreducibility has the invariant-subspace meaning recalled there, so an irreducible representation acts on a nonzero Hilbert space. The unitary dual $\widehat{G}$ is the set of unitary equivalence classes of irreducible strongly continuous unitary representations of $G$. The zero representation is not an element of $\widehat{G}$, since it is not irreducible.

• Equivalence is an equivalence relation. Identity intertwiners give reflexivity, inverses of unitary intertwiners give symmetry, and compositions of unitary intertwiners give transitivity; irreducibility is a class property, so the phrase "classes of irreducible representations" is unambiguous.
• Why the dual is a set. Hilbert spaces form no set, so the classes are not taken over all carriers. Instead, let $P_1(G)\subseteq {\mathbb{C}}^G$ be the set of normalized continuous functions of positive type. If $\pi$ is irreducible and $\xi \mathrm{\ne }0$, the closed linear span of $\{\pi (g)\xi :g\in G\}$ is a nonzero closed invariant subspace (Cyclic vector and cyclic unitary representation), hence all of $H$; so $\xi$ is cyclic, and its normalized diagonal coefficient lies in $P_1(G)$. By Normalized positive type and pointed cyclic unitary representations the map from equivalence classes of pointed cyclic triples to $P_1(G)$ is a bijection, with inverse given by the GNS construction (GNS construction for a continuous positive-type function). The subset $I\subseteq P_1(G)$ of those $\phi$ whose GNS representation is irreducible is then a set, and $\widehat{G}$ is, equivalently, the image of $I$ under the assignment $\phi \mapsto {\pi }_{\phi }$ followed by passage to unitary equivalence: the quotient identifies two functions when their GNS representations are unitarily equivalent after forgetting the distinguished vectors. An intertwiner gives matching unit vectors by transporting one chosen vector to the other carrier, and every irreducible class contains a normalized cyclic pointed representative. This realizes $\widehat{G}$ as a quotient of the set $I$, with no dimension bound assumed.
• Compact groups. When $G$ is compact the same construction applies verbatim and gives the usual dual of a compact group; no separability is assumed.
• Choice. The Axiom of Choice is inherited from the GNS construction and from Schur's lemma, which are the only steps of the construction that use it (The Axiom of Choice).

Let $G$ be a topological group and let $\pi$ and $\rho$ be strongly continuous unitary representations on Hilbert spaces $H_{\pi }$ and $H_{\rho }$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners). Write $\pi \prec \rho$ and say that $\pi$ is weakly contained in $\rho$ if every continuous function of positive type associated to $\pi$ can be approximated, uniformly on every compact subset of $G$, by finite sums of functions of positive type associated to $\rho$: for every $\xi \in H_{\pi }$, every compact $Q\subseteq G$ and every $ϵ>0$ there exist finitely many ${\eta }_1,\dots ,{\eta }_n\in H_{\rho }$ with ${\sup }_{g\in Q}\mid ⟨\pi (g)\xi ,\xi ⟩-{\sum }_{i=1}^n⟨\rho (g){\eta }_i,{\eta }_i⟩\mid <ϵ\mathrm{.}$ Write $\pi \sim \rho$ when both $\pi \prec \rho$ and $\rho \prec \pi$.

• Coefficient form. The vector $\xi$ of the definition is arbitrary, so the functions tested are exactly the diagonal matrix coefficients $c_{\xi ,\xi }(g)=⟨\pi (g)\xi ,\xi ⟩$ of $\pi$ (Matrix coefficient of a unitary representation); each is continuous and of positive type (Diagonal unitary coefficients have positive type), and so is each of the approximating functions (Continuous positive-type functions and normalization). Containment of a representation in another, when defined by subrepresentations, plainly implies weak containment; no multiplicity or dimension hypotheses are imposed, and the zero representation is allowed on either side.
• Reflexivity and invariance of the relation. Taking $n=1$ and ${\eta }_1=\xi$ shows $\pi \prec \pi$. If $U:H_{\pi }\to H_{{\pi }^{\mathrm{\prime }}}$ is a unitary intertwiner and $V:H_{\rho }\to H_{{\rho }^{\mathrm{\prime }}}$ is one, then $V$ carries every diagonal coefficient of $\rho$ to a diagonal coefficient of ${\rho }^{\mathrm{\prime }}$, so $\pi \prec \rho$ implies ${\pi }^{\mathrm{\prime }}\prec {\rho }^{\mathrm{\prime }}$: the relation is well defined on unitary equivalence classes.
• Transitivity. If $\pi \prec \rho$ and $\rho \prec \sigma$, then $\pi \prec \sigma$. Indeed, fix $\xi \in H_{\pi }$, compact $Q$ and $ϵ>0$. Since $\pi \prec \rho$, choose ${\eta }_1,\dots ,{\eta }_n\in H_{\rho }$ with ${\sup }_Q\mathrm{\mid }{c}_{\xi ,\xi }-{\sum }_j{c}_{{\eta }_j,{\eta }_j}\mathrm{\mid }<ϵ\mathrm{/}2$. Applying $\rho \prec \sigma$ to each of the finitely many vectors ${\eta }_j$ on the same compact $Q$ with tolerance $ϵ\mathrm{/}(2n)$ produces, for each $j$, finitely many vectors ${\zeta }_{j,k}\in H_{\sigma }$ with ${\sup }_Q\mathrm{\mid }{c}_{{\eta }_j,{\eta }_j}-{\sum }_k{c}_{{\zeta }_{j,k},{\zeta }_{j,k}}\mathrm{\mid }<ϵ\mathrm{/}(2n)$; summing the $n$ inequalities gives a finite family of vectors of $H_{\sigma }$ whose coefficient sum differs from $c_{\xi ,\xi }$ on $Q$ by less than $ϵ$.

