Projector Matrix Product Operators, Anyons and Higher Relative Commutants of Subfactors
PPro jector Matrix Product Operators, Anyons andHigher Relative Commutants of Subfactors
Yasuyuki Kawahigashi
Graduate School of Mathematical SciencesThe University of Tokyo, Komaba, Tokyo, 153-8914, Japane-mail: [email protected]
Kavli IPMU (WPI), the University of Tokyo5–1–5 Kashiwanoha, Kashiwa, 277-8583, JapanTrans-scale Quantum Science InstituteThe University of Tokyo, Bunkyo-ku, Tokyo 113-0033, JapanandiTHEMS Research Group, RIKEN2-1 Hirosawa, Wako, Saitama 351-0198,Japan
February 10, 2021
Dedicated to the memory of Vaughan Jones
Abstract
A bi-unitary connection in subfactor theory of Jones producing a subfactorof finite depth gives a 4-tensor appearing in a recent work of Bultinck-Mari¨ena-Williamson-S¸ahino˘glu-Haegemana-Verstraete on 2-dimensional topological or-der and anyons. In their work, they have a special projection called a projectormatrix product operator. We prove that the range of this projection of length k is naturally identified with the k th higher relative commutant of the sub-factor arising from the bi-unitary connection. This gives a further connectionbetween 2-dimensional topological order and subfactor theory. The Jones theory of subfactors [5] in operator algebras has found many profoundrelations to other topics in low-dimensional topology and mathematical physics.Here we present a new connection between subfactor theory and two-dimensionaltopological order. 1 a r X i v : . [ m a t h . OA ] F e b heory of topological phases of matter has recently caught much attention bothin mathematics and physics. A recent paper [2] on two-dimensional topologicalorder, tensor networks and anyons attracted recent interest of several researchersand this topic is closely related to theory of topological quantum computation [20].A certain operator P k on a finite dimensional Hilbert space called a projector matrixproduct operator (PMPO) [2, Section 3] plays a key role and its range is important instudies of gapped Hamiltonians and projected entangled pair states (PEPS) as in [2,Sections 4, 5] in connection to [9], [10]. The ranges of the projector matrix productoperators P k give an increasing sequence of finite dimensional Hilbert spaces indexedby k . We show here that this space has a natural meaning as the k th higher relativecommutant of a subfactor arising from a 4-tensor in the Jones theory through thenotion of a flat field of strings. (See Figure 13 for a matrix product operator O ka which is used in the definition of P k = (cid:88) a d a w O ka .)We note that flatness of a field of strings in the sense of [4, Theorems 11.15]is known to play an important role in subfactor theory and the higher relativecommutants are fundamental objects there. (This flatness was first introduced byOcneanu [15].)We have already seen a connection of the work [2] to subfactor theory and themeaning of anyons there in [8] and we now show a more direct and deeper connection.See [13] for another recent connection to theory of fusion categories, which is alsoclosely related to subfactor theory. See [7] for more general relations among subfactortheory, 2-dimensional conformal field theory and tensor categories.Recently we have much advance in operator algebraic classification of gappedHamiltonians on quantum spin chains [16] and we see some formal similarities ofmathematical structures there. It would be interesting to exploit this possible con-nection.This work was partially supported by JST CREST program JPMJCR18T6 andGrants-in-Aid for Scientific Research 19H00640 and 19K21832. We prepare notations and conventions on bi-unitary connections as in [1], [4, Chapter11], [6], [8] ,[14], [15].We have four finite unoriented connected bipartite graphs G , G (cid:48) , H , H (cid:48) . (Thesegraphs are allowed to have multiple edges between a pair of vertices.) The evenvertices of G and H are identified and we write V for the set of these vertices. Theodd vertices of H and G (cid:48) are identified and we write V for the set of