Group C*-Algebras: Amenability and Nuclearity

Intuition Beginner

A group is a collection of symmetries — moves you can undo and combine. From a single group you can build an algebra of operators in two different ways, and the gap between the two ways turns out to measure something deep about the group.

The first way is the most generous one imaginable. You promise to honour every possible way the group could ever act by unitary moves on any space at all, and you bundle all those ways together into one algebra. Nothing is left out; this is the freest, biggest algebra the group can produce. The second way is more modest. You let the group act on functions of its own elements — the group shuffling its own names around — and you keep only the algebra that this single self-action generates. This is a smaller, more concrete object.

There is always an honest map from the big algebra down to the small one, because the self-action is one of the many actions the big algebra promised to honour. The question is whether anything gets lost on the way down. For some groups nothing is lost: the two algebras are the very same object. For others the big algebra is strictly larger, carrying information the modest one cannot see.

Which groups lose nothing? Exactly the cooperative ones from the previous unit — the groups that admit a fair, shift-resistant average. Amenability, the averaging property, is precisely the dividing line where the generous and the modest constructions agree. When they agree, the resulting algebra is tame enough to be rebuilt from finite matrices.

Visual Beginner

The picture shows the two constructions side by side and the map between them. On the left sits a large box labelled the full algebra, drawn with many faint arrows feeding into it from all directions — each arrow is one way the group could act. On the right sits a smaller box labelled the reduced algebra, fed by a single bold arrow labelled the group acting on its own functions. A downward arrow connects the large box to the small one, labelled the canonical map.

Beneath the downward arrow a two-position switch reads: amenable on one side, where the map is a perfect match and the boxes are copies of one another; non-amenable on the other, where the map collapses some of the big box and the two boxes differ. The switch position is set entirely by whether the group has a fair average.

Worked example Beginner

Take the simplest infinite group of symmetries: the whole numbers under addition, where the basic move is "add one." We build the modest, reduced algebra and watch the add-one move become a concrete operator.

The group acts on functions of its own elements. A function here assigns a number to each whole number, with the numbers squared and summed to a finite total. The add-one move sends such a function to a new one, shifted over by a single step. Call this shift operator $S$: it takes the value sitting at position $k$ and moves it to position $k+1$.

Now compute what $S$ does and undo it. Applying $S$ shifts every value forward one step. Applying the reverse move, subtract one, shifts every value back one step; call that operator $T$. Doing $S$ then $T$ returns every value to where it started, so $T$ is the exact inverse of $S$. Both $S$ and $T$ preserve the squared-and-summed total of any function, because shifting only relabels positions and never changes the values themselves.

The reduced algebra for the whole numbers is the collection of all operators you can assemble from this single shift $S$ and its inverse $T$ by adding and scaling and combining. Because the whole numbers have a fair average, this modest algebra is the same as the generous one; nothing was lost.

What this tells us: a group's basic move becomes a concrete shift operator, and for an averaging group the modest algebra built from that shift already captures everything the generous construction promised.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $G$ is a discrete group with identity $e$, $\mathbb{C}[G]$ its complex group ring, and ${\ell }^2(G)$ the Hilbert space with orthonormal basis $\{{\delta }_t:t\in G\}$. The group ring carries the involution $({\sum }_t{c}_{t}t{)}^{\ast }={\sum }_t\overline{c_t}{t}^{-1}$ and the ${\ell }^1$-norm $\Vert {\sum }_t{c}_{t}t{\Vert }_1={\sum }_t|c_t|$, making ${\ell }^1(G)$ a Banach $\ast$-algebra under convolution.

Definition (full group C*-algebra). A unitary representation of $G$ is a homomorphism $\pi :G\to \mathcal{U}(H_{\pi })$ into the unitaries of a Hilbert space; it extends to a $\ast$-representation of ${\ell }^1(G)$. Define the universal norm on $\mathbb{C}[G]$ by $$ |x|{\mathrm{u}} = \sup\pi |\pi(x)|, $$ the supremum over all unitary representations $\pi$ (the supremum is finite because $\Vert \pi (x)\Vert \le \Vert x{\Vert }_1$). The full group C*-algebra $C^{\ast }(G)$ is the completion of $\mathbb{C}[G]$ in $\Vert \cdot {\Vert }_{\mathrm{u}}$. It carries the universal property: every unitary representation of $G$ extends uniquely to a $\ast$-representation of $C^{\ast }(G)$, and $\mathrm{Rep}(G)\cong \mathrm{Rep}(C^{\ast }(G))$ ^{[Pedersen §7.1]}.

