15d. Fixed points

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

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We know that the planar algebra associated to an Hadamard matrix [math]H\in M_N(\mathbb C)[/math] appears in fact as the planar algebra associated to a certain related quantum permutation group [math]G\subset S_N^+[/math]. In view of the various results from chapters 13-14, this suggests that the subfactor itself associated to [math]H[/math] should appear as a fixed point subfactor associated to [math]G[/math]. We will prove here that this is indeed the case. To be more precise, following and subsequent papers, regarding the subfactor itself, the result here is as follows:

Theorem

The subfactor associated to [math]H\in M_N(\mathbb C)[/math] is of the form

[[math]] A^G\subset(\mathbb C^N\otimes A)^G [[/math]]

with [math]A=R\rtimes\widehat{G}[/math], where [math]G\subset S_N^+[/math] is the associated quantum permutation group.

Show Proof

This is something more technical, the idea being that the basic construction procedure for the commuting squares, explained before Theorem 15.5, can be performed in an “equivariant setting”, for commuting squares having components as follows:

[[math]] D\otimes_GE=(D\otimes(E\rtimes\widehat{G}))^G [[/math]]

To be more precise, starting with a commuting square formed by such algebras, we obtain by basic construction a whole array of commuting squares as follows, with [math]\{D_i\},\{E_i\}[/math] being by definition Jones towers, and with [math]D_\infty,E_\infty[/math] being their inductive limits:

[[math]] \xymatrix@R=35pt@C35pt{ D_0\otimes_GE_\infty&D_1\otimes_GE_\infty&D_2\otimes_GE_\infty\\ D_0\otimes_GE_2\ar@.[u]\ar[r]&D_1\otimes_GE_2\ar@.[u]\ar[r]&D_2\otimes_GE_2\ar@.[u]\ar@.[r]&D_\infty\otimes_GE_2\\ D_0\otimes_GE_1\ar[u]\ar[r]&D_1\otimes_GE_1\ar[u]\ar[r]&D_2\otimes_GE_1\ar[u]\ar@.[r]&D_\infty\otimes_GE_1\\ D_0\otimes_GE_0\ar[u]\ar[r]&D_1\otimes_GE_0\ar[u]\ar[r]&D_2\otimes_GE_0\ar[u]\ar@.[r]&D_\infty\otimes_GE_0} [[/math]]

The point now is that this quantum group picture works in fact for any commuting square having [math]\mathbb C[/math] in the lower left corner. In the Hadamard matrix case, that we are interested in here, the corresponding commuting square is as follows:

[[math]] \xymatrix@R=35pt@C35pt{ \mathbb C\otimes_G\mathbb C^N\ar[r]&\mathbb C^N\otimes_G\mathbb C^N\\ \mathbb C\otimes_G\mathbb C\ar[u]\ar[r]&\mathbb C^N\otimes_G\mathbb C\ar[u] } [[/math]]

Thus, the subfactor obtained by vertical basic construction appears as follows:

[[math]] \mathbb C\otimes_GE_\infty\subset\mathbb C^N\otimes_GE_\infty [[/math]]

But this gives the conclusion in the statement, with the [math]{\rm II}_1[/math] factor appearing there being by definition [math]A=E_\infty\rtimes\widehat{G}[/math], and with the remark that we have [math]E_\infty\simeq R[/math].

■

All the above was of course quite brief, but we will discuss now all this with more details, directly in a more general setting, covering the Hadamard matrix situation. To be more precise, our claim is that the above fixed point subfactor techniques apply, more generally, to the commuting squares having [math]\mathbb C[/math] in the lower left corner:

[[math]] \xymatrix@R=40pt@C40pt{ E\ar[r]&X\\ \mathbb C\ar[u]\ar[r]&D\ar[u]} [[/math]]

In order to discuss this, let us go back to the fixed point subfactors, from chapter 13. In what concerns the fixed point factors, we know from there that we have:

Theorem

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 coactions

[[math]] \beta:B\to B\otimes A [[/math]]

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

is then a von Neumann subalgebra of [math]B\otimes P[/math]. Moreover, such algebras can be used in order to construct the generalized Wassermann subfactors, [math](B_0\otimes P)^G\subset(B_1\otimes P)^G[/math].