these vertices.The even vertices of G (cid:48) and H (cid:48) are identified and we write V for the set of thesevertices. The odd vertices of G and H (cid:48) are identified and we write V for the set ofthese vertices. They are depicted as in Figure 1. We assume that all of the fourgraphs have more than one edge. 2 H (cid:48) G (cid:48) G V V V V Figure 1: Four graphsLet ∆ G ,xy be the number of edges of G between x ∈ V and y ∈ V . Let ∆ G (cid:48) ,xy be the number of edges of G between x ∈ V and y ∈ V . Let ∆ H ,xy be the numberof edges of H between x ∈ V and y ∈ V . Let ∆ H (cid:48) ,xy be the number of edges of H (cid:48) between x ∈ V and y ∈ V . We assume that we have the following identities forsome positive numbers γ , γ . For each vertex x , we have a positive number µ x . Weassume the following identities. That is, for each of V , V , V , V , the vector givenby µ x gives a Perron-Frobenius eigenvector for the adjacency matrix of one of thefour graphs, and the numbers γ , γ are the Perron-Frobenius eigenvalues of thesematrices. Since all the four graphs have more than one edge, we have γ , γ > (cid:88) x ∆ G ,xy µ x = γ µ y , x ∈ V , y ∈ V , (cid:88) y ∆ G ,xy µ y = γ µ y , x ∈ V , y ∈ V , (cid:88) x ∆ G (cid:48) ,xy µ x = γ µ y , x ∈ V , y ∈ V , (cid:88) y ∆ G (cid:48) ,xy µ y = γ µ y , x ∈ V , y ∈ V , (cid:88) x ∆ H ,xy µ x = γ µ y , x ∈ V , y ∈ V , (cid:88) y ∆ H ,xy µ y = γ µ y , x ∈ V , y ∈ V , (cid:88) x ∆ H (cid:48) ,xy µ x = γ µ y , x ∈ V , y ∈ V , (cid:88) y ∆ H (cid:48) ,xy µ y = γ µ y , x ∈ V , y ∈ V , For an edge ξ of one of the graphs G , G (cid:48) , H , H (cid:48) , we regard it oriented, and write s ( ξ ) and r ( ξ ) for the source (starting vertex) and the range (ending vertex). Let ξ , ξ , ξ , ξ be oriented edges of G , H , G (cid:48) , H (cid:48) , respectively. If we have s ( ξ ) = x ∈ V , r ( ξ ) = x ∈ V , s ( ξ ) = x ∈ V , r ( ξ ) = x ∈ V , s ( ξ ) = x ∈ V , r ( ξ ) = x ∈ V , s ( ξ ) = x ∈ V , and r ( ξ ) = x ∈ V , then we call a combination of ξ i a cell , as inFigure 2.For each cell, we assign a complex number. We call this map a connection andwrite W for this. We also write as in Figure 3 for the number assigned by W to this3 ξ ξ ξ x x x x Figure 2: A cellcell. This setting is similar to an interaction-round-a-face (IRF) model in theory ofsolvable lattice models, where we also assign a complex number to each cell arisingfrom one graph. Wξ ξ ξ ξ x x x x Figure 3: A connection valueWe first require unitarity of W as in Figure 4, where the bar on the right celldenotes the complex conjugate. Wξ ξ ξ ξ zx wy Wξ ξ (cid:48) ξ ξ (cid:48) zx wy (cid:48) (cid:80) z,ξ ,ξ = δ y,y (cid:48) δ ξ ,ξ (cid:48) δ ξ ,ξ (cid:48) Figure 4: UnitarityWe define a new connection W (cid:48) as in Figure 5. We also require that this W (cid:48) satisfies unitarity. When unitarity holds for W and W (cid:48) , we say W satisfies bi-unitarity . Ocneanu and Haagerup found that a bi-unitary connection characterizesa non-degenerate commuting squares of finite dimensional C ∗ -algebras with a traceas in [4, Section 11.2],A typical example of a bi-unitary connection arises as follows. Fix one of theDynkin diagrams A n , D n , E , E , E and let N be its Coxeter number. Set all G , G (cid:48) , H , H (cid:48) to be this graph and set γ , γ to be 2 cos πN . We then have a bi-unitaryconnection as in [4, Figure 11.32]. (See [17] for the corresponding IRF models.)We assume this bi-unitarity for W from now on. (We do not assume flatness of W in the sense of [4, Definition 11.16].) 