Definition (left-regular representation and reduced group C*-algebra). The left-regular representation $\lambda :G\to \mathcal{U}({\ell }^2(G))$ is $\lambda (s){\delta }_t={\delta }_{st}$. It extends to a faithful $\ast$-representation $\lambda :\mathbb{C}[G]\to B({\ell }^2(G))$ by $\lambda ({\sum }_t{c}_{t}t)={\sum }_t{c}_t\lambda (t)$. The reduced group C*-algebra is the norm closure $$ C^*_r(G) = \overline{\lambda(\mathbb{C}[G])}^{,|\cdot|} \subseteq B(\ell^2(G)), $$ with reduced norm $\Vert x{\Vert }_r=\Vert \lambda (x)\Vert$. Since $\lambda$ is one of the representations contributing to the universal norm, $\Vert x{\Vert }_r\le \Vert x{\Vert }_{\mathrm{u}}$ for all $x\in \mathbb{C}[G]$.

Definition (canonical surjection). The inequality $\Vert \cdot {\Vert }_r\le \Vert \cdot {\Vert }_{\mathrm{u}}$ means the identity on $\mathbb{C}[G]$ extends to a surjective $\ast$-homomorphism $$ \lambda_* : C^(G) \longrightarrow C^r(G), \qquad \lambda|_{\mathbb{C}[G]} = \lambda, $$ the canonical surjection. It is the $$-homomorphisminducedbytheregularrepresentationthroughtheuniversalpropertyof$C^*(G)$.

Definition (weak containment). A representation $\sigma$ is weakly contained in $\pi$, written $\sigma \prec \pi$, if $\Vert \sigma (x)\Vert \le \Vert \pi (x)\Vert$ for all $x\in \mathbb{C}[G]$; equivalently every matrix coefficient of $\sigma$ is a limit, uniformly on finite sets, of sums of matrix coefficients of $\pi$. The unit representation $1_G:G\to \mathcal{U}(\mathbb{C})$, $1_G(s)=1$, is weakly contained in $\lambda$ precisely when $\Vert {\sum }_t{c}_{t}t{\Vert }_r\ge |{\sum }_t{c}_t|$ fails to be strict — when the constant functional survives in the reduced norm.

Counterexamples to common slips Intermediate+

• The canonical surjection is always defined; it need not be injective. Every group gives a surjection $C^{\ast }(G)\to C_r^{\ast }(G)$. Injectivity is the special feature; it fails exactly for non-amenable $G$, where $\ker {\lambda }_{\ast }\ne 0$ carries the representations the regular one cannot see.
• Reduced does not mean smaller-and-tamer in every respect. For the free group $F_2$, the reduced algebra $C_r^{\ast }(F_2)$ is simple with a unique trace, whereas $C^{\ast }(F_2)$ has many ideals; "reduced" refers to discarding non-regular representations, not to having fewer ideals.
• $\lambda$ is faithful on $\mathbb{C}[G]$ for every $G$. The regular representation never kills a nonzero group-ring element: $⟨\lambda (x){\delta }_e,{\delta }_t⟩=c_t$ recovers every coefficient. Faithfulness on the dense subalgebra does not upgrade to $\Vert \cdot {\Vert }_r=\Vert \cdot {\Vert }_{\mathrm{u}}$; the completions can still differ.
• Amenability is a property of $G$, not of a chosen representation. One does not "make $G$ amenable" by selecting a good representation; the equality $C^{\ast }(G)=C_r^{\ast }(G)$ is forced by, and forces, the existence of an invariant mean.

Key theorem with proof Intermediate+

Theorem (Hulanicki). For a discrete group $G$ the following are equivalent: (i) $G$ is amenable; (ii) the unit representation $1_G$ is weakly contained in the left-regular representation $\lambda$; (iii) the canonical surjection $\lambda_ : C^(G) \to C^_r(G)$ is an isomorphism.* ^{[Hulanicki 1966; Brown-Ozawa §2.5]}

Proof. (i) $\Rightarrow$ (ii). Let $(F_n)$ be a Følner sequence for $G$ 39.05.06 and set ${\xi }_n=|F_n{|}^{-1/2}{1}_{F_n}\in {\ell }^2(G)$, a unit vector. For $s\in G$, $$ |\lambda(s)\xi_n - \xi_n|2^2 = |F_n|^{-1}, |\mathbf{1}{sF_n} - \mathbf{1}_{F_n}|_2^2 = \frac{|sF_n \triangle F_n|}{|F_n|} \longrightarrow 0, $$ so $({\xi }_n)$ is a sequence of almost-invariant unit vectors. For $x={\sum }_t{c}_{t}t\in \mathbb{C}[G]$, $$ \langle \lambda(x)\xi_n, \xi_n\rangle = \sum_t c_t \langle \lambda(t)\xi_n, \xi_n\rangle \longrightarrow \sum_t c_t = 1_G(x), $$ because $⟨\lambda (t){\xi }_n,{\xi }_n⟩\to 1$ for each fixed $t$ by almost-invariance. Thus the state $1_G$ is a weak* limit of vector states of $\lambda$, which is the definition of $1_G\prec \lambda$.

