A few remarks on Pimsner-Popa bases and regular subfactors of depth 2
aa r X i v : . [ m a t h . OA ] F e b A FEW REMARKS ON PIMSNER-POPA BASES AND REGULARSUBFACTORS OF DEPTH 2
KESHAB CHANDRA BAKSHI AND VED PRAKASH GUPTA
In memory of Vaughan Jones, a true pioneer!
Abstract.
We prove that a finite index regular inclusion of II -factors with commutative firstrelative commutant is always a crossed product subfactor with respect to a minimal action of abiconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of II -factors which is of depth 2 and has simple first relative commutant (respectively, is regular andhas commutative or simple first relative commutant) admits a two-sided Pimsner-Popa basis(respectively, a unitary orthonormal basis). Introduction
Right from the early days of the evolution of the theory of operator algebras, the methodsof crossed product constructions and fixed point subalgebras with respect to actions by variousalgebraic objects on operator algebras have served extremely well to provide numerous exampleswith specific properties as well as to be considered as suitable candidates for structure resultsunder certain given hypotheses. One of the first such structure results (thanks to Ocneanu, Jones,Sutherland, Popa, Kosaki and Hong) states that every irreducible regular inclusion of factors oftype II with finite Jones index is a group subfactor of the form N ⊂ N ⋊ G , with respect to an outeraction of a finite group G on N . In particular, every such subfactor has depth
2. Further, it has alsobeen established (in a series of papers by Ocneanu, David, Szyma´nski and Nikshych-Vainerman)that every finite index inclusion of type II factors of depth 2 is of the form N ⊂ N ⋊ H , withrespect to a minimal action of some biconnected weak Hopf C ∗ -algebra H - see [15, 11, 7, 22, 12].More recently, Popa, Shlyakhtenko and Vaes, in [21], among various interesting results, classifiedregular subalgebras B of the hyperfinite II -factor R with B ′ ∩ R = Z ( B ). However, they do notprovide any structure for non-irreducible regular inclusions of factors of type II . This short noteis a first naive attempt in this direction, in which we prove the following: Theorem 4.6
Let N ⊂ M be a finite index regular inclusion of II -factors with commutativerelative commutant N ′ ∩ M . Then, there exists a biconnected weak Kac algebra K and a minimalaction of K on M such that N ⊂ M is isomorphic to N ⊂ N ⋊ K . It must be mentioned here that Ceccherini-Silberstein (in [6]) claimed to have proved that everyfinite index regular subfactor is a crossed product subfactor with respect to an outer action of afinite dimensional Hopf C ∗ -algebra. However, his assertion is incorrect and there is an obviousoversight in his proof as is pointed out in Remark 4.1.Theorem 4.6 is achieved by first proving that any finite index regular inclusion of II -factorswith commutative first relative commutant has depth 2 and then an appropriate application ofNikshych-Vainerman’s characterization of depth 2 subfactors yields the desired structure. In orderto take care of the first part, we utilize the notion of unitary orthonormal basis by Popa to show(in Theorem 4.3) that any regular subfactor with simple or commutative relative commutant is The first named author was supported through a DST INSPIRE faculty grant (reference no.DST/INSPIRE/04/2019/002754). of depth at most 2. It fits well to mention here that, in fact, Popa had recently asked (in [20])whether every integer index irreducible inclusion of II -factors admits a unitary orthonormal basisor not. It seems to be a difficult question to answer in full generality. In fact, the question can beasked for non-irreducible inclusions as well, and we provide a partial answer in: Theorem 3.21
Let N ⊂ M be a finite index regular inclusion of factors of type II . If N ′ ∩ M is either commutative or simple, then M admits a unitary orthonormal basis over N . Then, the second part of Theorem 4.6 is taken care of by a suitable application of the notion oftwo-sided basis for inclusions of finite von Neumann algebras. In fact, somewhat related to Popa’squestion, and equally fundamental in nature, is the question related to the existence of a two-sidedPimsner-Popa basis for any extremal inclusion of II -factors, which was asked by Vaughan Jonesaround a decade back at various places. This question too has tasted too little success. In [2], wehad shown that every finite index regular inclusion of II -factors admits a two-sided Pimsner-Popabasis and we have suitably adopted the idea of its proof in proving Theorem 3.21. We move onemore step closer towards answering Jones’ question by proving the following: Theorem 3.13
Let N ⊂ M be a finite index inclusion of type II -factors of depth with simplerelative commutant N ′ ∩ M . Then, M admits a two-sided Pimsner-Popa basis over M .Furthermore, M also admits a two-sided basis over N . The flow of the article is in the reverse order in the sense that, after some preliminaries inSection 2, we first make an attempt to partially answer Jones’ question regarding existence oftwo-sided basis in the first half of Section 3 and then move towards Popa’s question regardingexistence of unitary orthonormal basis in the second half of the same section. Finally, in Section4, we establish that any regular subfactor with commutative first relative commutant is given bycrossed product by a weak Kac algebra.2.