Assume the Axiom of Choice. Let $A$ be a complex C*-algebra, and let $B$ be $A$ itself when $A$ is unital and the minimal unitization $A^{+}$ otherwise; spectra of elements of $A$ are computed in $B$ (Minimal C star unitization, Spectrum and resolvent set in a Banach algebra). Then:

1. every self-adjoint $h\in B$ has real spectrum and a continuous functional calculus: for every $f\in C(\sigma (h))$ there is an element $f(h)\in C^{\ast }(1,h)\subseteq B$ with $\mathrm{\parallel }f(h)\mathrm{\parallel }={\sup }_{\lambda \in \sigma (h)}\mathrm{\mid }f(\lambda )\mathrm{\mid }$ and $\sigma (f(h))=f(\sigma (h))$, the assignment $f\mapsto f(h)$ is a unital $\ast$-homomorphism extending the polynomial calculus;
2. the algebraically positive elements $a^{\ast }a$ $(a\in B)$ are exactly the self-adjoint elements with nonnegative spectrum: with $P=\{h=h^{\ast }:\sigma (h)\subseteq [0,\mathrm{\infty })\}$ one has $P=\{a^{\ast }a:a\in B\}$, and $P$ is a closed convex cone with $P\cap (-P)=\{0\}$;
3. writing $a\le b$ for $b-a\in P$, conjugation preserves positivity and order, $x^{\ast }ax\in P$ whenever $a\in P$, and $0\le a\le b$ implies $\mathrm{\parallel }a\mathrm{\parallel }\le \mathrm{\parallel }b\mathrm{\parallel }$;
4. every star-homomorphism between C*-algebras is contractive; a unital star-homomorphism $\phi$ is natural for the calculus on normal elements, $\phi (f(a))=f(\phi (a))$; and if $f$ vanishes at $0$ while $h$ lies in a nonunital $A$, then $f(h)\in A$.

Assume the Axiom of Choice. Let $(X,\mathrm{\Sigma },\mu )$ be a $\sigma$-finite measure space and let $H=L^2(X,\mathrm{\Sigma },\mu ;\mathbb{C})$ with the integral pairing (The space $L^p(\mu )$ as the quotient by null functions, $L^2$ with the integral pairing is a Hilbert space, Finite, sigma-finite, and semifinite measures). For a bounded measurable $h$ write $M_{h}f=h\cdot f$ for the multiplication operator. Let $\mathcal{F}$ be a family of bounded real measurable functions generating $\mathrm{\Sigma }$ modulo null sets, in the sense that $\mathrm{\Sigma }$ is the completion of the $\sigma$-algebra $\sigma (\mathcal{F})$ generated by the sets $h^{-1}(B)$ $(h\in \mathcal{F},B$ Borel$)$. If $T\in \mathcal{B}(H)$ (A bounded linear operator between normed spaces) commutes with $M_h$ for every $h\in \mathcal{F}$, then $T=M_m$ for some bounded measurable $m$. If moreover $T$ commutes with the unitary operators $U_{S}f=f\circ S^{-1}$ induced by an ergodic family of invertible measure-preserving transformations $S$ with measurable inverses of $(X,\mathrm{\Sigma },\mu )$ (Measure-preserving transformations and systems, Ergodicity relative to an invariant measure), then $m$ is constant almost everywhere. Here ergodicity of the family means that every measurable set invariant modulo null sets under every member is null or conull.