(ii) $\Rightarrow$ (iii). It suffices to show $\Vert x{\Vert }_{\mathrm{u}}\le \Vert x{\Vert }_r$ for $x\in \mathbb{C}[G]$, the reverse inequality being automatic. Fix $x={\sum }_t{c}_{t}t$ and let $\pi$ be any unitary representation. By Fell's absorption principle (proved below), $\lambda \otimes \pi \cong \lambda \otimes 1_{H_{\pi }}$, so for every $\pi$ the representation $\lambda \otimes \pi$ has the same norm as a multiple of $\lambda$: $\Vert (\lambda \otimes \pi )(x)\Vert =\Vert \lambda (x)\Vert =\Vert x{\Vert }_r$. Hypothesis (ii) gives $1_G\prec \lambda$, hence $\pi =1_G\otimes \pi \prec \lambda \otimes \pi \cong \lambda \otimes 1$, and the norm of $\lambda \otimes 1$ equals $\Vert x{\Vert }_r$. Therefore $\Vert \pi (x)\Vert \le \Vert x{\Vert }_r$ for every $\pi$, and taking the supremum over $\pi$ yields $\Vert x{\Vert }_{\mathrm{u}}\le \Vert x{\Vert }_r$. Combined with $\Vert x{\Vert }_r\le \Vert x{\Vert }_{\mathrm{u}}$, the norms coincide and ${\lambda }_{\ast }$ is an isometric $\ast$-isomorphism.

(iii) $\Rightarrow$ (i). Suppose ${\lambda }_{\ast }$ is an isomorphism. Then $1_G$, which extends to a character of $C^{\ast }(G)$, factors through $C_r^{\ast }(G)$, so $1_G\prec \lambda$. Almost-invariant unit vectors ${\xi }_n$ for $\lambda$ then exist: $\Vert \lambda (s){\xi }_n-{\xi }_n{\Vert }_2\to 0$ for each $s$. Passing to $|{\xi }_n{|}^2\in \mathrm{Prob}(G)$ and applying the Powers-Størmer inequality $\Vert |{\xi }_n{|}^2-|\lambda (s){\xi }_n{|}^2{\Vert }_1\le \Vert {\xi }_n-\lambda (s){\xi }_n{\Vert }_2\Vert {\xi }_n+\lambda (s){\xi }_n{\Vert }_2$ gives $\Vert s\cdot |{\xi }_n{|}^2-|{\xi }_n{|}^2{\Vert }_1\to 0$, which is Reiter's property $(P_1)$ 39.05.06. Hence $G$ is amenable. $\square$

Bridge. This theorem is the foundational reason the group-theoretic averaging property and the operator-algebraic identification of two completions are one phenomenon: a Følner sequence is literally a sequence of almost-invariant vectors for $\lambda$, and weak containment of the unit representation converts that approximate invariance into the collapse of the universal norm onto the reduced norm. This is exactly the mechanism by which the invariant mean of 39.05.06 becomes the equality $C^{\ast }(G)=C_r^{\ast }(G)$, and it builds toward the nuclearity statement, where the same Følner vectors supply the completely positive approximation property of 39.05.04. The implication (ii) $\Rightarrow$ (iii) generalises the absorption phenomenon: Fell's principle is dual to weak containment in that it trades a tensor with an arbitrary $\pi$ for a tensor with the unit representation, and putting these together gives the equivalence. The bridge is between geometry (almost-invariant vectors) and the comparison of two C*-completions, and it appears again in the master-tier proof that nuclearity of either group algebra is one more equivalent face of amenability.

