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Back now to work, we would like to discuss all sorts of questions, for the most open, or at least difficult, in relation with groups and quantum groups, taken finite, discrete or compact, and with more general quantum manifolds and quantum spaces, in connection with the Murray-von Neumann factor [math]R[/math], amenability and hyperfiniteness. As a first such question, in relation with the considerations from chapter 10, we would like to understand which discrete quantum groups [math]\Gamma[/math] produce group algebras as follows:

[[math]] L(\Gamma)\simeq R [[/math]]

In terms of the compact quantum group duals [math]G=\widehat{\Gamma}[/math], the problem is that of understanding which compact quantum groups [math]G[/math] produce group algebras as follows:

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

In order to discuss this, we must first talk about amenability. We have here the following result, basically due to Woronowicz , and coming from the Peter-Weyl theory, extending to the discrete quantum groups the standard theory for discrete groups:

Theorem

Let [math](A,u)[/math] with [math]u\in M_N(A)[/math] be a Woronowicz algebra, as axiomatized before. Let [math]A_{full}[/math] be the enveloping [math]C^*[/math]-algebra of [math]\mathcal A= \lt u_{ij} \gt [/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 explicitly 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, as follows:

[[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}\quad,\quad 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}\quad,\quad 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, exactly as in the usual group case.

[math](1)\implies(5)[/math] Once again this is something well-known, which follows via some standard functional analysis arguments, exactly as in the usual group case.

■

Before getting further, with advanced amenability and hyperfiniteness questions, and as a first application of the above, we can now advance on a problem that we left open before, in chapter 7, when talking about cocommutative Woronowicz algebras. Indeed, we can now state and prove the following result, which clarifies the situation:

Proposition

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 the various findings from Theorem 12.16, following Woronowicz, the idea being to proceed in several steps, as follows:

(1) Theorem 12.16 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.

■

Let us focus now on the Kesten amenability criterion, from Theorem 12.16 (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:

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

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, with [math]v=u^{\otimes k}[/math] we obtain the result. As for the second assertion, if we assume [math]u\sim\bar{u}[/math], then we have [math]\chi=\chi^*[/math], and so [math]law(\chi)[/math] is a real probability measure, supported by the spectrum of [math]\chi[/math]. But, since the matrix [math]u\in M_N(A)[/math] is unitary, we have:

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

Thus the spectrum of the character satisfies [math]\sigma(\chi)\subset [-N,N][/math], as desired.

■

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(Re(\chi))\Big) [[/math]]

and 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 12.16 (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]] \int_GRe(\chi)^p =\int_G\left(\frac{\chi+\chi^*}{2}\right)^p =\frac{1}{2^p}\sum_{|k|=p}\int_G\chi^k [[/math]]

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

■

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 with the normalization [math]1\in u=\bar{u}[/math] made, we have 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 length [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]] \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\} [[/math]]

In the self-adjoint case now, [math]u\sim\bar{u}[/math], as in the statement, 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 length [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.

■

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.

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

(3) The Cayley graph assertion is something elementary as well.

(4) Finally, in the group dual case, 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:

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

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

■

Summarizing, we have a good understanding of what a discrete quantum group is. We can now formulate a generalization of Proposition 12.20, 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 12.18, and the Cayley graph interpretation comes from Theorem 12.21.

■

As an application of this, we can introduce the notion of growth, as follows:

Definition

Given a closed subgroup [math]G\subset U_N^+[/math], with [math]1\in u=\bar{u}[/math], consider the series whose coefficients are the ball volumes on the corresponding Cayley graph,

[[math]] f(z)=\sum_kb_kz^k\quad,\quad b_k=\sum_{l(v)\leq k}\dim(v)^2 [[/math]]

and call it growth series of the discrete quantum group [math]\widehat{G}[/math]. In the group dual case, [math]G=\widehat{\Gamma}[/math], we obtain in this way the usual growth series of [math]\Gamma[/math].

There are many things that can be said about the growth, and we will be back to this. As a first such result, in relation with the notion of amenability, we have:

Theorem

Polynomial growth implies amenability.

Show Proof

We recall from Theorem 12.21 that the Cayley graph of [math]\widehat{G}[/math] has by definition the elements of [math]Irr(G)[/math] as vertices, and the distance is as follows:

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

By taking [math]w=1[/math] and by using Frobenius reciprocity, the lenghts are given by:

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

By Peter-Weyl we have then a decomposition as follows, where [math]B_k[/math] is the ball of radius [math]k[/math], and where [math]m_k(v)\in\mathbb N[/math] are certain multiplicities:

[[math]] u^{\otimes k}=\sum_{v\in B_k}m_k(v)\cdot v [[/math]]

By using now Cauchy-Schwarz, we obtain the following inequality:

[[math]] \begin{eqnarray*} m_{2k}(1)b_k &=&\sum_{v\in B_k}m_k(v)^2\sum_{v\in B_k}\dim(v)^2\\ &\geq&\left(\sum_{v\in B_k}m_k(v)\dim(v)\right)^2\\ &=&N^{2k} \end{eqnarray*} [[/math]]

But shows that if [math]b_k[/math] has polynomial growth, then the following happens:

[[math]] \limsup_{k\to\infty}\, m_{2k}(1)^{1/2k}\geq N [[/math]]

Thus, the Kesten type criterion applies, and gives the result.

