7b. Tori, amenability

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In the remainder of this chapter we explore the whole new world opened by the [math]C^*[/math]-algebra theory, with the study of several key examples. We will first discuss the group duals, also called noncommutative tori. Let us start with a well-known result:

Theorem

The compact abelian groups [math]G[/math] are in correspondence with the discrete abelian groups [math]\Gamma[/math], via Pontrjagin duality,

[[math]] G=\widehat{\Gamma}\quad,\quad \Gamma=\widehat{G} [[/math]]

with the dual of a locally compact group [math]L[/math] being the locally compact group [math]\widehat{L}[/math] consisting of the continuous group characters [math]\chi:L\to\mathbb T[/math].

Show Proof

This is something very standard, the idea being that, given a group [math]L[/math] as above, its continuous characters [math]\chi:L\to\mathbb T[/math] form indeed a group, that we can call [math]\widehat{L}[/math]. The correspondence [math]L\to\widehat{L}[/math] constructed in this way has then the following properties:

(1) We have [math]\widehat{\mathbb Z}_N=\mathbb Z_N[/math]. This is the basic computation to be performed, before anything else, and which is something algebraic, with roots of unity.

(2) More generally, the dual of a finite abelian group [math]G=\mathbb Z_{N_1}\times\ldots\times\mathbb Z_{N_k}[/math] is the group [math]G[/math] itself. This comes indeed from (1) and from [math]\widehat{G\times H}=\widehat{G}\times\widehat{H}[/math].

(3) At the opposite end now, that of the locally compact groups which are not compact, nor discrete, the main example, which is standard, is [math]\widehat{\mathbb R}=\mathbb R[/math].

(4) Getting now to what we are interested in, it follows from the definition of the correspondence [math]L\to\widehat{L}[/math] that when [math]L[/math] is compact [math]\widehat{L}[/math] is discrete, and vice versa.

(5) Finally, in order to best understand this latter phenomenon, the best is to work out the main pair of examples, which are [math]\widehat{\mathbb T}=\mathbb Z[/math] and [math]\widehat{\mathbb Z}=\mathbb T[/math].

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Our claim now is that, by using operator algebra theory, we can talk about the dual [math]G=\widehat{\Gamma}[/math] of any discrete group [math]\Gamma[/math]. Let us start our discussion in the von Neumann algebra setting, where things are particularly simple. We have here:

Theorem

Given a discrete group [math]\Gamma[/math], we can construct its von Neumann algebra

[[math]] L(\Gamma)\subset B(l^2(\Gamma)) [[/math]]

by using the left regular representation. This algebra has a faithful positive trace, [math]tr(g)=\delta_{g,1}[/math], and when [math]\Gamma[/math] is abelian we have an isomorphism of tracial von Neumann algebras

[[math]] L(\Gamma)\simeq L^\infty(G) [[/math]]

given by a Fourier type transform, where [math]G=\widehat{\Gamma}[/math] is the compact dual of [math]\Gamma[/math].

Show Proof

There are many assertions here, the idea being as follows:

(1) The first part is standard, with the left regular representation of [math]\Gamma[/math] working as expected, and being a unitary representation, as follows:

[[math]] \Gamma\subset B(l^2(\Gamma))\quad,\quad \pi(g):h\to gh [[/math]]

(2) The positivity of the trace comes from the following alternative formula for it, with the equivalence with the definition in the statement being clear:

[[math]] tr(T)= \lt T1,1 \gt [[/math]]

(3) The third part is standard as well, because when [math]\Gamma[/math] is abelian the algebra [math]L(\Gamma)[/math] is commutative, and its spectral decomposition leads by delinearization to the group characters [math]\chi:\Gamma\to\mathbb T[/math], and so the dual group [math]G=\widehat{\Gamma}[/math], as indicated.

(4) Finally, the fact that our isomorphism transforms the trace of [math]L(\Gamma)[/math] into the Haar integration functional of [math]L^\infty(G)[/math] is clear. Moreover, the study of various examples show that what we constructed is in fact the Fourier transform, in its various incarnations.

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Getting back now to our quantum space questions, we have a beginning of answer, because based on the above, we can formulate the following definition:

Definition

Given a discrete group [math]\Gamma[/math], not necessarily abelian, we can construct its abstract dual [math]G=\widehat{\Gamma}[/math] as a quantum measured space, via the following formula:

[[math]] L^\infty(G)=L(\Gamma) [[/math]]

In the case where [math]\Gamma[/math] happens to be abelian, this quantum space [math]G=\widehat{\Gamma}[/math] is a classical space, namely the usual Pontrjagin dual of [math]\Gamma[/math], endowed with its Haar measure.