Exercises Intermediate+

Advanced results Master

Theorem 1 (Hulanicki, full equivalence). For a discrete group $G$ the following are equivalent: (i) $G$ is amenable; (ii) $1_G\prec \lambda$; (iii) ${\lambda }_{\ast }:C^{\ast }(G)\to C_r^{\ast }(G)$ is an isomorphism; (iv) $C_r^{\ast }(G)$ is nuclear; (v) $C^{\ast }(G)$ is nuclear; (vi) the group von Neumann algebra $L(G)=\lambda (G{)}^{\prime \prime }$ is injective; (vii) $\Vert \lambda (\mu )\Vert =1$ for every symmetric probability measure $\mu$ whose support generates $G$ (Kesten) ^{[Hulanicki 1966; Brown-Ozawa §2.6]}. The equivalence (i)$\iff$(ii)$\iff$(iii) is the Key theorem; (i)$\Rightarrow$(iv) runs through the Følner CPAP below; (iv)$\Rightarrow$(v) is immediate from (iii) once the algebras coincide; (v)$\Rightarrow$(i) recovers an invariant mean from a CPAP net; (iv)$\iff$(vi) is the bidual-injectivity bridge of 39.05.04 applied to $L(G)=C_r^{\ast }(G{)}^{\prime \prime }$; (i)$\iff$(vii) is Kesten's criterion from 39.05.06.

Theorem 2 (amenable $\Rightarrow$ nuclear, via Følner). If $G$ is amenable, $C_r^{\ast }(G)$ is nuclear. The Følner sequence $(F_n)$ yields completely positive contractive maps factoring the identity through matrix algebras. Define ${\phi }_n:C_r^{\ast }(G)\to M_{F_n}(\mathbb{C})=B({\ell }^2(F_n))$ by compression ${\phi }_n(x)=P_{n}x{P}_n$, where $P_n$ is the orthogonal projection of ${\ell }^2(G)$ onto ${\ell }^2(F_n)$, and ${\psi }_n:M_{F_n}(\mathbb{C})\to C_r^{\ast }(G)$ by ${\psi }_n(T)=\frac{1}{|F_n|}{\sum }_{s,t\in F_n}⟨T{\delta }_t,{\delta }_s⟩\lambda (s{t}^{-1})$. Both are unital completely positive; a direct estimate using $|s{F}_n△F_n|/|F_n|\to 0$ shows $\Vert {\psi }_n{\phi }_n(\lambda (g))-\lambda (g)\Vert \to 0$ for each $g$, so ${\psi }_n{\phi }_n\to \mathrm{id}$ in point-norm. Hence $C_r^{\ast }(G)$ has the completely positive approximation property and is nuclear 39.05.04.

Theorem 3 (Fell's absorption principle). For every unitary representation $\pi$ of $G$, $\lambda \otimes \pi \cong \lambda \otimes 1_{H_{\pi }}$ via the unitary $W({\delta }_t\otimes \xi )={\delta }_t\otimes \pi (t)\xi$. Consequently $\pi \otimes \lambda$ is quasi-equivalent to a multiple of $\lambda$ for every $\pi$, the regular representation absorbs tensoring, and $C_r^{\ast }(G)$ is closed under the comultiplication $\lambda (g)\mapsto \lambda (g)\otimes \lambda (g)$ — the structural fact underlying the compact-quantum-group / Hopf-algebra picture of group C*-algebras ^{[Fell 1962]}.

Theorem 4 (abelian groups: Gelfand-Pontryagin). For abelian $G$, both $C^{\ast }(G)$ and $C_r^{\ast }(G)$ equal $C(\hat{G})$, the continuous functions on the Pontryagin dual $\hat{G}=\mathrm{Hom}(G,\mathbb{T})$, via the Fourier transform $\lambda (g)\mapsto (\chi \mapsto \chi (g))$. The Gelfand spectrum of the commutative C*-algebra $C^{\ast }(G)$ is exactly $\hat{G}$, so the full/reduced coincidence is the Plancherel identification ${\ell }^2(G)\cong L^2(\hat{G})$. Compact groups (here, finite groups in the discrete setting) give $C^{\ast }(G)={⨁}_{\pi \in \hat{G}}{M}_{d_{\pi }}(\mathbb{C})$ by Peter-Weyl, manifestly nuclear; virtually abelian groups are amenable, hence give nuclear group algebras.

Theorem 5 (free groups: non-nuclear but exact). For the free group $F_n$ with $2\le n\le \infty$, the algebra $C_r^{\ast }(F_n)$ is not nuclear (since $F_n$ is non-amenable, Theorem 1) yet is exact: it embeds in a nuclear algebra, indeed $C_r^{\ast }(F_n)$ is exact because $F_n$ acts amenably on its Gromov boundary $\partial F_n$, giving a nuclear crossed product $C(\partial F_n){⋊}_r{F}_n$ into which $C_r^{\ast }(F_n)$ embeds 39.05.05. Moreover $C_r^{\ast }(F_n)$ is simple with unique trace (Powers ^{[Powers 1975]}) and has the completely bounded approximation property with constant $1$ via the Haagerup inequality $\Vert \lambda (f)\Vert \le (k+1)\Vert f{\Vert }_2$ for $f$ supported on words of length $k$ ^{[Haagerup 1979]}. Thus the free group separates nuclearity from exactness: it is the canonical exact-but-not-nuclear example.