Preliminaries
Since there are slightly varying (though equivalent) definitions available in literature, in orderto avoid any possible confusion, we quickly recall the definition that we shall be using here. Forfurther details, we refer the reader to [5, 14, 12, 13] and the references therein.
Definition 2.1. [14, 5](1) A weak bialgebra is a quintuple (
A, m, η, ∆ , ε ) so that ( A, m, η ) is an algebra, ( A, ∆ , ε ) isa coalgebra and the tuple satisfies the following compatibility conditions between algebraand coalgebra structures:(a) ∆ is an algebra homomorphism.(b) ε ( xyz ) = ε ( xy ) ε ( y z ) and ε ( xyz ) = ε ( xy ) ε ( y z ) for all x, y, z ∈ A .(c) ∆ (1) = (cid:0) ∆(1) ⊗ (cid:1)(cid:0) ⊗ ∆(1) (cid:1) = (cid:0) ⊗ ∆(1) (cid:1)(cid:0) ∆(1) ⊗ (cid:1) . (2) A weak Hopf algebra (or a quantum groupoid ) is a weak bialgebra ( A, m, η, ∆ , ε ) alongwith a k -linear map S : A → A , called an antipode , satisfying the following antipodeaxioms:(a) x S ( x ) = ε (1 x )1 ,(b) S ( x ) x = 1 ε ( x ) , (c) S ( x ) x S ( x ) = S ( x ) . (3) A weak Hopf algebra ( A, m, η, ∆ , ε ) is said to be a weak Hopf C ∗ -algebra if A is a finitedimensional C ∗ -algebra and the comultiplication map is ∗ -preserving, i.e., ∆( x ∗ ) = ∆( x ) ∗ . Definition 2.2. [5] A weak Kac algebra is a weak Hopf C ∗ -algebra ( A, m, η, ∆ , ε, S ) such that S = Id A and S is ∗ -preserving. Remark . (1) A weak Hopf algebra is a Hopf algebra if and only if the comultiplication isunit-preserving if and only if the counit is a homomorphism of algebras.In particular, every Kac algebra is a weak Kac algebra. ASES AND REGULAR SUBFACTORS 3 (2) The dual of a weak Kac algebra also admits a canonical weak Kac algebra.
Example 2.4.
Given a finite groupoid G , the associated groupoid algebra C [ G ] inherits a canonicalweak Kac algebra structure with respect to the comultiplication ∆, the counit ε and the antipode S satisfying ∆( g ) = g ⊗ g, ε ( g ) = 1 , S ( g ) = g − for g ∈ G . It is easily seen that C [ G ] (resp., C [ G ] ∗ ) is a cocommutative (resp., commutative) weak Kac algebra.And, conversely, it was proved by Yamanouchi that for every cocommutative weak Kac algebra H there exists a finite groupoid G such that H is isomorphic to C [ G ].Given any weak Kac algebra A , the target (resp., source) counital map ε t (resp., ε s ) on A , isgiven by ε t ( x ) = ε (1 (1) x )1 (2) (cid:0) resp., ε s ( x ) = 1 (1) ε ( x (2) ) (cid:1) for x ∈ A , where ∆(1) = 1 (1) ⊗ (2) inSweedler’s notation. These maps are idempotent, i.e., ε t ◦ ε t = ε t , ε s ◦ ε s = ε s , and their imagesare unital C ∗ -subalgebras (called the Cartan subalgebras ) of A : A t := { x ∈ A : ε t ( x ) = x } and A s := { x ∈ A : ε s ( x ) = x } .A is said to be connected if the inclusion A t ⊂ A is connected (see [8] for definition). And, A issaid to be biconnected if both A and its dual are connected. Remark . Given a finite groupoid G , the groupoid algebra C [ G ] is biconnected if and only if G is a group.2.1. Crossed product construction.
