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Let us discuss now some basic examples of subfactors, with concrete illustrations for all the above notions, constructions, and general theory. These examples will all come from group actions [math]G\curvearrowright Q[/math], which are assumed to be minimal, in the sense that:

[[math]] (Q^G)'\cap Q=\mathbb C [[/math]]

As a starting point, we have the following result, due to Jones :

Proposition

Assuming that [math]G[/math] is a compact group, acting minimally on a [math]{\rm II}_1[/math] factor [math]Q[/math], and that [math]H\subset G[/math] is a subgroup of finite index, we have a subfactor

[[math]] Q^G\subset Q^H [[/math]]

having index [math]N=[G:H][/math], called Jones subfactor.

Show Proof

This is something standard, the idea being that the factoriality of [math]Q^G,Q^H[/math] comes from the minimality of the action, and that the index formula is clear.

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Along the same lines, we have the following result, due to Ocneanu :

Proposition

Assuming that [math]G[/math] is a finite group, acting minimally on a [math]{\rm II}_1[/math] factor [math]Q[/math], we have a subfactor as follows,

[[math]] Q\subset Q\rtimes G [[/math]]

having index [math]N=|G|[/math], called Ocneanu subfactor.

Show Proof

This is standard as well, the idea being that the factoriality of [math]Q\rtimes G[/math] comes from the minimality of the action, and that the index formula is clear.

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We have as well a third result of the same type, due to Wassermann , namely:

Proposition

Assuming that [math]G[/math] is a compact group, acting minimally on a [math]{\rm II}_1[/math] factor [math]Q[/math], and that [math]G\to PU_n[/math] is a projective representation, we have a subfactor

[[math]] Q^G\subset (M_n(\mathbb C)\otimes Q)^G [[/math]]

having index [math]N=n^2[/math], called Wassermann subfactor.

Show Proof

As before, the idea is that the factoriality of [math]Q^G,(M_n(\mathbb C)\otimes Q)^G[/math] comes from the minimality of the action, and the index formula is clear.

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The above subfactors look quite related, and indeed they are, due to:

Theorem

The Jones, Ocneanu and Wassermann subfactors are all of the same nature, and can be written as follows,

[[math]] \left( Q^G\subset Q^H\right)\,\simeq\, \left( ({\mathbb C}\otimes Q)^G\subset (l^\infty(G/H)\otimes Q)^G\right) [[/math]]

[[math]] \left( Q\subset Q\rtimes G\right)\,\simeq\, \left( (l^\infty (G)\otimes Q)^G\subset ({\mathcal L} (l^2(G))\otimes Q)^G\right) [[/math]]

[[math]] \left( Q^G\subset (M_n(\mathbb C) \otimes Q)^G\right)\,\simeq\, \left( ({\mathbb C}\otimes Q)^G\subset (M_n(\mathbb C)\otimes Q)^G\right) [[/math]]

with standard identifications for the various tensor products and fixed point algebras.

Show Proof

This is something standard, from , modulo several standard identifications. We will explain all this more in detail later, after unifying these subfactors.

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In order to unify now the above constructions of subfactors, following , , the idea is quite clear. Given a compact group [math]G[/math], acting minimally on a [math]{\rm II}_1[/math] factor [math]Q[/math], and an inclusion of finite dimensional algebras [math]B_0\subset B_1[/math], endowed as well with an action of [math]G[/math], we would like to construct a kind of generalized Wassermann subfactor, as follows:

[[math]] (B_0\otimes Q)^G\subset (B_1\otimes Q)^G [[/math]]

In order to do this, we must talk first about the finite dimensional algebras [math]B[/math], and about inclusions of such algebras [math]B_0\subset B_1[/math]. Let us start with the following definition:

Definition

Associated to any finite dimensional algebra [math]B[/math] is its canonical trace, obtained by composing the left regular representation with the trace of [math]\mathcal L(B)[/math]:

[[math]] tr:B\subset\mathcal L(B)\to\mathbb C [[/math]]

We say that an inclusion of finite dimensional algebras [math]B_0\subset B_1[/math] is Markov if it commmutes with the canonical traces of [math]B_0,B_1[/math].

As a basic illustration for this, any inclusion of type [math]\mathbb C\subset B[/math] is Markov. In general, if we write [math]B_0=C(X_0)[/math] and [math]B_1=C(X_1)[/math], then the inclusion [math]B_0\subset B_1[/math] must come from a certain fibration [math]X_1\to X_0[/math], and the inclusion [math]B_0\subset B_1[/math] is Markov precisely when the fibration [math]X_1\to X_0[/math] commutes with the respective counting measures.

We will be back to Markov inclusions and their various properties on several occasions, in what follows. For our next purposes here, we just need the following result:

Proposition

Given a Markov inclusion of finite dimensional algebras [math]B_0\subset B_1[/math] we can perform to it the basic construction, as to obtain a Jones tower

[[math]] B_0\subset_{e_1}B_1\subset_{e_2}B_2\subset_{e_3}B_3\subset\ldots\ldots [[/math]]

exactly as we did in the above for the inclusions of [math]{\rm II}_1[/math] factors.