Theorem 6 (Haagerup property and a-T-menability). A group $G$ has the Haagerup property (is a-T-menable) if it admits a proper conditionally negative definite function, equivalently a proper affine isometric action on a Hilbert space. Amenable groups and free groups both have it; it is a weak amenability-like condition compatible with non-amenability, sitting opposite Kazhdan's property (T). The Haagerup property implies the Baum-Connes conjecture (Higson-Kasparov), tying the analytic flexibility of $C_r^{\ast }(G)$ to the $K$-theoretic computation of its $K$-groups. For free groups it is witnessed by the word-length function $\ell$, which is conditionally negative definite on $F_n$.

Synthesis. Amenability is the central insight that fuses the representation theory of a group with the structure of its operator algebras: the canonical surjection $C^{\ast }(G)\to C_r^{\ast }(G)$ is an isomorphism exactly when an invariant mean exists, and the foundational reason is that a Følner sequence is a sequence of almost-invariant vectors for $\lambda$, which both witnesses $1_G\prec \lambda$ and supplies the completely positive approximation property. The von Neumann-Day equivalences of 39.05.06 thereby acquire two new faces — nuclearity of the group C*-algebra and injectivity of $L(G)$ — so that amenability of $G$, equality of the two completions, nuclearity, and injectivity are one phenomenon viewed through group theory, representation theory, tensor-norm theory, and von Neumann algebra theory. Putting these together, Fell's absorption is dual to weak containment: it trades a tensor with an arbitrary representation for a multiple of the regular one, which is exactly what converts $1_G\prec \lambda$ into the collapse $\Vert \cdot {\Vert }_{\mathrm{u}}=\Vert \cdot {\Vert }_r$. The free group is where this all comes apart in a controlled way: $C_r^{\ast }(F_n)$ is non-nuclear yet exact, simple, and a-T-menable, generalising the integers' tame picture into a rigid one and feeding the exactness theory of 39.05.05 and the Baum-Connes machinery. The bridge from $\mathbb{Z}$, whose $C^{\ast }(\mathbb{Z})=C(\mathbb{T})$ is the Pontryagin-dual of the simplest amenable group, to $F_2$, whose reduced algebra is a simple non-nuclear factor-generating object, is the entire spectrum the nuclearity-exactness chapter measures.

Full proof set Master

Proposition 1 (the canonical surjection exists and is surjective). For every discrete group $G$ there is a unique surjective $\ast$-homomorphism ${\lambda }_{\ast }:C^{\ast }(G)\to C_r^{\ast }(G)$ restricting to $\lambda$ on $\mathbb{C}[G]$.

Proof. The regular representation $\lambda$ is a unitary representation of $G$, hence one of the representations entering the universal norm, so $\Vert x{\Vert }_r=\Vert \lambda (x)\Vert \le {\sup }_{\pi }\Vert \pi (x)\Vert =\Vert x{\Vert }_{\mathrm{u}}$ for all $x\in \mathbb{C}[G]$. The identity map on $\mathbb{C}[G]$ is therefore norm-decreasing from $(\mathbb{C}[G],\Vert \cdot {\Vert }_{\mathrm{u}})$ to $(\mathbb{C}[G],\Vert \cdot {\Vert }_r)$, and extends by continuity to a $\ast$-homomorphism ${\lambda }_{\ast }:C^{\ast }(G)\to C_r^{\ast }(G)$. Its image contains the dense subalgebra $\lambda (\mathbb{C}[G])$ and is closed (the image of a C*-algebra homomorphism is closed), so ${\lambda }_{\ast }$ is surjective. Uniqueness is forced by agreement on the dense subalgebra $\mathbb{C}[G]$. $\square$

Proposition 2 (Følner vectors are almost invariant). If $(F_n)$ is a Følner sequence for $G$ and ${\xi }_n=|F_n{|}^{-1/2}{1}_{F_n}\in {\ell }^2(G)$, then $\Vert {\xi }_n{\Vert }_2=1$ and $\Vert \lambda (s){\xi }_n-{\xi }_n{\Vert }_2\to 0$ for every $s\in G$.