Show Proof

This is something that we know from chapter 13, and for details, and comments in relation with the non-multiplicativity of [math]\beta\odot\pi[/math], we refer to the material there.

■

Let [math]\int_A:A\to\mathbb C[/math] be the Haar functional, let [math]l^2(A)[/math] be its [math]l^2[/math]-space and let [math]\widehat{A}\subset B(l^2(A))[/math] be the dual algebra. If [math]\alpha:E\to E\otimes\widehat A[/math] is a coaction of [math]\widehat{A}[/math] on a finite von Neumann algebra [math]E[/math], the crossed product [math]E\rtimes_\alpha\widehat{A}[/math] is the von Neumann subalgebra of [math]E\otimes B(l^2(A))[/math] generated by [math]\alpha(E)[/math] and by [math]1\otimes A[/math]. There exists a unique coaction [math]\widehat{\alpha}[/math] of [math]A[/math] on [math]E\rtimes_\alpha\widehat{A}[/math] such that [math](E\rtimes_\alpha\widehat{A})^{\widehat{\alpha}}=\alpha(E)[/math], and such that the copy [math]1\otimes A[/math] of [math]A[/math] is equivariant. With these conventions, again following and subsequent papers, we have the following result:

Proposition

Let [math]A[/math] be a Woronowicz algebra. If [math]\beta:D\to D\otimes A[/math] is a coaction on a finite dimensional finite von Neumann algebra and [math]\alpha:E\to E\otimes\widehat{A}_\sigma[/math] is a coaction on a finite von Neumann algebra then we have the equality

[[math]] (D\otimes(E\rtimes_\alpha\widehat{A}_\sigma))^{\beta\odot\widehat{\alpha}}=\overline{sp}^{\,w}\Big\{\beta(D)_{13}\cdot\alpha(E)_{23}\Big\} [[/math]]

as linear subspaces of [math]D\otimes E\otimes B(l^2(A_\sigma))[/math]. Moreover, the following diagram

[[math]] \begin{matrix} \alpha(E)_{23}&\subset&(D\otimes(E\rtimes_{\alpha}\widehat{A}_\sigma ))^{\beta\odot\widehat{\alpha}}\\ \cup&&\cup\\ \mathbb C&\subset&\beta(D)_{13} \end{matrix} [[/math]]

is a non-degenerate commuting square of finite von Neumann algebras.

Show Proof

By definition of the crossed product [math]E\rtimes_\alpha\widehat{A}_\sigma[/math], we have the following equalities between subalgebras of [math]D\otimes E\otimes B(l^2(A_\sigma))[/math]:

[[math]] \begin{eqnarray*} D\otimes(E\rtimes_\alpha\widehat{A}_\sigma) &=&D\otimes(\overline{sp}^{\,w}\{\alpha(E)\cdot (1\otimes A_\sigma)\})\\ &=&\overline{sp}^{\,w}\{(D\otimes A_\sigma )_{13}\cdot\alpha(E)_{23}\} \end{eqnarray*} [[/math]]

On the other hand, since the coactions on the finite dimensional algebras are automatically non-degenerate, we have as well the following equality:

[[math]] D\otimes A_\sigma=sp\{(1\otimes A_\sigma)\cdot\beta(D)\} [[/math]]

Thus, we have the following equality of algebras:

[[math]] D\otimes(E\rtimes_\alpha A_\sigma)=\overline{sp}^{\,w}\{(1\otimes 1\otimes A_\sigma)\cdot\beta(D)_{13}\cdot\alpha(E)_{23}\} [[/math]]

Let us compute now the restriction of the map [math]\beta\odot\widehat{\alpha}[/math] to the algebra [math]1\otimes1\otimes A_\sigma[/math], to the algebra [math]\beta(D)_{13}[/math], and to the algebra [math]\alpha (E)_{23}[/math]. This can be done as follows:

(1) The restriction of [math]\beta\odot\widehat{\alpha}[/math] to the algebra [math]1\otimes1\otimes A_\sigma[/math] is [math]1\otimes1\otimes\sigma\Delta[/math]. In particular the map [math]\beta\odot\widehat{\alpha}[/math] has no fixed points in this algebra [math]1\otimes1\otimes A_\sigma[/math].