Show Proof

This is something standard, from , by following the computations in the above, from the case of the [math]{\rm II}_1[/math] factors, and with everything extending well. It is of course possible to do something more general here, unifying the constructions for the inclusions of [math]{\rm II}_1[/math] factors [math]A_0\subset A_1[/math], and for the inclusions of Markov inclusions of finite dimensional algebras [math]B_0\subset B_1[/math], but we will not need this degree of generality, in what follows.

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With these ingredients in hand, getting back now to the Jones, Ocneanu and Wassermann subfactors, from Theorem 16.26, the point is that these constructions can be unified, and then further studied, the final result on the subject being as follows:

Theorem

Let [math]G[/math] be a compact group, and [math]G\to Aut(Q)[/math] be a minimal action on a [math]{\rm II}_1[/math] factor. Consider a Markov inclusion of finite dimensional algebras

[[math]] B_0\subset B_1 [[/math]]

and let [math]G\to Aut(B_1)[/math] be an action which leaves invariant [math]B_0[/math], and which is such that its restrictions to the centers of [math]B_0[/math] and [math]B_1[/math] are ergodic. We have then a subfactor

[[math]] (B_0\otimes Q)^G\subset (B_1\otimes Q)^G [[/math]]

of index [math]N=[B_1:B_0][/math], called generalized Wassermann subfactor, whose Jones tower is

[[math]] (B_1\otimes Q)^G\subset(B_2\otimes Q)^G\subset(B_3\otimes Q)^G\subset\ldots [[/math]]

where [math]\{ B_i\}_{i\geq 1}[/math] are the algebras in the Jones tower for [math]B_0\subset B_1[/math], with the canonical actions of [math]G[/math] coming from the action [math]G\to Aut(B_1)[/math], and whose planar algebra is given by:

[[math]] P_k=(B_0'\cap B_k)^G [[/math]]

These subfactors generalize the Jones, Ocneanu and Wassermann subfactors.

Show Proof

This is something which is routine, from , following Wassermann , and we will be back to this in a moment, with details, directly in a more general setting.

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In addition to the Jones, Ocneanu and Wassermann subfactors, discussed and unified in the above, we have the Popa subfactors, which are constructed as follows:

Proposition

Given a discrete group [math]\Gamma= \lt g_1,\ldots,g_n \gt [/math], acting faithfully via outer automorphisms on a [math]{\rm II}_1[/math] factor [math]P[/math], we have the following “diagonal” subfactor

[[math]] \left\{ \begin{pmatrix} g_1(q)\\ &\ddots\\ && g_n(q) \end{pmatrix} \Big| q\in P\right\} \subset M_n(P) [[/math]]

having index [math]N=n^2[/math], called Popa subfactor.

Show Proof

This is something standard, a bit as for the Jones, Ocneanu and Wassermann subfactors, with the result basically coming from the work of Popa , .

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In order to unify now Theorem 16.29 and Proposition 16.30, observe that the diagonal subfactors can be written in the following way, by using a group dual:

[[math]] (P\rtimes\Gamma)^{\widehat{\Gamma}}\subset(M_n(\mathbb C)\otimes (P\rtimes\Gamma))^{\widehat{\Gamma}} [[/math]]

Here the group dual [math]\widehat{\Gamma}[/math] acts on [math]Q=P\rtimes\Gamma[/math] via the dual of the action [math]\Gamma\subset Aut (P)[/math], and on [math]M_n(\mathbb C)[/math] via the adjoint action of the following formal representation:

[[math]] \oplus g_i :\widehat{\Gamma}\to {\mathbb C}^n [[/math]]

Summarizing, we are led into quantum groups. So, let us start with:

Definition

A coaction of a Woronowicz algebra [math]A[/math] on a finite von Neumann algebra [math]Q[/math] is an injective morphism [math]\Phi:Q\to Q\otimes A''[/math] satisfying the following conditions:

• Coassociativity: [math](\Phi\otimes id)\Phi=(id\otimes\Delta)\Phi[/math].
• Trace equivariance: [math](tr\otimes id)\Phi=tr(.)1[/math].
• Smoothness: [math]\overline{\mathcal Q}^{\,w}=Q[/math], where [math]\mathcal Q=\Phi^{-1}(Q\otimes_{alg}\mathcal A)[/math].

These conditions come from what happens in the commutative case, [math]A=C(G)[/math], where they correspond to the usual associativity, trace equivariance and smoothness of the corresponding action [math]G\curvearrowright Q[/math]. Along the same lines, we have as well:

Definition

A coaction [math]\Phi:Q\to Q\otimes A''[/math] as above is called:

• Ergodic, if the algebra [math]Q^\Phi=\left\{p\in Q\big|\Phi(p)=p\otimes1\right\}[/math] reduces to [math]\mathbb C[/math].
• Faithful, if the span of [math]\left\{(f\otimes id)\Phi(Q)\big|f\in Q_*\right\}[/math] is dense in [math]A''[/math].
• Minimal, if it is faithful, and satisfies [math](Q^\Phi)'\cap Q=\mathbb C[/math].

