14b. Amenability

[math] \newcommand{\mathds}{\mathbb}[/math]

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With this discussion made, and with the reiterated warning that a lot of modesty and basic common sense is needed, in order to deal with such questions, let us get now into the real thing, namely the understanding of the following projection map:

[[math]] \pi:A_{full}\to A_{red} [[/math]]

As already mentioned before, on numerous occasions, when the algebra [math]A[/math] is cocommutative, [math]A\sim C^*(\Gamma)[/math], and with the underlying group [math]\Gamma[/math] being assumed amenable, this projection map is an isomorphism. And the contrary happens when [math]\Gamma[/math] is not amenable.

This leads us into the amenability question for the general Woronowicz algebras [math]A[/math]. We have seen the basic theory here in chapter 3 above, in the form of a list of equivalent conditions, which altogether are called amenability. The theory presented there, worked out now in more detail, and with a few items added, is as follows:

Theorem

Let [math]A_{full}[/math] be the enveloping [math]C^*[/math]-algebra of [math]\mathcal A[/math], and let [math]A_{red}[/math] be the quotient of [math]A[/math] by the null ideal of the Haar integration. The following are then equivalent:

• The Haar functional of [math]A_{full}[/math] is faithful.
• The projection map [math]A_{full}\to A_{red}[/math] is an isomorphism.
• The counit map [math]\varepsilon:A\to\mathbb C[/math] factorizes through [math]A_{red}[/math].
• We have [math]N\in\sigma(Re(\chi_u))[/math], the spectrum being taken inside [math]A_{red}[/math].
• [math]||ax_k-\varepsilon(a)x_k||\to0[/math] for any [math]a\in\mathcal A[/math], for certain norm [math]1[/math] vectors [math]x_k\in L^2(A)[/math].

If this is the case, we say that the underlying discrete quantum group [math]\Gamma[/math] is amenable.

Show Proof

Before starting, we should mention that amenability and the present result are a bit like the Spectral Theorem, in the sense that knowing that the result formally holds does not help much, and in practice, one needs to remember the proof as well. For this reason, we will work out explicitely all the possible implications between (1-5), whenever possible, adding to the global formal proof, which will be linear, as follows:

[[math]] (1)\implies(2)\implies(3)\implies(4)\implies(5)\implies(1) [[/math]]

In order to prove these implications, and the other ones too, the general idea is that this is is well-known in the group dual case, [math]A=C^*(\Gamma)[/math], with [math]\Gamma[/math] being a usual discrete group, and in general, the result follows by adapting the group dual case proof.

[math](1)\iff(2)[/math] This follows from the fact that the GNS construction for the algebra [math]A_{full}[/math] with respect to the Haar functional produces the algebra [math]A_{red}[/math].

[math](2)\implies(3)[/math] This is trivial, because we have quotient maps [math]A_{full}\to A\to A_{red}[/math], and so our assumption [math]A_{full}=A_{red}[/math] implies that we have [math]A=A_{red}[/math].

[math](3)\implies(2)[/math] Assume indeed that we have a counit map [math]\varepsilon:A_{red}\to\mathbb C[/math]. In order to prove [math]A_{full}=A_{red}[/math], we can use the right regular corepresentation. Indeed, we can define such a corepresentation by the following formula:

[[math]] W(a\otimes x)=\Delta(a)(1\otimes x) [[/math]]

This corepresentation is unitary, so we can define a morphism as follows:

[[math]] \Delta':A_{red}\to A_{red}\otimes A_{full} [[/math]]

[[math]] a\to W(a\otimes1)W^* [[/math]]

Now by composing with [math]\varepsilon\otimes id[/math], we obtain a morphism as follows:

[[math]] (\varepsilon\otimes id)\Delta':A_{red}\to A_{full} [[/math]]

[[math]] u_{ij}\to u_{ij} [[/math]]

Thus, we have our inverse for the canonical projection [math]A_{full}\to A_{red}[/math], as desired.