(2) The algebra [math]\alpha(E)_{23}[/math] is by definition fixed by [math]\beta\odot\widehat{\alpha}[/math].

(3) We prove now that the algebra [math]\beta(D)_{13}[/math] is also fixed by [math]\beta\odot\widehat{\alpha}[/math]. For this purpose, let [math]\{u_{ij}\}[/math] be an orthonormal basis of [math]l^2(A_\sigma)[/math] consisting of coefficients of irreducible corepresentations of [math]A_\sigma[/math]. Since we have [math]\beta(D)\subset D\otimes_{alg}A_\sigma[/math], for any [math]b\in D[/math] we can use the notation [math]\beta(b)=\sum_{uij}b^u_{ij}\otimes u_{ij}[/math]. From the coassociativity of [math]\beta[/math] we obtain:

[[math]] \sum_{uij}\beta(b_{ij}^u)\otimes u_{ij}=\sum_{uijk}b_{ij}^u\otimes u_{kj}\otimes u_{ik} [[/math]]

Thus we have [math]\beta (b_{ik}^u)=\sum_j b_{ij}^u\otimes u_{kj}[/math] for any [math]u,i,k[/math], and so:

[[math]] \begin{eqnarray*} (id\otimes S)\beta(b^u_{ij}) &=&(id\otimes S)\left(\sum_s b_{is}^u\otimes u_{js}\right)\\ &=&\sum_sb_{is}^u\otimes u_{sj}^* \end{eqnarray*} [[/math]]

Also, we have [math]\widehat{\alpha}(1\otimes u_{ij})=\sum_k1\otimes u_{ik}\otimes u_{kj}[/math], and we obtain from this that we have:

[[math]] \begin{eqnarray*} (\beta\odot\widehat{\alpha})(\beta (b)_{13}) &=&\sum_{uij}\left(\sum_k1\otimes1\otimes u_{ik}\otimes u_{kj}\right)\left(\sum_sb_{is}^u\otimes1\otimes1\otimes u_{sj}^*\right)\\ &=&\sum_{uijks}b_{is}^u\otimes 1\otimes u_{ik}\otimes u_{kj}u_{sj}^* \end{eqnarray*} [[/math]]

By summing over [math]j[/math] the last term is replaced by [math](uu^*)_{ks}=\delta_{k,s}1[/math]. Thus we obtain, as desired, that our algebra consists indeed of fixed points:

[[math]] \begin{eqnarray*} (\beta\odot\widehat{\alpha})(\beta(b)_{13}) &=&\sum_{uik}b_{ik}^u\otimes 1\otimes u_{ik}\otimes 1\\ &=&(\beta(b)_{13})\otimes 1 \end{eqnarray*} [[/math]]

In order to finish now, observe that (1,2,3) above show that [math](D\otimes(E\rtimes_\alpha\widehat{A}_\sigma))^{\beta\odot\widehat{\alpha}}[/math], which is the fixed point algebra of [math]\overline{sp}^{\,w}\{ (1\otimes 1\otimes A_\sigma)\cdot\beta(D)_{13}\cdot\alpha(E)_{23}\}[/math] under the coaction [math]\beta\odot\widehat{\alpha}[/math], is equal to [math]\overline{sp}^{\,w}\{ \beta (D)_{13}\cdot\alpha(E)_{23}\}[/math]. This finishes the proof of the first assertion, and proves as well the non-degeneracy of the diagram in the statement.

Finally, observe that the diagram in the statement is the dual of the square on the left in the following diagram, where [math]P=E\rtimes_{\alpha}\widehat{A}_\sigma[/math] and [math]\pi=\widehat{\alpha}[/math]:

[[math]] \begin{matrix} D&\subset&(D\otimes P)^{\beta\odot\pi}&\subset&D\otimes P\\ \cup&&\cup&&\cup\\ \mathbb C&\subset&P^\pi&\subset&P \end{matrix} [[/math]]

Since [math]\pi[/math] is dual, the square on the right is a non-degenerate commuting square. We also know that the rectangle is a non-degenerate commuting square. Thus if we denote by [math]E_X:D\otimes P\to D\otimes P[/math] the conditional expectation onto [math]X[/math], for any [math]X[/math], then for any [math]b\in D[/math] we have [math]E_{P^\pi}(b)=E_P(b)=E_\mathbb C(b)[/math], and this proves the commuting square condition.