We now briefly recall the notion of the crossed product construction via an action of a weakHopf C ∗ -algebra, as in [14] (also see [13, 12]). Definition 2.6. (1) By a (left) action of a weak Hopf C ∗ -algebra A on a von Neumannalgebra M , we mean a linear map A ⊗ M ∋ a ⊗ x ( a ⊲ x ) ∈ M which defines a (left) module structure on M and satisfies the conditions(a) a ⊲ xy = ( a (1) ⊲ x )( a (2) ⊲ y ),(b) ( a ⊲ x ) ∗ = S ( a ) ∗ ⊲ x ∗ , and(c) a ⊲ ε t ( a ) ⊲ a ⊲ ε t ( a ) = 0for a ∈ A , x, y ∈ M .(2) Under such a (left) action, the crossed product algebra M ⋊ A is defined as follows:As a C -vector space it is the relative tensor product M ⊗ A t A , where A (resp., M )admits a canonical left (resp., right) A t -module structure so that x ( z ⊲ ⊗ a ∼ x ⊗ za, for all x ∈ M, a ∈ A, z ∈ A t . For each ( a, x ) ∈ A × M , [ x ⊗ a ] denotes the class of theelement x ⊗ a and a natural ∗ -algebra structure on M ⊗ A t A is given by:[ x ⊗ a ][ y ⊗ b ] = [ x ( a (1) ⊲ y ) ⊗ a (2) b ] , [ x ⊗ a ] ∗ = [( a ∗ (1) ⊲ x ∗ ) ⊗ a ∗ (2) ] , for all x, y ∈ M and a, b ∈ A .(3) The action is said to be minimal if A ′ ∩ ( M ⋊ A ) = A s . Remark . [14, 12, 13](1) M ⋊ A can be realized as a von Neumann algebra.(2) If M is a II -factor and A is a weak Hopf C ∗ -algebra acting minimally on M , then M ⋊ A is also a II -factor.Our interest in actions of weak Hopf C ∗ -algebras stems from the following beautiful charac-terization of depth 2 subfactors by Nikshych and Vainerman. Before stating them, it would beappropriate to recall the following definition. K C BAKSHI AND V P GUPTA
Definition 2.8.
Consider a finite index inclusion N ⊂ M of II -factors and suppose N ⊂ M ⊂ M ⊂ · · · ⊂ M k ⊂ · · · is its tower of Jones’ basic construction. Then, the inclusion N ⊂ M issaid to have finite depth if there exists a k such that N ′ ∩ M k − ⊂ N ′ ∩ M k − ⊂ N ′ ∩ M k is aninstance of basic construction. The least such k is defined as the depth of the inclusion.We urge the reader to see [8] for various other equivalent formulations of the notion of depth.For any finite index irreducible inclusion N ⊂ M of II -factors, i.e., N ′ ∩ M = C , it wasannounced by Ocneanu (in [15]) and proved later, separately, by Szyma´nski, David and Longo -see [22, 7, 11] - that if N ⊂ M is of depth 2, then there exists a Kac algebra K and a minimalaction of K on M such that M ∼ = M ⋊ K and M = M H . More generally, Nikshych andVainerman obtained the following characterization: Theorem 2.9. [12, 5]
A finite index inclusion N ⊂ M of II -factors is of depth if and only ifthere exists a biconnected weak Hopf C ∗ -algebra H and a minimal action of H on M such that M ∼ = M ⋊ H and M = M H . Pimsner-Popa Bases
Let
N ⊂ M be a unital inclusion of von Neumann algebras equipped with a faithful normalconditional expectation E from M onto N . Then, a finite set B := { λ , . . . , λ n } ⊂ M is calleda left (resp., right) Pimsner-Popa basis for M over N via E if every x ∈ M can be expressedas x = P ni =1 E ( xλ ∗ i ) λ i (resp., x = P nj =1 λ j E ( λ ∗ j x ). Further, such a basis { λ i } is said to be orthonormal if E ( λ i λ ∗ j ) = δ i,j for all i, j . And, a collection B is said to be a two-sided basis if it issimultaneously a left and a right Pimsner-Popa basis.In this article, when we do not use the adjectives left or right, by a basis we shall always meana right Pimsner-Popa basis (and not a two-sided basis).3.1. Two-sided basis.
About a decade back, Vaughan Jones asked the following question at various places . Question 3.1. (Vaughan Jones) Let N be a II -factor and N ⊂ M be an extremal subfactor offinite index. Then, does there always exist a two-sided Pimsner-Popa basis for M over N ? Example 3.2.