4 (cid:48) ξ ξ ˜ ξ ˜ ξ wy zx Wξ ξ ξ ξ zx wy = (cid:114) µ x µ w µ y µ z Figure 5: Renormalization (1)¯ W ˜ ξ ˜ ξ ξ ξ xz yw Wξ ξ ξ ξ zx wy = (cid:114) µ x µ w µ y µ z Figure 6: Renormalization (2)We also define new connections ¯ W and ¯ W (cid:48) as in Figures 6 and 7. They bothsatisfy bi-unitarity automatically. We also define a value of another diagram as inFigure 8. Note that we have Figure 9 due to Figures 5 and 8.We fix any vertex in V and write ∗ for this. As in [4, Section 11.3], we constructa double sequence of finite dimensional C ∗ -algebras { A nk } n,k =0 , ,... starting from ∗ and hyperfinite II factors A ∞ ,k and A n, ∞ , using W, W (cid:48) , ¯ W , ¯ W (cid:48) . (Here our µ ∗ is notnormalized to be 1, so we use µ x /µ ∗ to define a normalized trace on A nk as in [4,page 554].) Then the we have [ A ∞ , : A ∞ , ] = γ and [ A , ∞ : A , ∞ ] = γ for theJones index values as in [4, Theorem 11.9]. This construction is due to Ocneanu[14]. We now assume that one of these two subfactors has a finite depth in the senseof [4, Definition 9.41]. Note that in this case, the other subfactor also has a finitedepth by [18, Corollary 2.2].Let ˜ W be the (vertical) product of W and ¯ W as in Figure 10. That is, wemultiply two connection values and make a summation over all possible choices of ξ , like concatenation of tensors. We make irreducible decomposition of powers of˜ W . As in [1, Section 3], this product and irreducible decomposition correspond tothe relative tensor product and irreducible decompositions of A , ∞ - A , ∞ bimodulesarising from the subfactor A , ∞ ⊂ A , ∞ . (These bimodules are also understood interms of sectors as in [12].)Let { W a } a ∈ V be the set of representative of irreducible bi-unitary connections,up to equivalence, appearing in the irreducible decompositions of the powers of ˜ W .(See [1, Section 3] for the definition of equivalence of connections. This correspondsto an isomorphism of bimodules.) The finite depth assumption exactly means thatthe set V is finite. Each a corresponds to an irreducible A , ∞ - A , ∞ bimodules arisingfrom the subfactor A , ∞ ⊂ A , ∞ . Each a thus also corresponds to an even vertex ofthe principal graph of the subfactor A , ∞ ⊂ A , ∞ . Note that the horizontal graphof each W a is always the original graph G .5 W (cid:48) ˜ ξ ˜ ξ ˜ ξ ˜ ξ yw xz Wξ ξ ξ ξ zx wy =Figure 7: Renormalization (3) Wξ ξ ξ ξ wy zx Wξ ξ ξ ξ zx wy =Figure 8: Conjugate conventionLet d a be the Perron-Frobenius eigenvalue of the vertical graph correspondingto the bi-unitary connection W a . This is equal to the dimension of the bimodulecorresponding to W a as in [1, Section 3]. We define w = (cid:80) a ∈ V d a , which is some-times called the global index of the subfactor A , ∞ ⊂ A , ∞ . The Perron-Frobeniusvector ( µ x ) x ∈ V is unique up scalar. We now normalize this vector so that we have (cid:80) x ∈ V µ x = w .Let M yxa be the number of vertical edges with vertex x ∈ V at the upper leftcorner and y ∈ V at the lower left corner for the connection W a . Note that we have (cid:80) y ∈ V M yxa µ y = d a µ x by the Perron-Frobenius eigenvalue property. For a, b, c ∈ V ,let N cab be the multiplicity of W c in the irreducible decomposition of the product W a W b . This is also the structure constant of a relative tensor product of the corre-sponding bimodules over A , ∞ .We define ¯ a to be b ∈ V so that ¯ W a is equivalent to W b . We have M yxa = M xy ¯ a . The A , ∞ - A , ∞ bimodule corresponding to ¯ a is contragredient to the one correspondingto a .Finally, we recall the following elementary lemma about a conditional expecta-tion in the string algebra. (See [4, Definitions 11.1, 11.4] for string algebras and atrace there.) Lemma 2.1
Let A = C ⊂ B ⊂ C be an increasing sequence of string algebras oflength , , on a Bratteli diagram. We write ∗ for the initial vertex of the Brattelidiagram corresponding to A = C . We fix a faithful trace on C . The conditionalexpectation E from C onto B (cid:48) ∩ C is given as follows.Let ξ , ξ be edges of the Bratteli diagram corresponding to A ⊂ B , η , η beedges of the one corresponding to B ⊂ C . Assume r ( ξ ) = s ( η ) , r ( ξ ) = s ( η ) , ξ ξ ˜ ξ ˜ ξ zx wy Wξ ξ ξ ξ zx wy = (cid:114) µ x µ w µ y µ z Figure 9: Renormalization convention˜
W W ¯ Wξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ × = (cid:88) ξ Figure 10: The product connection ˜ Wr ( η ) = r ( η ) . We then have E (( ξ · η , ξ · η )) = δ ξ ,ξ K r ( ξ ) (cid:88) ξ ( ξ · η , ξ · η ) , where K r ( ξ ) is the number of edges from ∗ to r ( ξ ) on the Bratteli diagram corre-sponding to A ⊂ B . Proof.