Let us discuss now the same questions, in the [math]C^*[/math]-algebra setting. The situation here is more complicated than in the von Neumann algebra setting, as follows:

Proposition

Associated to any discrete group [math]\Gamma[/math] are several group [math]C^*[/math]-algebras,

[[math]] C^*(\Gamma)\to C^*_\pi(\Gamma)\to C^*_{red}(\Gamma) [[/math]]

which are constructed as follows:

[math]C^*(\Gamma)[/math] is the closure of the group algebra [math]\mathbb C[\Gamma][/math], with involution [math]g^*=g^{-1}[/math], with respect to the maximal [math]C^*[/math]-seminorm on this [math]*[/math]-algebra, which is a [math]C^*[/math]-norm.
[math]C^*_{red}(\Gamma)[/math] is the norm closure of the group algebra [math]\mathbb C[\Gamma][/math] in the left regular representation, on the Hilbert space [math]l^2(\Gamma)[/math], given by [math]\lambda(g)(h)=gh[/math] and linearity.
[math]C^*_\pi(\Gamma)[/math] can be any intermediate [math]C^*[/math]-algebra, but for best results, the indexing object [math]\pi[/math] must be a unitary group representation, satisfying [math]\pi\otimes\pi\subset\pi[/math].

Show Proof

This is something quite technical, with (2) being very similar to the von Neumann algebra construction from Theorem 7.20, with (1) being something new, with the norm property there coming from (2), and finally with (3) being an informal statement, that we will comment on later, once we will know about compact quantum groups.

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When [math]\Gamma[/math] is finite, or abelian, or more generally amenable, all the above group algebras coincide. In the abelian case, that we are particularly interested in here, the precise result is as follows, complementing the [math]L^\infty[/math] analysis from Theorem 7.20:

Theorem

When [math]\Gamma[/math] is abelian all its group [math]C^*[/math]-algebras coincide, and we have an isomorphism as follows, given by a Fourier type transform,

[[math]] C^*(\Gamma)\simeq C(G) [[/math]]

where [math]G=\widehat{\Gamma}[/math] is the compact dual of [math]\Gamma[/math]. Moreover, this isomorphism transforms the standard group algebra trace [math]tr(g)=\delta_{g,1}[/math] into the Haar integration of [math]G[/math].

Show Proof

Since [math]\Gamma[/math] is abelian, any of its group [math]C^*[/math]-algebras [math]A=C^*_\pi(\Gamma)[/math] is commutative. Thus, we can apply the Gelfand theorem, and we obtain [math]A=C(X)[/math], with [math]X=Spec(A)[/math]. But the spectrum [math]X=Spec(A)[/math], consisting of the characters [math]\chi:A\to\mathbb C[/math], can be identified by delinearizing with the Pontrjagin dual [math]G=\widehat{\Gamma}[/math], and this gives the results.

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At a more advanced level now, we have the following result:

Theorem

For a discrete group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], the following conditions are equivalent, and if they are satisfied, we say that [math]\Gamma[/math] is amenable:

The projection map [math]C^*(\Gamma)\to C^*_{red}(\Gamma)[/math] is an isomorphism.
The morphism [math]\varepsilon:C^*(\Gamma)\to\mathbb C[/math] given by [math]g\to1[/math] factorizes through [math]C^*_{red}(\Gamma)[/math].
We have [math]N\in\sigma(Re(g_1+\ldots+g_N))[/math], the spectrum being taken inside [math]C^*_{red}(\Gamma)[/math].

The amenable groups include all finite groups, and all abelian groups. As a basic example of a non-amenable group, we have the free group [math]F_N[/math], with [math]N\geq2[/math].

Show Proof

There are several things to be proved, the idea being as follows:

(1) The implication [math](1)\implies(2)[/math] is trivial, and [math](2)\implies(3)[/math] comes from the following computation, which shows that [math]N-Re(g_1+\ldots+g_N)[/math] is not invertible inside [math]C^*_{red}(\Gamma)[/math]:

[[math]] \begin{eqnarray*} \varepsilon[N-Re(g_1+\ldots+g_N)] &=&N-Re[\varepsilon(g_1)+\ldots+\varepsilon(g_n)]\\ &=&N-N\\ &=&0 \end{eqnarray*} [[/math]]

As for [math](3)\implies(1)[/math], this is something more advanced, that we will not need for the moment. We will be back to this later, directly in a more general setting.