[math](3)\implies(4)[/math] This implication is clear, because we have:

[[math]] \begin{eqnarray*} \varepsilon(Re(\chi_u)) &=&\frac{1}{2}\left(\sum_{i=1}^N\varepsilon(u_{ii})+\sum_{i=1}^N\varepsilon(u_{ii}^*)\right)\\ &=&\frac{1}{2}(N+N)\\ &=&N \end{eqnarray*} [[/math]]

Thus the element [math]N-Re(\chi_u)[/math] is not invertible in [math]A_{red}[/math], as claimed.

[math](4)\implies(3)[/math] In terms of the corepresentation [math]v=u+\bar{u}[/math], whose dimension is [math]2N[/math] and whose character is [math]2Re(\chi_u)[/math], our assumption [math]N\in\sigma(Re(\chi_u))[/math] reads:

[[math]] \dim v\in\sigma(\chi_v) [[/math]]

By functional calculus the same must hold for [math]w=v+1[/math], and then once again by functional calculus, the same must hold for any tensor power of [math]w[/math]:

[[math]] w_k=w^{\otimes k} [[/math]]

Now choose for each [math]k\in\mathbb N[/math] a state [math]\varepsilon_k\in A_{red}^*[/math] having the following property:

[[math]] \varepsilon_k(w_k)=\dim w_k [[/math]]

By Peter-Weyl we must have [math]\varepsilon_k(r)=\dim r[/math] for any [math]r\leq w_k[/math], and since any irreducible corepresentation appears in this way, the sequence [math]\varepsilon_k[/math] converges to a counit map:

[[math]] \varepsilon:A_{red}\to\mathbb C [[/math]]

[math](4)\implies(5)[/math] Consider the following elements of [math]A_{red}[/math], which are positive:

[[math]] a_i=1-Re(u_{ii}) [[/math]]

Our assumption [math]N\in\sigma(Re(\chi_u))[/math] tells us that [math]a=\sum a_i[/math] is not invertible, and so there exists a sequence [math]x_k[/math] of norm one vectors in [math]L^2(A)[/math] such that:

[[math]] \lt ax_k,x_k \gt \to 0 [[/math]]

Since the summands [math] \lt a_ix_k,x_k \gt [/math] are all positive, we must have, for any [math]i[/math]:

[[math]] \lt a_ix_k,x_k \gt \to0 [[/math]]

We can go back to the variables [math]u_{ii}[/math] by using the following general formula:

[[math]] ||vx-x||^2=||vx||^2 +2 \lt (1-Re(v))x,x \gt -1 [[/math]]

Indeed, with [math]v=u_{ii}[/math] and [math]x=x_k[/math] the middle term on the right goes to 0, and so the whole term on the right becomes asymptotically negative, and so we must have:

[[math]] ||u_{ii}x_k-x_k||\to0 [[/math]]

Now let [math]M_n(A_{red})[/math] act on [math]\mathbb C^n\otimes L^2(A)[/math]. Since [math]u[/math] is unitary we have:

[[math]] \sum_i||u_{ij}x_k||^2 =||u(e_j\otimes x_k)|| =1 [[/math]]

From [math]||u_{ii}x_k||\to1[/math] we obtain [math]||u_{ij}x_k||\to0[/math] for [math]i\neq j[/math]. Thus we have, for any [math]i,j[/math]:

[[math]] ||u_{ij}x_k-\delta_{ij}x_k||\to0 [[/math]]

Now by remembering that we have [math]\varepsilon(u_{ij})=\delta_{ij}[/math], this formula reads:

[[math]] ||u_{ij}x_k-\varepsilon(u_{ij})x_k||\to0 [[/math]]

By linearity, multiplicativity and continuity, we must have, for any [math]a\in\mathcal A[/math], as desired:

[[math]] ||ax_k-\varepsilon(a)x_k||\to0 [[/math]]

[math](5)\implies(1)[/math] This is something well-known, which follows via some standard functional analysis arguments, worked out in Blanchard's paper .