■

Let us denote now by [math]{\mathcal Alg}[/math] the category having as objects the finite dimensional [math]C^*[/math]-algebras and having as arrows the inclusions of [math]C^*[/math]-algebras which preserve the canonical traces. The above result suggests the following abstract definition:

Definition

Given objects [math](D,\beta)\in A-{\mathcal Alg}[/math] and [math](E,\alpha)\in\widehat{A}_\sigma -{\mathcal Alg}[/math], we let

[[math]] D\Box_AE=(D\otimes(E\rtimes_\alpha\widehat{A}_\sigma ))^{\beta\odot\widehat{\alpha}} [[/math]]

be the object in [math]{\mathcal Alg}[/math], constructed as in Proposition 15.18 above.

If [math](D',\beta')\subset(D,\beta )[/math] is an arrow in [math]A-{\mathcal Alg}[/math] and [math](E',\alpha')\subset(E,\alpha )[/math] is an arrow in [math]\widehat{A}_\sigma-{\mathcal Alg}[/math], then we have a canonical embedding, as follows:

[[math]] D'\Box_AE'\subset D\Box_AE [[/math]]

Now since both [math]D'\Box_AE'[/math] and [math]D\Box_AE[/math] are endowed with their canonical traces, this inclusion is Markov. Thus, we have constructed a bifunctor, as follows:

[[math]] \Box_A:A-{\mathcal Alg}\times\widehat{A}_\sigma-{\mathcal Alg}\to\mathcal Alg [[/math]]

With this convention, we have the following result:

Theorem

For any two arrows [math]D_0\subset D_1[/math] in [math]A-{\mathcal Alg}[/math] and [math]E_0\subset E_1[/math] in [math]\widehat{A}_\sigma -{\mathcal Alg}[/math],

[[math]] \begin{matrix} D_0\Box_AE_1&\subset&D_1\Box_AE_1\\ \cup&&\cup\\ D_0\Box_AE_0&\subset&D_1\Box_AE_0 \end{matrix} [[/math]]

is a non-degenerate commuting square of finite dimensional von Neumann algebras.

Show Proof

This can be proved in several steps, as follows:

\underline{Step I}. In the simplest case, [math]D_0=E_0=\mathbb C[/math], this follows from the above.

\underline{Step II}. We prove now the result in the general [math]E_0=\mathbb C[/math] case. Indeed, let [math]E=E_1[/math], and consider the following diagram:

[[math]] \begin{matrix} E&\subset&D_0\Box_AE&\subset&D_1\Box_AE\\ \cup&&\cup&&\cup\\ \mathbb C&\subset&D_0&\subset&D_1 \end{matrix} [[/math]]

By the result of Step I, both the square on the left and the rectangle are non-degenerate commuting squares. We want to prove that the square on the right is a non-degenerate commuting square. But the non-degeneracy condition follows from:

[[math]] D_1\Box_AE=sp\{E\cdot D_1\}\subset sp\{D_0\Box_AE\cdot D_1\} [[/math]]

Now let [math]x\in D_0\Box_AE[/math] and write [math]x=\sum_ib_ia_i[/math] with [math]b_i\in D_0[/math] and [math]a_i\in E[/math]. Then:

[[math]] \begin{eqnarray*} E_{D_1}(x) &=&\sum_ib_iE_{D_1}(a_i)\\ &=&\sum_ib_iE_{\mathbb C}(a_i)\\ &=&\sum_ib_iE_{D_0}(a_i)\\ &=&E_{D_0}(x) \end{eqnarray*} [[/math]]

But this proves the commuting square condition, and we are done.

\underline{Step III}. A similar argument shows that the result holds in the case [math]D_0=\mathbb C[/math].

