12b. Amenability

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The hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math], which is a quite fascinating object, was heavily investigated by Murray-von Neumann , and then by Connes . There are many things that can be said about it, which all interesting, but are usually quite technical as well.

As a central result here, in what regards advanced hyperfiniteness theory, we have the following theorem of Connes , whose proof is something remarkably heavy, and which is arguably the deepest result in operator algebra related functional analysis:

Theorem

For a finite von Neumann algebra [math]A[/math], the following are equivalent:

• [math]A[/math] is hyperfinite in the usual sense, namely it appears as the weak closure of an increasing limit of finite dimensional algebras:
  [[math]] A=\overline{\bigcup_iA_i}^{\,w} [[/math]]
• [math]A[/math] amenable, in the sense that the standard inclusion [math]A\subset B(H)[/math], with [math]H=L^2(A)[/math], admits a conditional expectation [math]E:B(H)\to A[/math].

Show Proof

This result, due to Connes , is something fairly heavy, that only a handful of people have really managed to understand, the idea being as follows:

[math](1)\implies(2)[/math] Assuming that the algebra [math]A[/math] is hyperfinite, let us write it as the weak closure of an increasing limit of finite dimensional subalgebras:

[[math]] A=\overline{\bigcup_iA_i}^{\,w} [[/math]]

Consider the inclusion [math]A\subset B(H)[/math], with [math]H=L^2(A)[/math]. In order to construct an expectation [math]E:B(H)\to A[/math], let us pick an ultrafilter [math]\omega[/math] on [math]\mathbb N[/math]. Given [math]T\in B(H)[/math], we can define the following quantity, with [math]\mu_i[/math] being the Haar measure on the unitary group [math]U(A_i)[/math]:

[[math]] \psi(T)=\lim_{i\to\omega}\int_{U(A_i)}UTU^*\,d\mu_i(U) [[/math]]

With this construction made, by using now the standard involution [math]J:H\to H[/math], given by the formula [math]T\to T^*[/math], we can further define a map as follows:

[[math]] E:B(H)\to A\quad,\quad E(T)=J\psi(T)J [[/math]]

But this is the expectation that we are looking for, with its left and right invariance properties coming from the left and right invariance of each Haar measure [math]\mu_i[/math].

[math](2)\implies(1)[/math] This is something heavy, using lots of advanced functional analysis, and for details here, we refer to Connes' original paper .

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We should mention that Connes' results in , besides proving the above implication [math](2)\implies(1)[/math], provide also a considerable extension of Theorem 2.10, with a number of further equivalent formulations of the notion of amenability, which are a bit more technical, but all good to know. The story here, still a bit simplified, is as follows: \begin{fact}[Connes] For a finite von Neumann algebra [math]A[/math], the following conditions are in fact equivalent:

• [math]A[/math] is hyperfinite, in the sense that it appears as the weak closure of an increasing limit of finite dimensional algebras:
  [[math]] A=\overline{\bigcup_iA_i}^{\,w} [[/math]]
• [math]A[/math] amenable, in the sense that the standard inclusion [math]A\subset B(H)[/math], with [math]H=L^2(A)[/math], admits a conditional expectation:
  [[math]] E:B(H)\to A [[/math]]
• There exist unit vectors [math]\xi_n\in L^2(A)\otimes L^2(A)[/math] such that, for any [math]x\in A[/math]:
  [[math]] ||x\xi_n-\xi_nx||_2\to0\quad,\quad \lt x\xi_n,\xi_n \gt \to tr(x) [[/math]]
• For any [math]x_1,\ldots,x_k\in A[/math] and [math]y_1,\ldots,y_k\in A[/math] we have:
  [[math]] \left|tr\left(\sum_ix_iy_i\right)\right|\leq\left|\left|\sum_ix_i\otimes y_i^{opp}\right|\right|_{min} [[/math]]

\end{fact} Again, this is something technical and advanced, that we won't get into, in this book. Let us mention however that the idea with all this is as follows:

[math](1)\implies(2)[/math] is elementary, as explained above.

[math](2)\implies(3)[/math] can be proved by using an inequality due to Powers-St\o rmer.

[math](3)\implies(4)[/math] is something quite technical, but doable as well.