[math](1)\implies(5)[/math] Once again this is something well-known, which follows via some standard functional analysis arguments, worked out in Blanchard's paper .

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This was for the basic amenability theory. We will be back to this on several occasions, with more specialized amenability conditions, which will add to the above list. As a first application of the above result, we can now advance on a problem left before, in chapter 3 above, and then in the beginning of the present chapter as well:

Theorem

The cocommutative Woronowicz algebras are the intermediate quotients of the following type, with [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math] being a discrete group,

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

and with [math]\pi[/math] being a unitary representation of [math]\Gamma[/math], subject to weak containment conditions of type [math]\pi\otimes\pi\subset\pi[/math] and [math]1\subset\pi[/math], which guarantee the existence of [math]\Delta,\varepsilon[/math].

Show Proof

We use Theorem 14.1 above, combined with Theorem 14.5 and then with Theorem 14.6, the idea being to proceed in several steps, as follows:

(1) Theorem 14.1 and standard functional analysis arguments show that the cocommutative Woronowicz algebras should appear as intermediate quotients, as follows:

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

(2) The existence of [math]\Delta:A\to A\otimes A[/math] requires our intermediate quotient to appear as follows, with [math]\pi[/math] being a unitary representation of [math]\Gamma[/math], satisfying the condition [math]\pi\otimes\pi\subset\pi[/math], taken in a weak containment sense, and with the tensor product [math]\otimes[/math] being taken here to be compatible with our usual maximal tensor product [math]\otimes[/math] for the [math]C^*[/math]-algebras:

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

(3) With this condition imposed, the existence of the antipode [math]S:A\to A^{opp}[/math] is then automatic, coming from the group antirepresentation [math]g\to g^{-1}[/math].

(4) The existence of the counit [math]\varepsilon:A\to\mathbb C[/math], however, is something non-trivial, related to amenability, and leading to a condition of type [math]1\subset\pi[/math], as in the statement.

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The above result is of course not the end of the story, because as formulated, with the above highly abstract conditions on [math]\pi[/math], it comes along with 0 non-trivial examples. We refer to Woronowicz and related papers for more on these topics.

Let us get back now to real life, and concrete mathematics, and focus on the Kesten amenability criterion, from Theorem 14.6 (4), which brings connections with interesting mathematics and physics, and which in practice will be our main amenability criterion. In order to discuss this, we will need the following standard fact:

Proposition

Given a Woronowicz algebra [math](A,u)[/math], with [math]u\in M_N(A)[/math], the moments of the main character [math]\chi=\sum_iu_{ii}[/math] are given by:

[[math]] \int_G\chi^k=\dim\left(Fix(u^{\otimes k})\right) [[/math]]

In the case [math]u\sim\bar{u}[/math] the law of [math]\chi[/math] is a usual probability measure, supported on [math][-N,N][/math].

Show Proof

There are two assertions here, the proof being as follows:

(1) The first assertion follows from the Peter-Weyl theory, which tells us that we have the following formula, valid for any corepresentation [math]v\in M_n(A)[/math]:

[[math]] \int_G\chi_v=\dim(Fix(v)) [[/math]]

Indeed, for the corepresentation [math]v=u^{\otimes k}[/math], the corresponding character is:

[[math]] \chi_v=\chi^k [[/math]]

Thus, we obtain the result, as a consequence of the above formula.

(2) As for the second assertion, if we assume [math]u\sim\bar{u}[/math] then we have [math]\chi=\chi^*[/math], and so the general theory, explained above, tells us that [math]law(\chi)[/math] is in this case a real probability measure, supported by the spectrum of [math]\chi[/math]. But, since [math]u\in M_N(A)[/math] is unitary, we have:

[[math]] \begin{eqnarray*} uu^*=1 &\implies&||u_{ij}||\leq 1,\forall i,j\\ &\implies&||\chi||\leq N \end{eqnarray*} [[/math]]

Thus the spectrum of the character satisfies the following condition:

[[math]] \sigma(\chi)\subset [-N,N] [[/math]]

Thus, we are led to the conclusion in the statement.