Given a finite group G and a subgroup H , by Hall’s Marriage Theorem, we canobtain a set of coset representatives which acts simultaneously as representatives of left and rightcosets of H in G . Therefore, if G acts outerly on a II -factor N , then N ⋊ G always possesses atwo-sided unitary orthonormal basis over N ⋊ H .This observation, therefore, allows us to think about the existence of a two-sided basis as asubfactor analogue of Hall’s Marriage Theorem.Recall that an inclusion Q ⊂ P of von Neumann algebras is said to be regular if its groupof normalizers N P ( Q ) := { u ∈ U ( P ) : u Q u ∗ = Q} generates P as von Neumann algebra, i.e., N P ( Q ) ′′ = P . Remark . To the best of our knowledge, till date, too little progress has been made in answeringQuestion 3.1.(1) If N ⊂ M is a regular irreducible subfactor of type II of finite index, then (as mentionedin the Introduction) it is a well-known fact that it is isomorphic to N ⊂ N ⋊ G , for someouter action of a finite group G on N . In particular, M has a two-sided basis over N . For instance, during the second talk by M. Izumi in the workshop organized in honour of V S Sunder’s 60thbirthday at IMSc, Chennai during March-April 2012. https://mathoverflow.net/questions/6647/do-subgroups-have-two-sided-bases. ASES AND REGULAR SUBFACTORS 5 (2) In [2], we could drop the irreducibility condition and showed, without depending uponany structure result, that every finite index regular subfactor N ⊂ M of type II admitsa two-sided basis. A little thought should convince the reader that the two-sided basis weconstructed in [2] is in fact orthonormal.A comment pertaining to an application of the notion of two-sided basis fits in well here: Remark . It is a known fact to the experts that any regular subfactor of type II has integerindex - see [8, Page 150]. However, there was no explicit proof easily accessible in literature untilCeccherini-Silberstein [6] suggested having one. Though, the argument provided in [6, Theorem4.5] seems incomplete as is indicated in Remark 3.19.To our satisfaction, we could do a little better (in [2]) by exhibiting that, for any finite indexregular subfactor N ⊂ M of type II , its index is given explicitly by(1) [ M : N ] = | G | dim( N ′ ∩ M ) , where G denotes the generalized Weyl group of the inclusion N ⊂ M , which is defined as thequotient group N M ( N ) U ( N ) U ( N ′ ∩ M ) .Depending upon the structure result of irreducible depth 2 subfactors by Szyma´nski and aresult by Kac which determines when a Kac algebra is a group algebra, Nikshych and Vainermanndeduced (in [12, Corollary 4.19]) that a depth 2 subfactor of type II with prime index p is nec-essarily a group subfactor with respect to an outer action of the cyclic group Z /p Z . Interestingly,it turns out that the formula in Equation (1) has the following analogous consequence. Proposition 3.5.
Let N ⊂ M be a finite index regular inclusion of II -factors. If [ M : N ] = p is prime, then N ⊂ M is irreducible.In particular, the cyclic group G := Z /p Z acts outerly on N and N ⊂ M is isomorphic to N ⊂ N ⋊ G .Proof. Suppose, on contrary, that N ⊂ M is not irreducible. Then, from Equation (1), it followsthat [ M : N ] = dim C ( N ′ ∩ M ) . Note that, if Λ denotes the inclusion matrix of the inclusion C ⊂ N ′ ∩ M , then k Λ k = dim C ( N ′ ∩ M ). In particular, k Λ k = [ M : N ], which then implies that C ⊂ N ′ ∩ M ⊂ N ′ ∩ M is an instanceof basic construction - see [8, Theorem 4.6.3 (vii)]. Thus, N ′ ∩ M ∼ = M n ( C ) for some n ≥
2; sothat [ M : N ] = n . This contradicts the hypothesis that [ M : N ] is a prime number. Hence, N ⊂ M must be irreducible.The asserted structure of N ⊂ M is then well-known. ✷ Further, employing appropriate two-sided bases for the inclusions N ⊂ N ∨ ( N ′ ∩ M ) and N ∨ ( N ′ ∩ M ) ⊂ M , the following useful observation was proved explicitly in the first two paragraphsof the proof of [2, Theorem 3.12]. We will be using it crucially in the proof of Theorem 4.6 andshall not repeat the details here. Proposition 3.6. [2]
Let N ⊂ M be a finite index regular inclusion of II -factors. Then, theWatatani index of the restriction of tr M to N ′ ∩ M is a scalar. Adding to the list, we shall provide, in the next section, an yet another application of the notionof two-sided basis for regular inclusions.
K C BAKSHI AND V P GUPTA
One more step towards Jones’ question.
Note that, any irreducible regular factorial inclusion of type II , being isomorphic to a crossedproduct subfactor by a group, must be of depth 2 (see [8] or Definition 2.8 for definition). Thus,it is natural to ask the following question: Question 3.7.
Let N ⊂ M be a depth subfactor of type II of finite index. Then, does M/N always have a two-sided basis?
We do not know the answer yet in this generality. However, we provide a partial answer inTheorem 3.13, for which we require some preparation.First, we need (a mild generalization of) a useful result of Popa [19, § D, C, B, A ) of von Neumannalgebras is said to be non-degenerate ifspan[ CB ] S . O . T . = A = span[ BC ] S . O . T . . Lemma 3.8. (Popa) Let M be a finite von Neumann algebra with a faithful normal tracial stateand ( N , K , L , M ) be a non-degenerate commuting square of von Neumann subalgebras of M .Then, any right basis for K / N is also a right basis for M / L . Proof.