[math](4)\implies(2)[/math] is again something technical, but doable as well.

[math](2)\implies(1)[/math] is, as before in Theorem 12.10, the difficult implication.

Regarding the difficult implication, [math](2)\implies(1)[/math], the difficulty here comes of course from the fact that, no matter what beautiful abstract functional analysis things you know about [math]A[/math], at some point you will have to get to work, and construct that finite dimensional subalgebras [math]A_i\subset A[/math], and it is not even clear where to start from. For a solution to this problem, and for more, we refer to Connes's article , and also to his book .

Getting back now to more everyday mathematics, the above results as stated remain something quite abstract, and advanced, and understanding their concrete implications will be our next task. In the case of the [math]{\rm II}_1[/math] factors, we have the following result:

Theorem

For a [math]{\rm II}_1[/math] factor [math]R[/math], the following are equivalent:

• [math]R[/math] amenable, in the sense that we have an expectation, as follows:
  [[math]] E:B(L^2(R))\to R [[/math]]
• [math]R[/math] is the Murray-von Neumann hyperfinite [math]{\rm II}_1[/math] factor.

Show Proof

This follows indeed from Theorem 12.10, when coupled with the Murray-von Neumann uniqueness result for the hyperfinite [math]{\rm II}_1[/math] factor, from Theorem 12.6.

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As another application, getting back now to the general case, that of the finite von Neumann algebras, from Theorem 12.10 as stated, a first question is about how all this applies to the group von Neumann algebras, and more generally to the quantum group von Neumann algebras [math]L(\Gamma)[/math]. In order to discuss this, let us start with the case of the usual discrete groups [math]\Gamma[/math]. We will need the following result, which is standard:

Theorem

For a discrete group [math]\Gamma[/math], the following two conditions are equivalent, and if they are satisfied, we say that [math]\Gamma[/math] is amenable:

• [math]\Gamma[/math] admits an invariant mean [math]m:l^\infty(\Gamma)\to\mathbb C[/math].
• The projection map [math]C^*(\Gamma)\to C^*_{red}(\Gamma)[/math] is an isomorphism.

Moreover, the class of amenable groups contains all the finite groups, all the abelian groups, and is stable under taking subgroups, quotients and products.

Show Proof

This is something very standard, the idea being as follows:

(1) The equivalence [math](1)\iff(2)[/math] is standard, with the amenability conditions (1,2) being in fact part of a much longer list of amenability conditions, including well-known criteria of F\o lner, Kesten and others. We will be back to this, with details, in a moment, directly in a more general setting, that of the discrete quantum groups.

(2) As for the last assertion, regarding the finite groups, the abelian groups, and then the stability under taking subgroups, quotients and products, this is something elementary, which follows by using either of the above definitions of the amenability.

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Getting back now to operator algebras, we can complement Theorem 12.10 with:

Theorem

For a group von Neumann algebra [math]A=L(\Gamma)[/math], the following conditions are equivalent:

• [math]A[/math] is hyperfinite.
• [math]A[/math] amenable.
• [math]\Gamma[/math] is amenable.

Show Proof

The group von Neumann algebras [math]A=L(\Gamma)[/math] being by definition finite, Theorem 12.10 above applies, and gives the equivalence [math](1)\iff(2)[/math]. Thus, it remains to prove that we have [math](2)\iff(3)[/math], and we can prove this as follows:

[math](2)\implies(3)[/math] This is something clear, because if we assume that [math]A=L(\Gamma)[/math] is amenable, we have by definition a conditional expectation [math]E:B(L^2(A))\to A[/math], and the restriction of this conditional expectation is the desired invariant mean [math]m:l^\infty(\Gamma)\to\mathbb C[/math].