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In relation now with the notion of amenability, we have:

Theorem

A Woronowicz algebra [math](A,u)[/math], with [math]u\in M_N(A)[/math], is amenable when:

[[math]] N\in supp\Big(law\left[Re(\chi)\right]\Big) [[/math]]

Also, the support on the right depends only on [math]law(\chi)[/math].

Show Proof

There are two assertions here, the proof being as follows:

(1) According to the Kesten amenability criterion, from Theorem 14.6 (4), the algebra [math]A[/math] is amenable when the following condition is satisfied:

[[math]] N\in\sigma(Re(\chi)) [[/math]]

Now since [math]Re(\chi)[/math] is self-adjoint, we know from spectral theory that the support of its spectral measure [math]law(Re(\chi))[/math] is precisely its spectrum [math]\sigma(Re(\chi))[/math], as desired:

[[math]] supp(law(Re(\chi)))=\sigma(Re(\chi)) [[/math]]

(2) Regarding the second assertion, once again the variable [math]Re(\chi)[/math] being self-adjoint, its law depends only on the moments [math]\int_GRe(\chi)^p[/math], with [math]p\in\mathbb N[/math]. But, we have:

[[math]] \begin{eqnarray*} \int_GRe(\chi)^p &=&\int_G\left(\frac{\chi+\chi^*}{2}\right)^p\\ &=&\frac{1}{2^p}\sum_{|k|=p}\int_G\chi^k \end{eqnarray*} [[/math]]

Thus [math]law(Re(\chi))[/math] depends only on [math]law(\chi)[/math], and this gives the result.

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Let us work out now in detail the group dual case. Here we obtain a very interesting measure, called Kesten measure of the group, as follows:

Proposition

In the case [math]A=C^*(\Gamma)[/math] and [math]u=diag(g_1,\ldots,g_N)[/math], and by enlarging if necessary the generating set [math]g_1,\ldots,g_N[/math], as for the following to happen,

[[math]] 1\in u=\bar{u} [[/math]]

the moments of the main character are given by the formula

[[math]] \int_{\widehat{\Gamma}}\chi^p=\#\left\{i_1,\ldots,i_p\Big|g_{i_1}\ldots g_{i_p}=1\right\} [[/math]]

counting the loops based at [math]1[/math], having lenght [math]p[/math], on the corresponding Cayley graph.

Show Proof

Consider indeed a discrete group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math]. The main character of [math]A=C^*(\Gamma)[/math], with fundamental corepresentation [math]u=diag(g_1,\ldots,g_N)[/math], is then:

[[math]] \chi=g_1+\ldots+g_N [[/math]]

Given a colored integer [math]k=e_1\ldots e_p[/math], the corresponding moment is given by:

[[math]] \begin{eqnarray*} \int_{\widehat{\Gamma}}\chi^k &=&\int_{\widehat{\Gamma}}(g_1+\ldots+g_N)^k\\ &=&\#\left\{i_1,\ldots,i_p\Big|g_{i_1}^{e_1}\ldots g_{i_p}^{e_p}=1\right\} \end{eqnarray*} [[/math]]

In the self-adjoint case, [math]u\sim\bar{u}[/math], we are only interested in the moments with respect to usual integers, [math]p\in\mathbb N[/math], and the above formula becomes:

[[math]] \int_{\widehat{\Gamma}}\chi^p=\#\left\{i_1,\ldots,i_p\Big|g_{i_1}\ldots g_{i_p}=1\right\} [[/math]]

Assume now that we have in addition [math]1\in u[/math], so that the condition [math]1\in u=\bar{u}[/math] in the statement is satisfied. At the level of the generating set [math]S=\{g_1,\ldots,g_N\}[/math] this means:

[[math]] 1\in S=S^{-1} [[/math]]

Thus the corresponding Cayley graph is well-defined, with the elements of [math]\Gamma[/math] as vertices, and with the edges [math]g-h[/math] appearing when the following condition is satisfied:

[[math]] gh^{-1}\in S [[/math]]

A loop on this graph based at 1, having lenght [math]p[/math], is then a sequence as follows:

[[math]] (1)-(g_{i_1})-(g_{i_1}g_{i_2})-\ldots-(g_{i_1}\ldots g_{i_{p-1}})-(g_{i_1}\ldots g_{i_p}=1) [[/math]]

Thus the moments of [math]\chi[/math] count indeed such loops, as claimed.