Proof. Normalisation: $\Vert {\xi }_n{\Vert }_2^2=|F_n{|}^{-1}{\sum }_{t\in F_n}1=1$. Since $\lambda (s)1_{F_n}=1_{s{F}_n}$ and $1_{s{F}_n},1_{F_n}$ take values in $\{0,1\}$, $$ |\lambda(s)\xi_n - \xi_n|2^2 = |F_n|^{-1}\sum{t}\big(\mathbf{1}{sF_n}(t) - \mathbf{1}{F_n}(t)\big)^2 = \frac{|sF_n \triangle F_n|}{|F_n|}, $$ because $(1_A-1_B{)}^2=1_{A△B}$ pointwise. The Følner condition sends the right-hand side to $0$. $\square$

Proposition 3 (Fell absorption). For any unitary representation $\pi$ of $G$ on $H_{\pi }$, the unitary $W({\delta }_t\otimes \xi )={\delta }_t\otimes \pi (t)\xi$ on ${\ell }^2(G)\otimes H_{\pi }$ satisfies $W(\lambda (s)\otimes 1)W^{\ast }=\lambda (s)\otimes \pi (s)$ for all $s$, so $\lambda \otimes 1_{H_{\pi }}\cong \lambda \otimes \pi$.

Proof. $W$ is unitary: it preserves the orthonormal basis structure, acting on the fibre over $t$ by the unitary $\pi (t)$, with adjoint $W^{\ast }({\delta }_t\otimes \xi )={\delta }_t\otimes \pi (t{)}^{-1}\xi$. For $s,t\in G$ and $\xi \in H_{\pi }$, $$ W(\lambda(s)\otimes 1)(\delta_t \otimes \xi) = W(\delta_{st}\otimes \xi) = \delta_{st}\otimes \pi(st)\xi, $$ $$ (\lambda(s)\otimes \pi(s))W(\delta_t \otimes \xi) = (\lambda(s)\otimes \pi(s))(\delta_t \otimes \pi(t)\xi) = \delta_{st}\otimes \pi(s)\pi(t)\xi = \delta_{st}\otimes\pi(st)\xi. $$ The two coincide on a basis, so $W(\lambda (s)\otimes 1)=(\lambda (s)\otimes \pi (s))W$, giving the stated equivalence. $\square$

Proposition 4 (amenable $\Rightarrow$ $C_r^{\ast }(G)$ nuclear). If $G$ is amenable then the maps ${\phi }_n,{\psi }_n$ of Theorem 2 are UCP and witness the CPAP, so $C_r^{\ast }(G)$ is nuclear.

Proof. Let $P_n$ be the projection onto ${\ell }^2(F_n)$. The compression ${\phi }_n(x)=P_{n}x{P}_n\in B({\ell }^2(F_n))\cong M_{|F_n|}(\mathbb{C})$ is unital completely positive, being a compression of a $\ast$-representation 39.05.02. The map ${\psi }_n(T)=|F_n{|}^{-1}{\sum }_{s,t\in F_n}⟨T{\delta }_t,{\delta }_s⟩\lambda (s{t}^{-1})$ is UCP: it is the composition of the embedding $M_{|F_n|}(\mathbb{C})\hookrightarrow B({\ell }^2(G))$ with the averaging $T\mapsto |F_n{|}^{-1}{\sum }_{s,t}⟨T{\delta }_t,{\delta }_s⟩\lambda (s{t}^{-1})$, which is a state-weighted sum of the completely positive maps $T\mapsto \lambda (s{)}^{\ast }{T}^{\prime }\lambda (t)$ collapsed onto $\lambda (s{t}^{-1})$, and a computation on matrix units confirms complete positivity and ${\psi }_n(1)=|F_n{|}^{-1}{\sum }_{t\in F_n}\lambda (e)=1$. For a generator $\lambda (g)$, $$ \psi_n\varphi_n(\lambda(g)) = \frac{1}{|F_n|}\sum_{s,t\in F_n}\langle P_n\lambda(g)P_n \delta_t, \delta_s\rangle \lambda(st^{-1}) = \frac{1}{|F_n|}\sum_{\substack{t\in F_n\ gt\in F_n}} \lambda(gt,t^{-1}) = \frac{|F_n \cap g^{-1}F_n|}{|F_n|},\lambda(g), $$ since $⟨P_n\lambda (g)P_n{\delta }_t,{\delta }_s⟩={\delta }_{s,gt}{1}_{F_n}(gt)$ for $t\in F_n$ and $s{t}^{-1}=gt{t}^{-1}=g$. The coefficient $|F_n\cap g^{-1}{F}_n|/|F_n|\to 1$ because $|F_n△g^{-1}{F}_n|/|F_n|\to 0$. Hence $\Vert {\psi }_n{\phi }_n(\lambda (g))-\lambda (g)\Vert =|1-|F_n\cap g^{-1}{F}_n|/|F_n||\to 0$, and by linearity and uniform boundedness ($\Vert {\psi }_n{\phi }_n\Vert \le 1$) the convergence ${\psi }_n{\phi }_n\to \mathrm{id}$ holds point-norm on all of $C_r^{\ast }(G)$. Thus $C_r^{\ast }(G)$ has the CPAP and is nuclear 39.05.04. $\square$

Proposition 5 ($C^{\ast }(\mathbb{Z})=C_r^{\ast }(\mathbb{Z})=C(\mathbb{T})$). The full and reduced C*-algebras of $\mathbb{Z}$ are both isometrically isomorphic to $C(\mathbb{T})$.