(2) The fact that any finite group [math]G[/math] is amenable is clear, because all the group [math]C^*[/math]-algebras are equal to the usual group [math]*[/math]-algebra [math]\mathbb C[G][/math], in this case. As for the case of the abelian groups, these are all amenable as well, as shown by Theorem 7.23.

(3) It remains to prove that [math]F_N[/math] with [math]N\geq2[/math] is not amenable. By using [math]F_2\subset F_N[/math], it is enough to do this at [math]N=2[/math]. So, consider the free group [math]F_2= \lt g,h \gt [/math]. In order to prove that [math]F_2[/math] is not amenable, we use [math](1)\implies(3)[/math]. To be more precise, it is enough to show that 4 is not in the spectrum of the following operator:

[[math]] T=\lambda(g)+\lambda(g^{-1})+\lambda(h)+\lambda(h^{-1}) [[/math]]

This is a sum of four terms, each of them acting via [math]\delta_w\to\delta_{ew}[/math], with [math]e[/math] being a certain length one word. Thus if [math]w\neq 1[/math] has length [math]n[/math] then [math]T(\delta_w)[/math] is a sum of four Dirac masses, three of them at words of length [math]n+1[/math] and the remaining one at a length [math]n-1[/math] word. We can therefore decompose [math]T[/math] as a sum [math]T_++T_-[/math], where [math]T_+[/math] adds and [math]T_-[/math] cuts:

[[math]] T=T_++T_- [[/math]]

That is, if [math]w\neq 1[/math] is a word, say beginning with [math]h[/math], then [math]T_\pm[/math] act on [math]\delta_w[/math] as follows:

[[math]] T_+(\delta_w)=\delta_{gw}+\delta_{g^{-1}w}+\delta_{hw}\quad,\quad T_-(\delta_w)=\delta_{h^{-1}w} [[/math]]

It follows from definitions that we have [math]T_+^*=T_-[/math]. We can use the following trick:

[[math]] (T_++T_-)^2+\left( i(T_+-T_-)\right)^2=2(T_+T_-+T_-T_+) [[/math]]

Indeed, this gives [math](T_++T_-)^2\leq 2(T_+T_-+T_-T_+)[/math], and we obtain in this way:

[[math]] ||T||^2=||T_++T_-||^2\leq 2||T_+T_-+T_-T_+|| [[/math]]

Let [math]w\neq 1[/math] be a word, say beginning with [math]h[/math]. We have then:

[[math]] T_-T_+(\delta_w)=T_-(\delta_{gw}+\delta_{g^{-1}w}+\delta_{hw})=3\delta_w [[/math]]

The action of [math]T_-T_+[/math] on the remaining vector [math]\delta_1[/math] is computed as follows:

[[math]] T_-T_+(\delta_1)=T_-(\delta_g+\delta_{g^{-1}}+\delta_{h}+\delta_{h^{-1}})=4\delta_1 [[/math]]

Summing up, with [math]P:\delta_w\to\delta_1[/math] being the projection onto [math]\mathbb C\delta_1[/math], we have:

[[math]] T_-T_+=3+P [[/math]]

On the other hand we have [math]T_+T_-(\delta_1)=T_+(0)=0[/math], so the subspace [math]\mathbb C\delta_1[/math] is invariant under the operator [math]T_+T_-+T_-T_+[/math]. We have the following norm estimate:

[[math]] ||T||^2 \leq2||T_+T_-+T_-T_+|| \leq2\cdot\max\left\{||3+P||,\,\,\, ||(T_+T_-+T_-T_+)(1-P)||\right\} [[/math]]

The norm of [math]3+P[/math] is equal to [math]4[/math], and the other norm is estimated as follows:

[[math]] \begin{eqnarray*} ||(T_+T_-+T_-T_+)(1-P)|| &\leq&||T_+T_-||+||(3+P)(1-P)||\\ &=&||T_-T_+||+3\\ &=&7 \end{eqnarray*} [[/math]]

Thus we have [math]||T||\leq\sqrt{14} \lt 4[/math], and this finishes the proof.

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General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].