Moreover, he showed that all these numbers are realized as indices of subfactors of the hyperfinite II1 factor. In [Sch90], Schou showed that subfactors of the hyperfinite II1 factor could also be constructed from certain infinite-dimensional commuting squares and found examples of irreducible hyperfinite subfactors that cannot be constructed from finite-dimensional commuting squares. In Chapter 1 we describe the method for constructing an irreducible subfactor of the hyperfinite II1factor from a (finite-dimensional or infinite-dimensional) symmetric commuting square as shown in [Sch90].
In this construction, the resulting subfactor has an index equal to the square of the norm of a given connected, locally finite, bipartite graph on which the commute square is based. In [Bis94b], Bisch used an infinite-dimensional symmetric commuting square to construct the first example of an irreducible subfactor of the hyperfinite II1 factor whose index is a rational, non-integer and thus not an algebraic integer.
Preliminaries
If A⊂GB are (finite-dimensional or infinite-dimensional) multimatrix algebras with dimensional vectors~ain~b and trace vectors~α and~β defining finite traces matching on A, then ~α =G~β and ~ b=Gt~a. We will consider only inclusions A⊂GB of multimatrix algebras with inclusive graphs ΓG, which are:. i) locally finite, i.e. all matrix polynomials in G0tG0. Note that every unitary, injective∗-homomorphismφ :A→Bbetween (both finite-dimensional or both infinite-dimensional) multimatrix algebras A and B, up to unitary conjugation, is of the diagonal form (1.1).
We call a square finite-dimensional (resp. infinite-dimensional) motion if the multi-matrix algebras involved are all finite-dimensional (resp. infinite-dimensional). Then these are equivalent: i) There exists a symmetric moving square of multi-matrix algebras A, B, C, D. with respect to a faithful and finite tracer D. ii).
Construction of hyperfinite subfactors
To construct hyperfinite subfactors from infinite-dimensional symmetric commutative squares, a generalization of the Perron-Frobenius theorem is needed:. This eigenvalue and its positive eigenvectors are called the Perron-Frobenius eigenvalue and Perron-Frobenius eigenvectors of T, as in the finite case. Let T be a finite or countably infinite, positive-semidefinite matrix and ~ξ its Perron-Frobenius eigenvector such that k~ξk2=1.
This relation between Perron-Frobenius eigenvectors and Markov traces can be used to obtain the following result about the symmetric squares of the motion:. be a moving symmetric square of multimatrix algebras with respect to a normalized one, . faithful tracing trD in D such that ΓG,ΓH,ΓK,ΓL are connected and locally finite. i) kKk=kHk,trDis trace(kHk−2)-Markov of the inclusion B⊂H D and trD C is the trace(kKk−2)-Markov of the inclusion A⊂KC. ii) kGk=kLk,trDis trace(kLk−2)-Markov of inclusion C⊂L D and trD. B is the (kGk−2)-Markov trace of the embedding A⊂GB. iii) ~δ is a Perron-Frobenius eigenvector of HtH and LtL.~γ is a Perron-Frobenius eigenvector of LLt and KtK.~β is a Perron-Frobenius vector of HHand GtG.~α is a Perron-Frobenius eigenvector of GGt and KKt.
Irreducibility
Note that the index of a subfactor of the hyperfinite II1 factor constructed in this way from affine-dimensional symmetric commutative square is always an algebraic integer. If A is an II1factor, trits is normalized, faithful, normal trace, and a,b∈A is self-adjoint such that:. i) a=∑mi=1αipi where pi∈A are mutually orthogonal projections, such that tr(pi) =ti, for1≤i≤m, and. ii) b has strictly less than m spectral values enkbk ≤ kak, thenkb−ak22≥ε. For j∈N, letq2lj be the minimal central projection of B2l contained in its jth simple summation, i.e.
Let A⊂B be the hyperfinite II1 factors constructed from this commuting square, as described after Note 1.11. If H or K contains a sequence with exactly one non-zero entry equal to 1, then A⊂B is irreducible.
Ocneanu compactness
This was achieved by an explicit construction of the symmetric commutation square of infinite-dimensional multimatrix algebras based on the inclusion graph given by a 4-star with an A∞-tail. Our goal is to find symmetric driving squares based on the inclusion graphs given by aN-star with an A∞-tail, so in the following we consider fixed N≥4 due to Proposition 1.10 and Remark 2.2. Fork∈N,Gk is again an infinite matrix with well-defined finite entries because ΓN is locally finite.
Let G be a bipartite adjacency matrix ΓN such that its rows are numbered by the upper vertices of ΓN and its columns are numbered by the lower vertices of ΓN. If G has an eigenvector `2 whose entries are all positive, then by Theorem 1.5 this eigenvector is a Perron-Frobenius eigenvector and the corresponding eigenvalue is its Perron-Frobenius eigenvalue. Since we want to use it to define the final trace, we also need to have this Perron-Frobenius eigenvector '1.