Suppose { λ i : i ∈ I } is a right basis for K / N . Then, P i λ i e KN λ ∗ i = 1, where e KN denotesthe Jones projection corresponding to the inclusion N ⊂ K . Let Ω denote the canonical cyclicvector for L ( M ). Then, for any x ∈ L and y ∈ K , we have X i λ i e ML λ ∗ i ( yx Ω) = X i λ i E ML ( λ ∗ i yx )Ω= X i λ i E ML ( λ ∗ i y ) x Ω= X i λ i E KN ( λ ∗ i y ) x Ω [by commuting square condition]= yx Ω . As the commuting square is non-degenerate, we have span LK SOT = M = span KL SOT . Inparticular, [span LK ]Ω k·k = L ( M ) = [span KL ]Ω k·k . Therefore, we conclude that P i λ i e ML λ ∗ i =1 and the proof is complete. ✷ Some specific conditions guarantee non-degeneracy of some commuting squares.
Lemma 3.9.
Let M be a finite von Neumann algebra with a faithful normal tracial state and ( N , P , Q , M ) be a commuting square consisting of von Neumann subalgebras of M . If, either (1) Q ⊂ M is an inclusion of II -factors with finite index and N ⊂ P is a connected inclusionof finite dimensional C ∗ -algebras with [ M : Q ] = k Λ k , where Λ denotes the inclusionmatrix of N ⊂ P ; or (2) both
N ⊂ P and
Q ⊂ M are connected inclusions of finite dimensional C ∗ -algebras with k Λ k = k Γ k , where Λ and Γ denote the respective inclusion matrices,then ( N , P , Q , M ) is non-degenerate.Proof. A proof can be obtained on similar lines as that of [3, Lemma 18] based on the character-ization of a basis illustrated in [1, Theorem 2.2]. ✷ The next useful observation is a straight forward adaptation of [2, Proposition 3.3], which usesthe notion of path algebras associated to inclusions of finite dimensional C ∗ -algebras by Sunderand Ocneanu. We skip the details. ASES AND REGULAR SUBFACTORS 7
Lemma 3.10. [2]
Let A be a finite dimensional C ∗ -algebra and tr be a faithful tracial state on A . Then, A has a two-sided orthonormal basis over C with respect to tr . The following interesting observation is a folklore.
Proposition 3.11.
Let N ⊂ M be a finite index depth subfactor of type II and M ⊃ N ⊃ N − ⊃ N − ⊃ · · · ⊃ N − k ⊃ · · · be a tunnel construction for N ⊂ M . Then, N − k ⊂ N − k +1 hasdepth for all k ≥ .Moreover, M k − ⊂ M k is also of depth for all k ≥ .Proof. For the tunnel part, it suffices to show that N − ⊂ N − has depth 2.Let Γ and Ω denote the inclusion matrices for the inclusions ( N ′− ∩ N − ⊂ N ′− ∩ N ) and( N ′− ∩ N ⊂ N ′− ∩ M ), respectively. Then, by [8, Theorem 4.6.3], it will follow that N − ⊂ N − has depth 2 if we can show that k Γ k < [ N − : N − ] = k Ω k .Consider the Jones’ basic construction tower N − ⊂ N − ⊂ N ⊂ M ⊂ M ⊂ M ⊂ M · · · ⊂ M k ⊂ · · · . By [4, Theorem 2.13], there exists a ∗ -isomorphism (the shift operator) ϕ : N ′− ∩ M → N ′ ∩ M such that ϕ ( N ′− ∩ N ) = N ′ ∩ M and ϕ ( N ′− ∩ N − ) = N ′ ∩ M . Thus, the truncated towers[ N ′− ∩ N − ⊂ N ′− ∩ N ⊂ N ′− ∩ M ] and [ N ′ ∩ M ⊂ N ′ ∩ M ⊂ N ′ ∩ M ] are isomorphic. Inparticular, if Λ i denotes the inclusion matrix for the inclusion ( N ′ ∩ M i ⊂ N ′ ∩ M i +1 ), then Λ = Γand Λ = Ω; so, by [8, Theorem 4.6.3], we obtain k Ω k = k Λ k = [ M : N ] = [ N − : N − ] and k Γ k = k Λ k < [ M : N ] = [ N − : N − ].For the basic construction part, it suffices to show that M ⊂ M has depth 2. The shiftoperator ψ : N ′ ∩ M → M ′ ∩ M does the job as above. ✷ Corollary 3.12.
Let N ⊂ M be a finite index depth subfactor of type II . Then, M k ⊂ M k +1 also has depth for all k ≥ .In particular, N − k ⊂ N − k +1 has depth for all k ≥ , for any tunnel construction M ⊃ N ⊃ N − ⊃ N − ⊃ · · · ⊃ N − k ⊃ · · · of N ⊂ M .Proof. It suffices to show that M ⊂ M is of depth 2.Fix a 2-step downward basic construction N − ⊂ N − ⊂ N of N ⊂ M . Then, by the precedingproposition, N − ⊂ N − is also of depth 2. So, by [12], there exists a biconnected weak Hopf C ∗ -algebra H with a minimal action on N such that ( N H ⊂ N ) ∼ = ( N − ⊂ N ). Thus, N − ⊂ N is also of depth 2, by [5] (also see [13, Section 8.1]). Thus, by Proposition 3.11 again, M ⊂ M isalso of depth 2. ✷ We are now all set for the theorem of this subsection.