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In order to generalize the above result to arbitrary Woronowicz algebras, we can use the discrete quantum group philosophy. The fundamental result here is as follows:

Theorem

Let [math](A,u)[/math] be a Woronowicz algebra, and assume, by enlarging if necessary [math]u[/math], that we have [math]1\in u=\bar{u}[/math]. The following formula

[[math]] d(v,w)=\min\left\{k\in\mathbb N\Big|1\subset\bar{v}\otimes w\otimes u^{\otimes k}\right\} [[/math]]

defines then a distance on [math]Irr(A)[/math], which coincides with the geodesic distance on the associated Cayley graph. In the group dual case we obtain the usual distance.

Show Proof

The fact that the lengths are finite follows from Woronowicz's analogue of Peter-Weyl theory, and the other verifications are as follows:

(1) The symmetry axiom is clear from definitions.

(2) The triangle inequality is elementary to establish as well.

(3) Finally, the last assertion is elementary as well.

In the group dual case now, where our Woronowicz algebra is of the form [math]A=C^*(\Gamma)[/math], with [math]\Gamma= \lt S \gt [/math] being a finitely generated discrete group, our normalization condition [math]1\in u=\bar{u}[/math] means that the generating set must satisfy the following condition:

[[math]] 1\in S=S^{-1} [[/math]]

But this is precisely the normalization condition made before for the discrete groups, and the fact that we obtain the same metric space is clear.

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Summarizing, we have a good understanding of what a discrete quantum group is. We can now formulate a generalization of Proposition 14.10, as follows:

Theorem

Let [math](A,u)[/math] be a Woronowicz algebra, with the normalization assumption [math]1\in u=\bar{u}[/math] made. The moments of the main character,

[[math]] \int_G\chi^p=\dim\left(Fix(u^{\otimes p})\right) [[/math]]

count then the loops based at [math]1[/math], having lenght [math]p[/math], on the corresponding Cayley graph.

Show Proof

Here the formula of the moments, with [math]p\in\mathbb N[/math], is the one coming from Proposition 14.8 above, and the Cayley graph interpretation comes from Theorem 14.11.

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Here is a related useful result, in relation with the notion of amenability:

Theorem

A Woronowicz algebra [math](A,u)[/math] is amenable precisely when

[[math]] ||X||=N [[/math]]

where [math]X[/math] is the principal graph of the associated planar algebra

[[math]] P_k=End(u^{\otimes k}) [[/math]]

obtained by deleting the reflections in the Bratteli diagram of [math]P=(P_k)[/math].

Show Proof

This is something which might look quite complicated, but the idea is very simple. First of all, it is well-known that the spaces in the statement form indeed a planar algebra in the sense of Jones , and we refer here to , and related papers, but we will not really need this here. What we need to know, which is something quite elementary, and for which we refer again to , and related papers, is that in the following sequence of inclusions, we have a copy of Jones' basic construction at every step, so that we can delete the corresponding reflections, as in the statement:

[[math]] P_0\subset P_1\subset P_2\subset\ldots [[/math]]

With this done, via some standard identifications and rescalings, we have:

[[math]] \begin{eqnarray*} ||X|| &=&||M_{\chi_u}||_{A_{central}}\\ &=&||\chi_u||_{A_{central}}\\ &=&||\chi_u||_{A_{red}} \end{eqnarray*} [[/math]]

Thus, the result follows from the Kesten amenability criterion.

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There are many concrete illustrations for the above results, and we will be back to this, on several occasions, in what follows.

General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].