We have this identity by a direct computation. (cid:3)
We define projector matrix product operators [2, Section 3.1], which was originallydefined in terms of 4-tensors, with language of bi-unitary connections in the previousSection.We define a 4-tensor a as in Figure 11 and [8, Figure 11]. Note that we have ahorizontal concatenation of the connections W a and W (cid:48) a here, since we have consid-ered only symmetric bi-unitary connections in [8, Section 2] while we do not assumethis symmetric condition here. (If we have s ( ξ ) (cid:54) = s ( ξ ), then the value of the4-tensor is set to be 0. Similarly, if the edges do not make a cell for one of the twosquares, the value of the 4-tensor is 0.) 7 ξ ξ ξ · ξ ξ · ξ W a W (cid:48) a ξ ξ ξ ξ ξ ξ zx wy = (cid:114) µ x µ w µ y µ z Figure 11: The 4-tensor a and the connection W a Remark 3.1
When we concatenate edges ξ , ξ , . . . , ξ k taken from the horizontalgraph of W a , we impose the condition r ( ξ m ) = s ( ξ m +1 ) for m = 1 , , . . . , k −
1. Inthe 4-tensor setting, we do not impose such a condition for concatenation of edges,but this difference does not cause any problem here. If we have r ( ξ m ) (cid:54) = s ( ξ m +1 ), thepath ξ . . . ξ · · · ξ k is mapped to zero by any matrix product operator and we canignore this path, since we are interested in the range of a matrix product operator.Fix a positive integer k . Let Path k ( G ) be the C -vector space with a basisconsisting of paths of length 2 k on G starting at an even vertex of G . We definea matrix product operator O k,ya,x from Path kx ( G ) to Path kx ( G ), where Path kx ( G ) is a C -linear space spanned by paths of length 2 k starting from x on G , as in Figure 12,where ξ and ξ have length k each. W a W (cid:48) a · · · · · · W (cid:48) a W a η η ξ ξ η · η ζ ζxy xyO k,ya,x ( ξ · ξ ) = (cid:88) ζ,η ,η Figure 12: The operator O k,ya,x We next define a matrix product operator O ka by O ka ( (cid:77) x ξ x ) = (cid:77) y (cid:88) x O k,ya,x ξ x , where ξ x ∈ Path kx ( G ). Note that this is the same as the matrix product operator O ka defined by Figure 13 as in [2, Section 3.2]. We have different normalizationconvention for the tensor a and the connection W a as in Figure 11, but these coeffi-cients cancel out due to the horizontal periodicity of the picture. (Remark 3.1 againapplies here about the domains of the two operators O ka .)We then have O ka O kb = (cid:80) c N cab O kc . We further define a projector matrix productoperator P k = (cid:88) a d a w O ka as in [2, Section 3.1]. (This is a projection as shown there.)For a path ξ · ξ with r ( ξ ) = s ( ξ ) and | ξ | = | ξ | = k , we define Φ k ( ξ · ξ ) = µ s ( ξ ) µ r ( ξ ) ( ξ , ˜ ξ ), which is a map from Path k ( G ) to B k , where ˜ ξ is the reversed path of8 a · · · aη ξ η ξ η k ξ k (cid:88) ξ ,ξ ,...,ξ k ,η ,η ,...,η k | ξ ξ · · · ξ k (cid:105)(cid:104) η η · · · η k | Figure 13: The matrix product operator O ka ξ , B k = (cid:76) x Str kx ( G ) and Str kx ( G ) is the string algebra on G with length k startingat a vertex x ∈ V of G . (See [4, Definitions 11.1, 11.4] for string algebras.)We define a matrix product operator ˜ O k,ya,x from Str kx ( G ) to Str ky ( G ) as in Figure14. W a W (cid:48) a · · · · · · W (cid:48) a W a η η ξ ξ ( η , η ) ζ ζxy xy ˜ O k,ya,x (( ξ , ξ )) = (cid:88) ζ,η ,η Figure 14: The operator ˜ O k,ya,x We next define a matrix product operator ˜ O ka by˜ O ka ( (cid:77) x ξ x ) = (cid:77) y (cid:88) x ˜ O k,ya,x ξ x , where ξ x ∈ Path kx ( G ). We again have ˜ O ka ˜ O kb = (cid:80) c ∈ V N cab ˜ O kc . We further define aprojector matrix product operator ˜ P k = (cid:88) a ∈ V d a w ˜ O ka again as in [2, Section 3.1].We then have Φ k O ka = ˜ O ka Φ k because of the normalization in Figure 9 and( ˜ P k ) = ˜ P k for the same reason as ( P k ) = P k .Each Str kx ( G ) has a standard normalized trace tr x as in [4, page 554]. We settr( σ ) = (cid:88) x ∈ V µ x w tr x ( σ x ) for σ = (cid:76) x ∈ V σ x ∈ Str kx ( G ). We let (cid:107) σ (cid:107) st , = (cid:112) tr( σ ∗ σ ) for σ ∈ B k .Let C be the maximum of the number of x ∈ V , the number of a ∈ V and (cid:107) ˜ O ka (cid:107) over all a ∈ V . Here the norm (cid:107) ˜ O ka (cid:107) is the operator norm on B k with respect to (cid:107) · (cid:107) st , .Let K nx be the number of paths from ∗ to x on H of length 2 n . Let α n = (cid:112)(cid:80) x ( K nx ) , and κ nx = K nx /α n . By the Perron-Frobenius theorem, we have κ nx → µ x / √ w as n → ∞ for all x ∈ V .For a positive integer n , let ˜ W n ∼ = (cid:80) a L na W a , β n = (cid:112)(cid:80) a ( L na ) and λ na = L na /β n .By the Perron-Frobenius theorem again, we have λ na → d a / √ w as n → ∞ for all a ∈ V .We recall the following elementary lemma.9 emma 3.2 Let M be a von Neumann algebra with a normalized trace tr and P beits subalgebra. For σ ∈ M and ε < , if we have |(cid:107) σ (cid:107) − (cid:107) E P ( σ ) (cid:107) | < ε (cid:107) σ (cid:107) , thenwe have (cid:107) σ − E P ( σ ) (cid:107) < √ √ ε (cid:107) σ (cid:107) . Proof.
Since (cid:107) E P ( σ ) (cid:107) > (1 − ε ) (cid:107) σ (cid:107) and (cid:107) σ (cid:107) = (cid:107) E P ( σ ) (cid:107) + (cid:107) σ − E P ( σ ) (cid:107) , wehave the conclusion. (cid:3) With these preparations, we are going to prove the following main result of thispaper.
Theorem 3.3
The range of the projector matrix product operator P k of length k is naturally identified with the k th higher relative commutant A (cid:48)∞ , ∩ A ∞ ,k for thesubfactor A ∞ , ⊂ A ∞ , arising from the original connection W . Proof.
Note that the map Φ k gives a linear isomorphism from the range of P k tothat of ˜ P k in B k .We first construct a linear isomorphism ∆ from A (cid:48)∞ , ∩ A ∞ ,k to the range of ˜ P k .By [4, Theorem 11.15], an arbitrary element in A (cid:48)∞ , ∩ A ∞ ,k is given by a flat field (cid:76) x σ x ∈ B k and identified with σ ∗ ∈ A ,k .We define an operator T k,ya,x,ζ ,ζ from Str kx ( G ) to Str ky ( G ) as in Figure 15. W a W (cid:48) a · · · · · · W (cid:48) a W a η η ξ ξ ( η , η ) ζ ζ xy xyT k,ya,x,ζ ,ζ (( ξ , ξ )) = (cid:88) η ,η Figure 15: The operator T k,ya,x,ζ ,ζ Then flatness of the field [4, Theorems 11.15] gives the equality T k,ya,x,ζ ,ζ ( σ x ) = δ ζ ,ζ σ y . (This holds as in [4, Figure 11.19]. Though flatness of the bi-unitaryconnection is not assumed here, flatness of the fields works instead.) This impliesthat ˜ O ka σ x = (cid:76) y M yxa σ y . Note we have (cid:88) x ∈ V ,a ∈ V d a µ x M yxa = (cid:88) a ∈ V d a (cid:88) x ∈ V µ x M xy ¯ a = (cid:88) a ∈ V d a µ y = wµ y . We then have P k ( (cid:77) x ∈ V µ x σ x ) = (cid:88) x ∈ V d a w ˜ O ka µ x σ x = (cid:77) y ∈ V (cid:88) x ∈ V ,a ∈ V d a w µ x M yxa σ y = (cid:77) y ∈ V µ y σ y so the map ∆ assigning (cid:76) x ∈ V µ x σ x to σ ∗ gives a linear injection from A (cid:48)∞ , ∩ A ∞ ,k to the range of ˜ P k in B k . 