Theorem 3.13.
Let N ⊂ M be a finite index inclusion of type II -factors of depth with simplerelative commutant N ′ ∩ M . Then, M admits a two-sided orthonormal basis over M .Furthermore, M also admits a two-sided orthonormal basis over N .Proof. Although some of the arguments below are well-known (see [19]), we provide sufficientdetails for the sake of self-containment and convenience of the reader.
Step I:
Any (left/right) basis for M ′ ∩ M over M ′ ∩ M is also a (left/right) basis for M over M .Note that, by Lemma 3.8, it suffices to show that the quadruple M ⊂ M ∪ ∪ M ′ ∩ M ⊂ M ′ ∩ M is a non-degenerate commuting square. Towards this direction, first, recall that the quadruple G := M ⊂ M ∪ ∪ N ′ ∩ M ⊂ N ′ ∩ M is a commuting square - see, for instance, [8, Proposition 4.2.7], wherein the bottom inclusion isconnected.Let Λ denote the inclusion matrix for the inclusion N ′ ∩ M ⊂ N ′ ∩ M . Since N ⊂ M is ofdepth 2, as was recalled in Proposition 3.11, we have [ M : M ] = [ M : N ] = k Λ k . Therefore,by Lemma 3.9, the quadruple G is a non-degenerate commuting square. Thus, its extension (asdefined in [19, § . . G := M ⊂ M ∪ ∪ N ′ ∩ M ⊂ N ′ ∩ M ;and, by the proposition in § . . G is a non-degenerate commuting square as well. Onthe other hand, note that if Γ denotes the inclusion matrix for M ′ ∩ M ⊂ M ′ ∩ M , then since M ⊂ M is also of depth 2 (see Corollary 3.12), we have k Γ k = [ M : M ] = [ M : N ] = k Λ T k ;so, the commuting square G := N ′ ∩ M T ⊂ N ′ ∩ M ∪ ∪ M ′ ∩ M ⊂ M ′ ∩ M is also non-degenerate, by Lemma 3.9. In particular, concatenating G and G , we deduce from[19, § . .
5] that the quadruple M ⊂ M ∪ ∪ M ′ ∩ M ⊂ M ′ ∩ M is a non-degenerate commuting square. Step II: M ′ ∩ M has a two-sided orthonormal basis over M ′ ∩ M .We assert that (cid:0) M ′ ∩ M ⊂ M ′ ∩ M (cid:1) is isomorphic to (cid:16) M ′ ∩ M ⊂ ( M ′ ∩ M ) ⊗ Q (cid:17) for someunital ∗ -subalgebra Q of ( M ′ ∩ M ) ′ ∩ ( M ′ ∩ M ). Once this is established, we can then readilydeduce from Lemma 3.10 that M ′ ∩ M has a two-sided orthonormal basis over M ′ ∩ M .Since N ′ ∩ M ∋ x Jx ∗ J ∈ M ′ ∩ M is an anti-isomorphism and N ′ ∩ M is simple, so is M ′ ∩ M . Again, since M ⊂ M is also of depth 2, the tower M ′ ∩ M ⊂ M ′ ∩ M ⊂ M ′ ∩ M is an instance of basic construction. So, M ′ ∩ M is also simple. Thus, it follows from [8, Lemma2.2.2] that ( M ′ ∩ M ) ′ ∩ ( M ′ ∩ M ) is simple and that M ′ ∩ M ∼ = ( M ′ ∩ M ) ⊗ (cid:2) ( M ′ ∩ M ) ′ ∩ ( M ′ ∩ M ) (cid:3) . Suppose that M ′ ∩ M ∼ = M n ( C ) and that M ′ ∩ M ∼ = M n ( C ) ⊗ M k ( C ) . Denote the intermediatesubalgebra corresponding to M ′ ∩ M by P . It is well-known that P is of the form M n ( C ) ⊗ Q ,where Q is some unital ∗ -subalgebra of M k ( C ). We provide the details for the convenience of thereader. By [8, Proposition 4.2.7] again, the quadruple G := P ⊂ M n ( C ) ⊗ M k ( C ) ∪ ∪ (cid:0) M n ( C ) ⊗ (cid:1) ′ ∩ P ⊂ ⊗ M k ( C ) . ASES AND REGULAR SUBFACTORS 9 is also a commuting square. Note that, there exists a unital ∗ -subalgebra Q of M k ( C ) such that (cid:0) M n ( C ) ⊗ (cid:1) ′ ∩ P = 1 ⊗ Q . Clearly, M n ( C ) ⊗ Q ⊆ P . To see the reverse inclusion, consider x = P i a i ⊗ b i ∈ P ⊂ M n ( C ) ⊗ M k ( C ) . Then, we have x = P ( a i ⊗ E P (1 ⊗ b i ). Since G is acommuting square, we immediately see that E P (1 ⊗ b i ) ∈ ⊗ Q and hence x ∈ M n ( C ) ⊗ Q. Inconclusion, we have P = M n ( C ) ⊗ Q , as was asserted.Thus, from Steps I and II, we deduce that M has a two-sided orthonormal basis over M .Finally, fix any 2-step downward basic construction N − ⊂ N − ⊂ N ⊂ M for N ⊂ M .Then, by Proposition 3.11, N − ⊂ N − also has depth 2. Further, as seen in Proposition 3.11, N ′− ∩ N − ∼ = N ′ ∩ M is simple. Hence, we readily deduce from the preceding discussion that M must admit a two-sided orthonormal basis over N . ✷ As an immediate consequence we deduce the following.