Proof. A unitary representation of $\mathbb{Z}$ is determined by the single unitary $U=\pi (1)$; conversely any unitary $U$ on any $H$ gives such a representation. Thus $C^{\ast }(\mathbb{Z})$ is the universal C*-algebra generated by one unitary $u$. For a Laurent polynomial $p(u)={\sum }_k{c}_k{u}^k$, the continuous functional calculus identifies $\Vert \pi (p)\Vert ={\sup }_{z\in \sigma (U)}|p(z)|\le {\sup }_{z\in \mathbb{T}}|p(z)|$, with equality attained by the multiplication representation $U=M_z$ on $L^2(\mathbb{T})$ whose spectrum is all of $\mathbb{T}$. Hence $\Vert p{\Vert }_{\mathrm{u}}=\Vert p{\Vert }_{C(\mathbb{T})}$ and $C^{\ast }(\mathbb{Z})\cong C(\mathbb{T})$ by Gelfand, which is $C(\hat{\mathbb{Z}})$ by Pontryagin duality $\hat{\mathbb{Z}}=\mathbb{T}$. On the reduced side $\lambda (1)$ is the bilateral shift on ${\ell }^2(\mathbb{Z})$; via the Fourier transform ${\ell }^2(\mathbb{Z})\cong L^2(\mathbb{T})$ it becomes multiplication by $z$, with full spectrum $\mathbb{T}$, so $C_r^{\ast }(\mathbb{Z})\cong C(\mathbb{T})$ as well. The two coincide, consistent with the amenability of $\mathbb{Z}$. $\square$

Proposition 6 ($C_r^{\ast }(F_2)$ is non-nuclear). The reduced group C*-algebra of the free group $F_2$ is not nuclear.

Proof. By Theorem 1, $C_r^{\ast }(G)$ is nuclear if and only if $G$ is amenable. The free group $F_2$ is non-amenable: it admits a paradoxical decomposition, equivalently no invariant mean 39.05.06. Hence $C_r^{\ast }(F_2)$ is not nuclear. Concretely, if it were nuclear, the canonical surjection ${\lambda }_{\ast }$ would be an isomorphism (the CPAP net would transport to $C^{\ast }(F_2)$), forcing $1_{F_2}\prec \lambda$ and producing almost-invariant vectors, hence a Følner sequence — which $F_2$ does not possess. The obstruction is detected against the partner $B=C_r^{\ast }(F_2{)}^{\mathrm{op}}$, where $\min \ne \max$ on the tensor product, exactly the Takesaki example referenced in 39.05.04. $\square$

Connections Master

• Amenable groups, Følner sequences, and invariant means 39.05.06. Hulanicki's theorem is the operator-algebraic continuation of the von Neumann-Day equivalences: the Følner sequence that witnesses amenability is literally the sequence of almost-invariant vectors ${\xi }_n=|F_n{|}^{-1/2}{1}_{F_n}$ for the regular representation, and weak containment $1_G\prec \lambda$ is the analytic shadow of the invariant mean. The foundational reason the two completions $C^{\ast }(G)$ and $C_r^{\ast }(G)$ coincide is the same averaging property that makes $G$ amenable, so this unit is exactly where the group-theoretic dividing line becomes a comparison of C*-completions.

• Nuclear C*-algebras and the completely positive approximation property 39.05.04. The implication amenable $\Rightarrow$ nuclear is realised by the Følner CPAP net ${\phi }_n,{\psi }_n$ of Theorem 2; the compressions onto ${\ell }^2(F_n)$ are the completely positive finite-rank factorisations of the identity. This is dual to the general theory: nuclearity of $C^{\ast }(G)$ is injectivity of $L(G)$, the group instance of the bidual-injectivity circle. The free group supplies the chapter's canonical non-nuclear algebra, the same obstruction that makes $\min \ne \max$ against a free-group partner.