Due to Remark 2.3, we are only interested in solutions such that λ ≥2 and that implies x1≥1≥x2>0. Due to Theorem 1.5, we are only interested in solutions such as k~ξk2<∞so we must have c1=0. Since we are looking for an eigenvector~ξ, we can setec2=λ (or any other arbitrary non-zero scalar).
Since all entries of ~ξ are positive, it follows that λ is the Perron-Frobenius eigenvalue and ~ξ is a Perron-Frobenius eigenvector of G due to Theorem 1.5.
Bi-unitary condition
If we are to have a moving symmetric square of this form, then by Proposition 1.10 we must have ~δ =~t (up to a scalar) since it is the Markov trace (after normalization) for the inclusionB⊂GD. Also, note that ~t is `1 so it defines a finite trace overD from the entries of the Dare dimension vector bounded by a constant. These traces are not normalized, but we can apply the results from Chapter 1 to normalize them when necessary.
In the case ofv(bj,cl) we label the blocks with(ai,dk) and writev(bj,cl). For (ai,di+1) all S-paths go through bN+i−1 and all T-paths go through cN+i−1, so u(ai,di+1) consists of a single elementary aryn×n block. For (ai+1,di) all S-paths go through bN+i−2 and all T-paths go through cN+i−2, so u(ai+1,di) consists of a single elementary array×n block.
Here for each (bj,cl) there are blocks that make up (bj,cl)and each represents some elements×nblockv(b(aj,cl) . i,dk). For each elementaryn×nblock inv(oru) we now use (2.5) to calculate the corresponding valueq. i.e. the multiplicative inverse of the calculated value) at the appropriate place within the block structure form v(oru) we get a matrix that tells us what we need to multiply each elementaryyn×nblock ofv by to get the corresponding elementaryyn×nblock ofu. Note that for each remaining non-zero elementaryn×nblock in allv(bj,cl)ofv for j, l≥N (marked by *) we can choose an arbitrary n×nunitary matrix.
This is because every such∗in vis is equal to some non-zero elementaryn×nblock in some(ai,dk)vulllari+k>2 (again denoted by ∗) since the corresponding q. Thus, every such choice is consistent and all the resulting v(bj,cl)for j, l≥Nandu(ai,dk)fori+k>2 are uniform due to Lemma 2.4. Note that if the biunitary matrices are of this form and we fix their entries, then for each choice of n × units for∗the entries in the resulting matrices u and var are bi-unitarily equivalent (see [JS97]), i.e.
Case n = 1, general N
Now, without loss of generality (by rearranging rows and columns if necessary), Remark 2.9 implies that. Thus, according to proposition 2.5, there are no symmetric commuting squares of the form (2.3) for N =4 and n=1. This is why Bisch's construction of an irreducible hyperfinite subfactor with index 4.5 is more involved and is based on a symmetric commutation.
It follows from Proposition 2.5 that there exists a symmetric commutation square of the form (2.3) for N=5 and n=1.
All equilateral pentagons (5-gons) of Φ (from Lemma 2.8) are the same (up to the permutation of sides). For N =7 there exist at least two non-equivalent unit matrices U of the form as in Remark 2.7. We do not know whether the subfactors obtained from the two non-equivalent symmetric commutative squares based on the above connections are isomorphic.
Case n = 1, odd N
One possible matrixΦof the form in (2.17) resulting from this solution is characterized by values.
Summary and remarks
For N≥11 it does not seem feasible to obtain a solution in this way that is so simple that it can be written out explicitly. All these indices are rational non-integers and thus not algebraic integers, implying that these subfactors cannot be obtained from finite-dimensional symmetric pendulum squares. Numerical calculations suggest that there are no pendulum squares of the form (2.3) for N=8 and n=1, but that such pendulum squares do exist for n=1 and several consecutive numbers N≥10.
Regarding the existence of such commuting squares forn=1 and oddN whose corresponding matrix Φ is of the form as in (2.17), numerical calculations indicate that they exist for many consecutive odd numbersN≥11. There exists a symmetric commutative square based on the inclusion graph 4-star withA∞-tail of the form (2.3) for n=2, this is the commutative square that gives rise to the subfactor of index 4.5 of Bisch from [Bis94b]. Subfactors constructed in this thesis arising from commuting squares based on inclusion graphsN-star with A∞-tail of the form (2.3) forN≥5 all have indices greater than 5.25 and are therefore outside the scope of the current classification of small index subfactors ( [AMP15]).
It is an open problem to determine their main graphs, or more generally, their standard invariant. There has been a lot of interest lately in quantum Fourier analysis and the biunitary compounds we have calculated are explicit examples of what is called quantum Fourier transform in [JJL+20]. It's not clear what the quantum symmetries are that transform them, so more interesting work can be done here.
Bisch,An example of an irreducible subfactor of the hyperfinite factor II1 with rational, incomplete index, J. Ocneanu, Quantized groups, string algebras and Galois theory of algebras, in: Operator Algebras and Applications, Vol.