\underline{Step IV}. General case. We will use many times the following diagram, in which all the rectangles and all the squares, except possibly for the square in the statement, are non-degenerate commuting squares, cf. the conclusions of Steps I, II, III:

[[math]] \begin{matrix} E_1&\subset&D_0\Box_AE_1&\subset&D_1\Box_AE_1\\ \cup&&\cup&&\cup\\ E_0&\subset&D_0\Box_AE_0&\subset&D_1\Box_AE_0\\ \cup&&\cup&&\cup\\ \mathbb C&\subset&D_0&\subset&D_1 \end{matrix} [[/math]]

The non-degeneracy condition follows from:

[[math]] D_1\Box_AE_1=sp\{E_1\cdot D_1\}\subset sp\{D_0\Box_AE_1\cdot D_1\Box_AE_0\} [[/math]]

Now let [math]x\in D_0\Box_AE_1[/math] and write [math]x=\sum_ib_ia_i[/math] with [math]b_i\in D_0[/math] and [math]a_i\in E_1[/math]. Then:

[[math]] \begin{eqnarray*} E_{D_1\Box_AE_0}(x) &=&\sum_ib_iE_{D_1\Box_AE_0}(a_i)\\ &=&\sum_ib_iE_{E_0}(a_i)\\ &=&\sum_ib_iE_{D_0\Box_AE_0}(a_i)\\ &=&E_{D_0\Box_AE_0}(x) \end{eqnarray*} [[/math]]

But this proves the commuting square condition, and we are done.

■

We show now that the bifunctor [math]\Box_A[/math] behaves well with respect to basic constructions. If [math]A[/math] is a Woronowicz algebra, a sequence of two arrows [math]D_0\subset D_1\subset D_2[/math] in [math]A-{\mathcal Alg}[/math] is called a basic construction if [math]D_0\subset D_1\subset D_2[/math] is a basic construction in [math]{\mathcal Alg}[/math] and if its Jones projection [math]e\in D_2[/math] is a fixed by the coaction [math]D_2\to D_2\otimes A[/math]. An infinite sequence of basic constructions in [math]A-{\mathcal Alg}[/math] is called a Jones tower in [math]A-{\mathcal Alg}[/math]. We have:

Proposition

If [math]D_0\subset D_1\subset D_2\subset D_3\subset\ldots[/math] is a Jones tower in [math]A-{\mathcal Alg}[/math] and [math]E_0\subset E_1\subset E_2\subset E_3\subset\ldots[/math] is a Jones tower in [math]\widehat{A}_\sigma-{\mathcal Alg}[/math] then

[[math]] \begin{matrix} \vdots &\ & \vdots &\ & \vdots &\ & \ \\ \cup &\ &\cup &\ &\cup\\ D_0\Box_AE_2&\subset&D_1\Box_AE_2&\subset&D_2\Box_AE_2&\subset&\cdots\\ \cup &\ &\cup &\ &\cup\\ D_0\Box_AE_1&\subset&D_1\Box_AE_1&\subset&D_2\Box_AE_1&\subset&\cdots\\ \cup &\ &\cup &\ &\cup\\ D_0\Box_AE_0&\subset&D_1\Box_AE_0&\subset&D_2\Box_AE_0&\subset&\cdots \end{matrix} [[/math]]

is a lattice of basic constructions for non-degenerate commuting squares.

Show Proof

We prove only that the rows are Jones towers, the proof for the columns being similar. By restricting the attention to a pair of consecutive inclusions, it is enough to prove that if [math]D_0\subset D_1\subset D_2[/math] is a basic construction in [math]A-{\mathcal Alg}[/math] and [math]E[/math] is an object of [math]\widehat{A}_\sigma-{\mathcal Alg}[/math] then [math]D_0\Box_AE\subset D_1\Box_AE\subset D_2\Box_AE[/math] is a basic construction in [math]{\mathcal Alg}[/math].

For this purpose, we will use many times the following diagram, in which all squares and rectangles are non-degenerate commuting squares:

[[math]] \begin{matrix} E&\subset&D_0\Box_AE&\subset&D_1\Box_AE&\subset&D_2\Box_AE\\ \cup &\ &\cup &\ &\cup &\ &\cup\\ \mathbb C&\subset&D_0&\subset&D_1&\subset&D_2 \end{matrix} [[/math]]

We will use the abstract characterization of the basic construction, stating that [math]N\subset M\subset P[/math] is a basic construction, with Jones projection [math]e\in P[/math], precisely when:

(1) [math]P=sp\{ M\cdot e\cdot M\}[/math].

(2) [math][e,N]=0[/math].

(3) [math]exe=E_N(x)e[/math] for any [math]x\in M[/math].