Corollary 3.14.
Every finite index irreducible inclusion of II factors of depth 2 admits a two-sided orthonormal basis.Remark . Note that a subfactor as in Theorem 3.13 need not be regular. For instance, anyKac algebra K , which is not a group algebra, acts outerly on the hyperfinite factor R and yieldsa non-regular irreducible depth 2 subfactor.3.2. Unitary orthonormal basis.
We now move towards unitary orthonormal bases and touch upon another fundamental questionasked recently by Sorin Popa in [20, § Question 3.16. (Sorin Popa) Does there always exist an orthonormal basis consisting of n manyunitaries for an integer index ( = n ) irreducible inclusion of II -factors? Example 3.17.
Let H be a subgroup of a finite group G . If G acts outerly on a II -factor N ,then N ⋊ G has a unitary orthonormal basis over N ⋊ H . Example 3.18.
Every irreducible regular subfactor, being isomorphic to a group subfactor, ad-mits a unitary orthonormal basis.In view of the preceding example, it is natural to ask whether we can drop the irreducibilitycondition or not. The following remarks fall in place here:
Remark . (1) The question of existence of unitary orthonormal basis for a general finiteindex regular subfactor which is not necessarily irreducible (thus modifying Question 3.16)was discussed by Ceccherini-Silberstein [6]. In fact, he asserted (in [6, Theorem 4.5]) thatif N ⊂ M is a regular subfactor with finite index, then M/N has a unitary orthonormalbasis. However, his proof depends on a technique of Popa ([18, Theorem 2.3]) which holdsfor Cartan subalgebras. Since Popa’s proof depends crucially on maximal abelian-ness ofthe subalgebra, it is not clear whether it holds, more generally, for regular subalgebras ornot. So, the proof of [6, Theorem 4.5] seems to be incomplete; although, the statementmay still be true, which we rephrase in Conjecture 3.20.(2) Furthermore, Ceccherini-Silberstein (in [6, Theorem 4.7]) had also asserted that if
M/N has a unitary orthonormal basis then the subfactor M ⊂ M is of the form M ⊂ M ⋊ H and hence has depth 2 (see for instance [12]), which then implies that N ⊂ M is alsoof depth 2 - see Proposition 3.11. However, this is well known to be incorrect as everygroup-subgroup subfactor ( R × H ) ⊂ ( R ⋊ G ) always has a unitary orthonormal basis (seeExample 3.17) whereas it is not necessarily of depth 2.(3) Though not directly related to the present discussion, the characterization of index 3subfactors provided in Corollary 3.19 of [6] is known to be incorrect. Conjecture 3.20.
Let N ⊂ M be a finite index regular inclusion of factors of type II . Then, M/N has a unitary orthonormal basis.
As a partial progress in the resolution of this conjecture, we prove the following:
Theorem 3.21.
Let N ⊂ M be a finite index regular inclusion of factors of type II . If N ′ ∩ M is either commutative or simple, then M admits a unitary orthonormal basis over N . We will need the following couple of results to achieve this.
Lemma 3.22.