• Exactness and exact C*-algebras 39.05.05. Free groups separate nuclearity from exactness: $C_r^{\ast }(F_n)$ is non-nuclear yet exact, because $F_n$ acts amenably on its Gromov boundary, embedding $C_r^{\ast }(F_n)$ into the nuclear crossed product $C(\partial F_n){⋊}_r{F}_n$. This unit furnishes the motivating example for the exactness theory developed there — exactness is the subalgebra-stable weakening of nuclearity that the reduced free-group algebra satisfies but nuclearity does not.

• Pontryagin duality and the Fourier transform on locally compact abelian groups 22.04.03. For abelian $G$, the identification $C^{\ast }(G)=C(\hat{G})$ is precisely Pontryagin duality read through the Gelfand transform: the spectrum of the commutative group C*-algebra is the dual group, and the Plancherel theorem ${\ell }^2(G)\cong L^2(\hat{G})$ is the unitary equivalence making $\lambda$ multiplication by characters. The amenability of abelian groups is the structural reason this commutative picture is complete.

Historical & philosophical context Master

The full and reduced group C*-algebras emerged from the harmonic analysis of locally compact groups in the 1950s and 1960s, as the C*-completions of the convolution algebra $L^1(G)$ that organise, respectively, all unitary representations and the regular representation. The decisive structural theorem is due to Andrzej Hulanicki, whose 1964 and 1966 Studia Mathematica papers proved that amenability of $G$ is equivalent to weak containment of the unit representation in the regular representation, and hence to the coincidence of the full and reduced norms ^{[Hulanicki 1966]}. The weak-containment language itself was developed by James M. G. Fell, whose 1962 papers introduced the topology on the dual and the absorption principle $\lambda \otimes \pi \cong \lambda \otimes 1$ that makes the regular representation absorb tensoring ^{[Fell 1962]}.

The free group became the proving ground for everything amenability is not. Robert Powers showed in 1975 that $C_r^{\ast }(F_2)$ is simple with a unique trace ^{[Powers 1975]}, the first example of a simple C*-algebra arising so directly from a group, and the prototype for the rigidity of reduced free-group algebras. Uffe Haagerup's 1979 Inventiones paper proved that $C_r^{\ast }(F_n)$ has the metric approximation property despite being non-nuclear, via the inequality controlling the norm of convolution by length-$k$ functions ^{[Haagerup 1979]}; this separated approximation properties from nuclearity and seeded the theory of weak amenability and the Haagerup property. The identification of amenability with nuclearity of $C^{\ast }(G)$, with injectivity of $L(G)$ through Connes' classification, and with the completely positive approximation property is the organising framework that Brown and Ozawa develop, placing group C*-algebras at the centre of the finite-dimensional approximation programme.

Bibliography Master

@article{Hulanicki1966,
  author  = {Hulanicki, Andrzej},
  title   = {Means and F{\o}lner condition on locally compact groups},
  journal = {Studia Mathematica},
  volume  = {27},
  year    = {1966},
  pages   = {87--104}
}

@article{Fell1962,
  author  = {Fell, James M. G.},
  title   = {Weak containment and induced representations of groups},
  journal = {Canadian Journal of Mathematics},
  volume  = {14},
  year    = {1962},
  pages   = {237--268}
}

@article{Powers1975,
  author  = {Powers, Robert T.},
  title   = {Simplicity of the C*-algebra associated with the free group on two generators},
  journal = {Duke Mathematical Journal},
  volume  = {42},
  year    = {1975},
  pages   = {151--156}
}

@article{Haagerup1979,
  author  = {Haagerup, Uffe},
  title   = {An example of a non-nuclear C*-algebra which has the metric approximation property},
  journal = {Inventiones Mathematicae},
  volume  = {50},
  year    = {1979},
  pages   = {279--293}
}

@article{Connes1976b,
  author  = {Connes, Alain},
  title   = {Classification of injective factors. Cases $II_1$, $II_\infty$, $III_\lambda$, $\lambda \neq 1$},
  journal = {Annals of Mathematics},
  volume  = {104},
  year    = {1976},
  pages   = {73--115}
}

@book{Pedersen1979,
  author    = {Pedersen, Gert K.},
  title     = {C*-Algebras and Their Automorphism Groups},
  publisher = {Academic Press},
  year      = {1979}
}

@book{BrownOzawa2008b,
  author    = {Brown, Nathanial P. and Ozawa, Narutaka},
  title     = {C*-Algebras and Finite-Dimensional Approximations},
  series    = {Graduate Studies in Mathematics},
  volume    = {88},
  publisher = {American Mathematical Society},
  year      = {2008}
}

@book{Dixmier1977,
  author    = {Dixmier, Jacques},
  title     = {C*-Algebras},
  publisher = {North-Holland},
  year      = {1977}
}