[math](3)\implies(2)[/math] Assume that we are given a discrete amenable group [math]\Gamma[/math]. In view of Theorem 12.13, this means that [math]\Gamma[/math] has an invariant mean, as follows:

[[math]] m:l^\infty(\Gamma)\to\mathbb C [[/math]]

Consider now the Hilbert space [math]H=l^2(\Gamma)[/math], and for any operator [math]T\in B(H)[/math] consider the following map, which is a bounded sesquilinear form:

[[math]] \varphi_T:H\times H\to\mathbb C [[/math]]

[[math]] (\xi,\eta)\to m\left[\gamma\to \lt \rho_\gamma T\rho_\gamma^*\xi,\eta \gt \right] [[/math]]

By using the Riesz representation theorem, we conclude that there exists a certain operator [math]E(T)\in B(H)[/math], such that the following holds, for any two vectors [math]\xi,\eta[/math]:

[[math]] \varphi_T(\xi,\eta)= \lt E(T)\xi,\eta \gt [[/math]]

Summarizing, to any operator [math]T\in B(H)[/math] we have associated another operator, denoted [math]E(T)\in B(H)[/math], such that the following formula holds, for any two vectors [math]\xi,\eta[/math]:

[[math]] \lt E(T)\xi,\eta \gt =m\left[\gamma\to \lt \rho_\gamma T\rho_\gamma^*\xi,\eta \gt \right] [[/math]]

In order to prove now that this linear map [math]E[/math] is the desired expectation, observe that for any group element [math]g\in\Gamma[/math], and any two vectors [math]\xi,\eta\in H[/math], we have:

[[math]] \begin{eqnarray*} \lt \rho_gE(T)\rho_g^*\xi,\eta \gt &=& \lt E(T)\rho_g^*\xi,\rho_g^*\eta \gt \\ &=&m\left[\gamma\to \lt \rho_\gamma T\rho_\gamma^*\rho_g^*\xi,\rho_g^*\eta \gt \right]\\ &=&m\left[\gamma\to \lt \rho_{g\gamma}T\rho_{g\gamma}^*\xi,\eta \gt \right]\\ &=&m\left[\gamma\to \lt \rho_\gamma T\rho_\gamma^*\xi,\eta \gt \right]\\ &=& \lt E(T)\xi,\eta \gt \end{eqnarray*} [[/math]]

Since this is valid for any [math]\xi,\eta\in H[/math], we conclude that we have, for any [math]g\in\Gamma[/math]:

[[math]] \rho_gE(T)\rho_g^*=E(T) [[/math]]

But this shows that the element [math]E(T)\in B(H)[/math] is in the commutant of the right regular representation of [math]\Gamma[/math], and so belongs to the left regular group algebra of [math]\Gamma[/math]:

[[math]] E(T)\in L(\Gamma) [[/math]]

Summarizing, we have constructed a certain linear map [math]E:B(H)\to L(\Gamma)[/math]. Now by using the above explicit formula of it, in terms of [math]m:l^\infty(\Gamma)\to\mathbb C[/math], which was assumed to be an invariant mean, we conclude that [math]E[/math] is indeed an expectation, as desired.

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As a very concrete application of all this technology, in relation now with the discrete group algebras which are [math]{\rm II}_1[/math] factors, the results that we have lead to:

Theorem

For a discrete group [math]\Gamma[/math], the following conditions are equivalent:

• [math]\Gamma[/math] is amenable, and has the ICC property.
• [math]A=L(\Gamma)[/math] is the hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math].

Show Proof

This follows indeed from Theorem 12.14, coupled with the standard fact, that we know well from chapter 10, that a group algebra [math]A=L(\Gamma)[/math] is a factor, and so a [math]{\rm II}_1[/math] factor, precisely when the group [math]\Gamma[/math] has the ICC property.

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As a comment here, this result, coming from Connes' Theorem 12.10, is far better than what we knew to come from Murray-von Neumann's Theorem 12.6, and with the statement itself being something elementary, not involving any kind of advanced functional analysis, such as the notion of amenability for von Neumann algebras. In fact, Murray-von Neumann knew about this statement, but their hunt for a proof proved to be unsuccessful, with the only possible proof being the one above, via advanced functional analysis.

Summarizing, and to put things in context, Murray-von Neumann did great work with their papers , , , , , but were stuck with 3 questions, namely reduction theory, type [math]{\rm III}[/math] factors, and solutions of [math]L(\Gamma)=R[/math]. And these questions were solved later by von Neumann himself , then Connes , and Connes again .

Beautiful times these must have been, for mathematics and for mankind, and job for us, future generations, at least to write a complete von Neumann algebra book, clearly explaining all this fundamental material. And with many older people giving up, for various technical reasons, this will be most likely a job for you, young reader.

General references

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