■

There are many other things that can be said, as a continuation of the above, notably with explicit computations of growth exponents for all the discrete quantum groups that we know, and with some further generalities too, of functional analytic nature, and in relation with Lie theory and its generalizations too. For more on all this, you can check my quantum group textbook , and the quantum group literature cited there.

To summarize now, we have a decent understanding of what a discrete quantum group is, and also of what amenability means, in the discrete quantum group setting. However, all this does not exactly solve the von Neumann algebra questions, and we have: \begin{question} Which discrete quantum groups [math]\Gamma[/math] have the property [math]L(\Gamma)\simeq R[/math]? Equivalently, which compact quantum groups [math]G[/math] have the property [math]L^\infty(G)\simeq R[/math]? \end{question} Here the equivalence between the above two questions comes from the fact that, with [math]\Gamma=\widehat{G}[/math], we have [math]L(\Gamma)=L^\infty(G)[/math]. As for the questions themselves, normally the hyperfiniteness part can be dealt with as in the classical group case, by using the amenability theory developed above, and the problem is with the ICC property, guaranteeing factoriality, with no one presently knowing what this “quantum ICC” property is.

As a funny comment here, the equation [math]L(\Gamma)\simeq R[/math] is precisely the one Murray and von Neumann were stuck with, in the classical group case, some 90 years ago. Some sort of Connes is needed, coming and solving this problem, with new ideas.

Finally, let us mention that in connection with amenability and hyperfiniteness, we have as well a series of further questions, in relation with the actions of quantum groups. To be more precise, the problems that we would like to solve are as follows:

(1) We would like to understand, given a compact group or quantum group acting on a von Neumann algebra, [math]G\curvearrowright P[/math], when the fixed point algebra [math]P^G[/math] is a factor.

(2) More generally, we would like to understand under which assumptions on [math]G\curvearrowright P[/math] the fixed point algebra [math](B\otimes P)^G[/math] is a factor, for any finite dimensional algebra [math]B[/math].

(3) In fact, we would like to understand when the fixed point algebra [math]P^G[/math], or more generally all the fixed point algebras [math](B\otimes P)^G[/math], are the hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math].

These questions are all of interest in subfactor theory, the idea being that a quite standard construction of subfactors is [math](B_0\otimes P)^G\subset(B_1\otimes P)^G[/math], coming from a von Neumann algebra [math]P[/math], an inclusion of finite dimensional algebras [math]B_0\subset B_1[/math], and a compact quantum group [math]G[/math] acting on everything, provided that the fixed point algebras involved are indeed factors. And then, once such a subfactor constructed and studied, the main problem is that of understanding if this subfactor can be taken to be hyperfinite.

These are quite technical questions, to be discussed in chapters 13-16 below, when doing subfactors. Let us mention however, coming a bit in advance, that we have: \begin{fact} Assuming that [math]\Gamma=\widehat{G}[/math] has an outer action on the hyperfinite [math]{\rm II}_1[/math] factor

[[math]] \Gamma\curvearrowright R [[/math]]

we can set [math]P=R\rtimes\Gamma[/math], and the answer to the above questions is yes. \end{fact} Which brings us into the very interesting question on whether we have such outer actions [math]\Gamma\curvearrowright R[/math], with the status of the subject being as follows:

(1) All this goes back to work in the 80s of Ocneanu, and Wassermann too, with Ocneanu eventually conjecturing that any discrete group [math]\Gamma[/math], and more generally any discrete quantum group [math]\Gamma[/math], should have such an action. This question is still open.

(2) In practice, the result is known in the finite case, [math]|\Gamma| \lt \infty[/math], and more generally in the case where [math]C^*(\Gamma)[/math] has an inner faithful matrix model, in the sense of chapter 11, with this being worked out in and its follow-ups, and then by Vaes in .

(3) And there has been quite some work on this, since then. For the status of the question, and relations with other questions, such as the Connes embedding problem, Voiculescu microstates and more, we refer to Brannan-Chirvasitu-Freslon .

Summarizing, many things going on here, with the philosophy being somehow that, once we want our factors or subfactors to be hyperfinite, isomorphic to [math]R[/math], we are all of the sudden into all sorts of interesting questions, in relation with advanced mathematics and physics. But more on this later, in chapters 13-16 below, when doing subfactors.

General references

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