10e next construct an injective linear map for the converse direction. Let (cid:76) x ∈ V µ x σ x be in the range of ˜ P k in B k . For a positive integer n , we set σ ( n ) = (cid:88) x ∈ V (cid:88) ξ ( ξ, ξ ) · σ x ∈ A n,k , where ξ gives all paths from ∗ to x on H with length 2 n . We assume that n issufficiently large so that the numbers K nx are all nonzero.Suppose that we have the following two estimates for sufficiently small ε > − ε ) µ x √ w < κ nx < (1 + ε ) µ x √ w , for all x ∈ V (1)(1 − ε ) d a √ w < λ ma < (1 + ε ) d a √ w , for all V ∈ a (2)A computation shows that E A (cid:48) n +2 m, ∩ A n +2 m,k ( σ ( n ) ) is equal to (cid:77) y ∈ V (cid:88) a ∈ V,x ∈ V K nx K n + my L ma ˜ O k,ya,x ( σ x )by Lemma 2.1, since we have ˜ W ∼ = (cid:80) a ∈ V L ma W a . Here for large n and m , K nx isalmost equal to α n µ x √ w , L ma is almost equal to β m d a √ w , and K n + my is almost equal to (cid:88) a ∈ V,x ∈ V α n µ x √ w β m d a √ w M yxa = α n β m w (cid:88) a ∈ V,x ∈ V d a µ x M xy ¯ a = α n β m w (cid:88) a ∈ V d a µ y = α n β m µ y . If we had exact equalities for all these three pairs, then we would have (cid:77) y (cid:88) a ∈ V,x ∈ V K nx K n + my L ma ˜ O k,ya,x ( σ x ) = (cid:77) y ∈ V (cid:88) x ∈ V µ x √ wβ m µ y β m d a √ w ˜ O k,ya,x ( σ x )= (cid:77) y ∈ V d a wµ y ˜ O k,ya,x ( µ x σ x )= (cid:77) y ∈ V σ y and hence E A (cid:48) n +2 m, ∩ A n +2 m,k ( σ ( n ) ) would be equal to σ ( n + m ) . Now we take the ap-proximation errors into account. Suppose we have the estimates (1) and (2). Wethen have (1 − C ε ) α n β m µ y < K n + my < (1 + 3 C ε ) α n β m µ y , for all y − C ε ) α n β m µ y < K nx K n + my < (1 + 4 C ε ) α n β m µ y , for all x, y. We further have (1 − C ε ) µ x d a µ y w < K nx K n + my L ma < (1 + 5 C ε ) µ x d a µ y w . This shows (cid:107) E A (cid:48) n +2 m, ∩ A n +2 m,k ( σ ( n ) ) − σ ( n + m ) (cid:107) ≤ C ε (cid:107) σ ( n + m ) (cid:107) ≤ C ε (cid:107) σ (cid:107) st , , since we have(1 − Cε ) (cid:107) σ (cid:107) st , < √ − Cε (cid:107) σ (cid:107) st , < (cid:107) σ ( n ) (cid:107) < √ Cε (cid:107) σ (cid:107) st , < (1 + Cε ) (cid:107) σ (cid:107) st , and(1 − Cε ) (cid:107) σ (cid:107) st , < √ − Cε (cid:107) σ (cid:107) st , < (cid:107) σ ( n + m ) (cid:107) < √ Cε (cid:107) σ (cid:107) st , < (1 + Cε ) (cid:107) σ (cid:107) st , . We now have |(cid:107) E A (cid:48) n +2 m, ∩ A n +2 m,k ( σ ( n ) ) (cid:107) − (cid:107) σ ( n ) (cid:107) | < C ε (cid:107) σ (cid:107) st , . By Lemma 3.2, we then have (cid:107) E A (cid:48) n +2 m, ∩ A n +2 m,k ( σ ( n ) ) − σ ( n ) (cid:107) < C √ ε (cid:107) σ (cid:107) st , . We now have (cid:107) σ ( n + m ) − σ ( n ) (cid:107) ≤ C √ ε (cid:107) σ (cid:107) st , . We first choose n so that we have (1) with n = n and ε = 1100 C · l = 1, we make the following procedure inductively. We choose m l so thatwe have (2) with m = m l and ε = 1100 C · l and (1) with n = n l + m l and ε = 1100 C · l +1 . We next set n l +1 = n l + m l .Then we have (cid:107) σ ( n l ) − σ ( n l +1 ) (cid:107) ≤ l (cid:107) σ (cid:107) st , . Because of this estimate, we know