(4) [math]tr(xe)=\lambda tr(x)[/math] for any [math]x\in M[/math], where [math]\lambda[/math] is the inverse of the index of [math]N\subset M[/math].

Let [math]e\in D_2[/math] be the Jones projection for the basic construction [math]D_0\subset D_1\subset D_2[/math]. With [math]N=D_0[/math], [math]M=D_1[/math] and [math]P=D_2[/math] the verification of (1-4) is as follows:

(1) This follows from the following computation:

[[math]] \begin{eqnarray*} D_2\Box_AE &=&sp\{D_2\cdot E\}\\ &=&sp\{D_1\cdot e\cdot D_1\cdot E\}\\ &=&sp\{D_1\cdot e\cdot D_1\Box_AE\} \end{eqnarray*} [[/math]]

(2) This follows from [math]D_0\Box_AE=sp\{D_0\cdot E\}[/math], from [math][e,E]=0[/math] and from [math][e,D_0]=0[/math].

(3) Let [math]x\in D_1\Box_AE[/math], and write [math]x=\sum_ib_ia_i[/math] with [math]b_i\in D_1[/math] and [math]a_i\in E[/math]. Then:

[[math]] \begin{eqnarray*} exe &=&\sum_ieb_ia_ie\\ &=&\sum_ieb_iea_i\\ &=&\sum_iE_{D_0}(b_i)ea_i\\ &=&\sum_iE_{D_0}(b_i)a_ie \end{eqnarray*} [[/math]]

On the other hand, we have as well the following computation:

[[math]] \begin{eqnarray*} E_{D_0\Box_AE}(x)e &=&\sum_iE_{D_0\Box_AE}(b_ia_i)e\\ &=&\sum_iE_{D_0\Box_AE}(b_i)a_ie\\ &=&\sum_iE_{D_0}(b_i)a_ie \end{eqnarray*} [[/math]]

(4) With the above notations, we have that:

[[math]] \begin{eqnarray*} E_{D_2}(xe) &=&\sum_iE_{D_2}(b_ia_ie)\\ &=&\sum_ib_iE_{D_2}(a_i)e\\ &=&\sum_ib_iE_\mathbb C(a_i)e \end{eqnarray*} [[/math]]

We also have [math]b_iE_\mathbb C(a_i)\in D_1[/math] for every [math]i[/math], and so:

[[math]] \begin{eqnarray*} tr_{D_2\Box_AE}(xe) &=&tr_{D_2}(E_{D_2}(xe))\\ &=&\lambda\sum_itr_{D_1}(b_iE_\mathbb C(a_i)) \end{eqnarray*} [[/math]]

On the other hand, we have as well the following computation:

[[math]] \begin{eqnarray*} tr_{D_1\Box_AE}(x) &=&tr_{D_1}(E_{D_1}(x))\\ &=&\sum_itr_{D_1}(b_iE_{D_1}(a_i))\\ &=&\sum_itr_{D_1}(b_iE_\mathbb C(a_i)) \end{eqnarray*} [[/math]]

Thus, the fourth condition for a basic construction is verified, as desired.

■

With standard coaction conventions, from chapter 13, we have:

Proposition

Given a corepresentation and a representation, as follows,

[[math]] v\in M_n(\mathbb C)\otimes A\quad,\quad \pi:A_\sigma\to M_k(\mathbb C) [[/math]]

consider, via some standard identifications, the associated objects

[[math]] (M_n(\mathbb C),\iota_v)\in A-{\mathcal Alg}\quad,\quad (M_k(\mathbb C),\iota_{\check{\pi}})\in \widehat{A}_\sigma-{\mathcal Alg} [[/math]]

and form the corresponding algebra [math]M_n(\mathbb C)\Box_AM_k(\mathbb C)[/math]. Then there exists an isomorphism

[[math]] \begin{pmatrix} M_k(\mathbb C)&\subset&M_n(\mathbb C)\Box_AM_k(\mathbb C)\\ \cup &\ &\cup\\ \mathbb C&\subset & M_n(\mathbb C)\end{pmatrix} \simeq \begin{pmatrix} \mathbb C\otimes M_k(\mathbb C) &\subset & M_n(\mathbb C)\otimes M_k(\mathbb C)\\ \cup &\ &\cup\\ \mathbb C&\subset &u(M_n(\mathbb C)\otimes\mathbb C)u^*\end{pmatrix} [[/math]]

sending [math]z\to 1\otimes z[/math] for [math]z\in M_k(\mathbb C)[/math] and [math]y\to\iota_u (y)[/math] for [math]y\in M_n(\mathbb C)[/math], where [math]u=(id\otimes\pi )v[/math].