Observe that the minimality of the action implies in particular that the fixed point algebra [math]Q^\Phi[/math] is a factor. Thus, we are getting here to the case that we are interested in, actions producing factors, via their fixed point algebras. Following , we have:

Proposition

Consider a Woronowicz algebra [math]A=(A,\Delta,S)[/math], and denote by [math]A_\sigma[/math] the Woronowicz algebra [math](A,\sigma\Delta ,S)[/math], where [math]\sigma[/math] is the flip. Given two coactions

[[math]] \beta:B\to B\otimes A\quad,\quad \pi:Q\to Q\otimes A_\sigma [[/math]]

with [math]B[/math] being finite dimensional, the following linear map, while not being multiplicative in general, is coassociative with respect to the comultiplication [math]\sigma\Delta[/math] of [math]A_\sigma[/math],

[[math]] \beta\odot\pi:B\otimes Q\to B\otimes Q\otimes A_\sigma [[/math]]

[[math]] b\otimes p\to \pi (p)_{23}((id\otimes S)\beta(b))_{13} [[/math]]

and its fixed point space, which is by definition the following linear space,

[[math]] (B\otimes Q)^{\beta\odot\pi}=\left\{x\in B\otimes Q\Big|(\beta\odot\pi )x=x\otimes 1\right\} [[/math]]

is then a von Neumann subalgebra of [math]B\otimes Q[/math].

Show Proof

This is something standard, which follows from a straightforward algebraic verification, explained in . As mentioned in the statement, to be noted is that the tensor product coaction [math]\beta\odot\pi[/math] is not multiplicative in general. See .

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Our first task is to investigate the factoriality of such algebras, and we have here:

Theorem

If [math]\beta:B\to B\otimes A[/math] is a coaction and [math]\pi:Q\to Q\otimes A_\sigma[/math] is a minimal coaction, then the following conditions are equivalent:

• The von Neumann algebra [math](B\otimes Q)^{\beta\odot\pi}[/math] is a factor.
• The coaction [math]\beta[/math] is centrally ergodic, [math]Z(B)\cap B^\beta=\mathbb C[/math].

Show Proof

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

(1) Our first claim, which is something whose proof is a routine verification, explained in , is that the following diagram is a non-degenerate commuting square:

[[math]] \begin{matrix} Q&\subset&B\otimes Q\\ \cup &\ &\cup \\ Q^\pi&\subset&(B\otimes Q)^{\beta\odot\pi} \end{matrix} [[/math]]

(2) In order to prove now the result, it is enough to check the following equality, between subalgebras of the von Neumann algebra [math]B\otimes Q[/math]:

[[math]] Z((B\otimes Q)^{\beta\odot\pi})=(Z(B)\cap B^\beta)\otimes 1 [[/math]]

But this follows from the non-degeneracy of the above commuting square. See .

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With the above results in hand, we can now formulate our main theorem regarding the fixed point subfactors, of the most possible general type, as follows:

Theorem

Let [math]G[/math] be a compact quantum group, and [math]G\to Aut(Q)[/math] be a minimal action on a [math]{\rm II}_1[/math] factor. Consider a Markov inclusion of finite dimensional algebras

[[math]] B_0\subset B_1 [[/math]]

and let [math]G\to Aut(B_1)[/math] be an action which leaves invariant [math]B_0[/math] and which is such that its restrictions to the centers of [math]B_0[/math] and [math]B_1[/math] are ergodic. We have then a subfactor

[[math]] (B_0\otimes Q)^G\subset (B_1\otimes Q)^G [[/math]]

of index [math]N=[B_1:B_0][/math], called generalized Wassermann subfactor, whose Jones tower is

[[math]] (B_1\otimes Q)^G\subset(B_2\otimes Q)^G\subset(B_3\otimes Q)^G\subset\ldots [[/math]]

where [math]\{ B_i\}_{i\geq 1}[/math] are the algebras in the Jones tower for [math]B_0\subset B_1[/math], with the canonical actions of [math]G[/math] coming from the action [math]G\to Aut(B_1)[/math], and whose planar algebra is given by:

[[math]] P_k=(B_0'\cap B_k)^G [[/math]]

These subfactors generalize the Jones, Ocneanu, Wassermann and Popa subfactors.

Show Proof

This is something routine, based on the above general theory and results, and for the full story here, and technical details, we refer to , .

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The above result is important in connection with probability questions, because our usual character computations for [math]G[/math], for instance in the case where [math]G\subset U_N^+[/math] is easy, take place in the associated planar algebra [math]P_k=(B_0'\cap B_k)^G[/math]. More on this later.

This was for the basic theory of the fixed point subfactors. Many more things can be said about them, notably with an axiomatization of the planar algebras that we can obtain in this way, as being the subalgebras of Jones' bipartite graph planar algebras from , and also with a number of results and open questions regarding amenability. For more on all this, and for further details on the above, we refer to , , .

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].