that the sequence { σ ( n l ) } l converges in A ∞ ,k inthe strong operator topology. We set Γ( (cid:76) x ∈ V µ x σ x ) = lim l →∞ σ ( n l ) . Since σ ( n l ) ∈ A (cid:48) n l , ∩ A ∞ ,k , we have Γ( (cid:76) x ∈ V µ x σ x ) ∈ A (cid:48)∞ , ∩ A ∞ ,k . This Γ is clearly a linearmap. We have (cid:107) Γ( (cid:76) x ∈ V µ x σ x ) (cid:107) = (cid:107) (cid:76) x ∈ V σ x (cid:107) st , , so Γ is injective. This showsthe dimension of the range of ˜ P k is smaller than or equal to dim( A (cid:48)∞ , ∩ A ∞ ,k ). Wethus conclude that the map ∆ constructed above is a linear isomorphism. (Thisactually shows that (cid:76) x ∈ V σ x is a flat field and all σ ( n ) are equal in A ∞ ,k .) (cid:3) emark 3.4 The range of the projector matrix product operator of length k hasa clear invariance under rotation of 2 π/k . This passes to invariance of flat fields ofstrings of length k under rotation of 2 π/k . Such invariance was observed by Ocneanuin early days of the theory and this rotation was called a Fourier transform of a flatfield of strings. See [11] for a recent progress of this notion of the Fourier transform. Remark 3.5
Replace the initial bi-unitary connection W with W (cid:48) . The resultingsubfactor A , ∞ ⊂ A , ∞ does not change, so the set { a, b, . . . } of labels of the ir-reducible bi-unitary connections does not change, but the subfactor A ∞ , ⊂ A ∞ , changes to its dual subfactor. So the range of the projector matrix product oper-ator also changes from the higher relative commutant to the dual higher relativecommutant of a subfactor in this process. Remark 3.6
Recall that the Drinfel (cid:48) d center of the fusion category of A , ∞ - A , ∞ bimodules arising from the subfactor A , ∞ ⊂ A , ∞ is a modular tensor categoryrelated to the 2-dimensional topological order appearing in [2, Section 5], as shownin [8, Theorem 3.2]. Note that the higher relative commutants of the other subfactor A ∞ , ⊂ A ∞ , appear here in this paper. Relations between these two subfactors areclarified in [19, Theorem 3.3]. Remark 3.7
The range of P k does not have a natural algebra structure, but weknow from the above Theorem that it has a natural structure of a ∗ -algebra. Example 3.8
An almost trivial example is given as follows. All the sets V , V , V , V are one-point sets and identified with { x } . All the graphs G , G (cid:48) , H , H (cid:48) consist of d multiple edges from x to x and they are all identified. We have µ x = 1, γ = γ = d and the connection W is given as in Figure 16. Wξ ξ ξ ξ = δ ξ ,ξ δ ξ ,ξ . Figure 16: An almost trivial exampleThis is a flat connection, and ths set V is identified with { x } . We have d x = 1and w = 1.In this case, the range of ˜ P k is M d ( C ) ⊗ k , where M d ( C ) is the d × d full matrixalgebra with complex entries.In this example, the natural C ∗ -algebra appearing in the inductive limit is aUHF algebra. In the general case, we have an AF algebra instead.13 xample 3.9 An easy example of a bi-unitary connection arises from a finite group G as in [4, Figure 10.25]. This corresponds to a trivial 3-cocycle case considered in[2, Section 6].We have γ = γ = (cid:112) | G | and this is a flat connection. The sets V and V areboth identified with G as sets. All d a are 1 and w = | G | . Example 3.10
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