Show Proof

Consider the following [math]*[/math]-morphism of algebras:

[[math]] \Phi :M_n(\mathbb C)\otimes M_k(\mathbb C)\to M_n(\mathbb C)\otimes M_k(\mathbb C)\otimes B (l^2(A_\sigma )) [[/math]]

[[math]] x\to ad(v_{13}\check{\pi}_{23}u_{12}^*)(x\otimes 1) [[/math]]

Since both the squares in the statement are non-degenerate commuting squares, all the assertions are consequences of the following formulae, that we will prove now:

[[math]] \Phi (1\otimes z)=\iota_{\check{\pi}} (z)_{23}\quad,\quad \Phi (\iota_u (y))=\iota_v(y)_{13} [[/math]]

The second formula follows from the following computation:

[[math]] \Phi (u(y\otimes 1)u^*) =v_{13}(y\otimes 1\otimes 1)v_{13}^* =(v(y\otimes 1)v^*)_{13} [[/math]]

For the first formula, what we have to prove is that:

[[math]] v_{13}\check{\pi}_{23}u_{12}^*(1\otimes z\otimes 1)u_{12}\check{\pi}_{23}^*v_{13}^*=(\check{\pi}(z\otimes 1)\check{\pi}^*)_{23} [[/math]]

By moving the unitaries to the left and to the right we have to prove that:

[[math]] \check{\pi}_{23}^*v_{13}\check{\pi}_{23}u_{12}^*\in (\mathbb C\otimes M_k(\mathbb C)\otimes\mathbb C)^\prime =M_n(\mathbb C)\otimes\mathbb C\otimes B(l^2(H_\sigma )) [[/math]]

Let us call this unitary [math]U[/math]. Since [math]\check{\pi}=(\pi\otimes id )V^\prime[/math] we have:

[[math]] U=(id\otimes\pi\otimes id)(V_{23}^*v_{13}V_{23}v_{12}^*) [[/math]]

The comultiplication of [math]H_\sigma[/math] is given by [math]\Delta(y)=V^*(1\otimes y)V[/math]. On the other hand since [math]v^*[/math] is a corepresentation of [math]H_\sigma[/math], we have [math](id\otimes\Delta )(v^*)=v^*_{12}v^*_{13}[/math]. We get that:

[[math]] \begin{eqnarray*} V_{23}^*v_{13}V_{23} &=&(V_{23}^*v_{13}^*V_{23})^*\\ &=&((id\otimes\Delta )(v^*))^*\\ &=&(v^*_{12}v^*_{13})^*\\ &=&v_{13}v_{12} \end{eqnarray*} [[/math]]

Thus we have [math]U=v_{13}[/math], and we are done.

■

We are now ready to formulate our main result, as follows:

Theorem

Any commuting square having [math]\mathbb C[/math] in the lower left corner,

[[math]] \begin{matrix} E&\subset & X\\ \cup &\ &\cup\\ \mathbb C &\subset&D\end{matrix} [[/math]]

must appear as follows, for a suitable Woronowicz algebra [math]A[/math], with actions on [math]D,E[/math],

[[math]] \begin{matrix} E&\subset&D\Box_AE\\ \cup &\ &\cup\\ \mathbb C&\subset&D\end{matrix} [[/math]]

and the vertical subfactor associated to it is isomorphic to

[[math]] R\subset(D\otimes(R\rtimes_{\gamma_\pi}\widehat{A}_\sigma ))^{\beta_v\odot \widehat{\gamma_\pi}} [[/math]]

which is a fixed point subfactor, in the sense of chapter 13.

Show Proof

This is something quite technical, which basically follows by combining the above results, and for full details on this, we refer to and related papers.

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Summarizing, all the commuting squares having [math]\mathbb C[/math] in the lower left corner are described by quantum groups. This is of course something quite special, and we will study more general commuting squares, not coming from quantum groups, in the next chapter.

General references

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