Let A := C ⊕ C ⊕ · · · ⊕ C ( n -copies). (1) The Markov trace tr : A → C for the unital inclusion C ⊂ A is given by tr(( z , . . . , z n )) = 1 n X i z i . (2) There exists a unitary orthonormal basis for A over C with respect to the Markov trace ifand only if there exists a unitary matrix U = [ u ij ] ∈ M n ( C ) such that | u ij | = √ n for all ≤ i, j ≤ n .Proof. (1) This follows easily from [8, Proposition 2.7.2].(2): ( ⇒ ) Let { λ i = ( z (1) i , z (2) i , . . . , z ( n ) i ) : 1 ≤ i ≤ n } be a unitary orthonormal basis for A over C . Consider the matrix U = [ u ij ] ∈ M n whose entries are given by u ij = √ n z ( j ) i , i.e., whose i -thcolumn constitutes of the complex entries in λ i . Then, | u ij | = √ n for all 1 ≤ i, j ≤ n and U ∗ U = tr( λ ∗ λ ) tr( λ ∗ λ ) · · · tr( λ ∗ λ n )tr( λ ∗ λ ) tr( λ ∗ λ ) · · · tr( λ ∗ λ n )... ... . . . ...tr( λ ∗ n λ ) tr( λ ∗ n λ ) · · · tr( λ ∗ n λ n ) = I n . ( ⇐ ) Let U = [ u ij ] ∈ U ( n ) be such that | u ij | = √ n for all 1 ≤ i, j ≤ n . Consider λ i := √ n ( u i , u i , . . . , u ni ) ∈ A, i = 1 , , . . . , n. Since | u ij | = √ n for all 1 ≤ i, j ≤ n , it follows that λ ∗ i λ i = (1 , , . . . ,
1) for all 1 ≤ i ≤ n , i.e., each λ i is a unitary in A . Further, note that I n = U ∗ U = [ u ij ] ∗ [ u ij ] = tr( λ ∗ λ ) tr( λ ∗ λ ) · · · tr( λ ∗ λ n )tr( λ ∗ λ ) tr( λ ∗ λ ) · · · tr( λ ∗ λ n )... ... . . . ...tr( λ ∗ n λ ) tr( λ ∗ n λ ) · · · tr( λ ∗ n λ n ) . Hence, tr( λ ∗ i λ j ) = δ i,j for all 1 ≤ i, j ≤ n , which implies that { λ , . . . , λ n } forms a unitaryorthonormal basis for A over C with respect to above tracial state. ✷ Recall that a unitary error basis for a matrix algebra M n ( C ) is a Hamel basis that is orthogonalwith respect to the inner product induced by the canonical trace of M n ( C ). Little is known abouttheir structure. There are two popular methods of construction of unitary error bases. One isalgebraic in nature (due to Knill) and the other combinatorial (due to Werner). Proposition 3.23.
Let A be a finite dimensional C ∗ -algebra which is either simple or commuta-tive. Then, A/ C has a unitary orthonormal basis with respect to the Markov trace for the unitalinclusion C ⊂ A . ASES AND REGULAR SUBFACTORS 11
Proof.
Suppose first that A = M n ( C ) for some n ≥
2. The existence of a unitary orthonormalbasis follows from the known construction of a unitary error basis. We include the details for thereader’s convenience.We first recall such a basis for n = 2 (because of its importance and popularity in quantuminformation theory). The Pauli spin matrices (unitary error bases in dimension 2) are defined asfollows: σ x = (cid:20) (cid:21) , σ y = (cid:20) − ii (cid:21) , σ z = (cid:20) − (cid:21) . It is an amazing fact that the set { I , σ x , σ y , σ z } forms an orthonormal basis consisting of unitariesfor M ( C ) . For higher dimensions, consider the following two important matrices due to Sylvester andWeyl: U = · · · ω · · ·
00 0 ω · · · · · · ω n − and V = · · · · · · · · · · · · · · · , where ω := e − πi/n (a primitive root of unity). Then, it is known that the set { U i V j : 1 ≤ i, j ≤ n } forms a unitary error basis (in fact, a nice error basis) for M n ( C ). These matrices also appeared ina work of Popa ([17]) (see also [6]). This proves that A/ C has unitary orthonormal basis whenever A is simple.Next, let A be isomorphic to C ⊕ C ⊕ · · · ⊕ C ( n -copies). Now, for a primitive root of unity ω as above, consider the well-known unitary DFT matrix U := 1 √ n · · · ω ω ω · · · ω n − ω ω ω · · · ω n − ω ω ω · · · ω n − ... ... ... ... . . . ...1 ω n − ω n − ω n − · · · ω ( − n − . Clearly, each entry of U has modulus 1 / √ n . So, by proposition 3.23, there exists a unitaryorthonormal basis for A/ C with respect to the Markov trace. ✷ The preceding observation will prove to be very crucial in the next section. Thus, it seemsit is worthwhile to investigate in detail the existence of unitary basis of the finite dimensionalinclusions B ⊂ A , which, in turn, may prove to be useful in answering the question of Popa forhyperfinite irreducible subfactors. Lemma 3.24.
Let N ⊂ M be a finite index regular subfactor of type II . Then, tr M | N ′∩ M is theMarkov trace for the inclusion C ⊂ N ′ ∩ M .Proof. As pointed out in Proposition 3.6, it can be extracted from the proof of [2, Theorem 3.12]that the Watatani index ([23]) of Ind(tr M ) is a scalar. Thus, it follows from [23, Corollary 2.4.3] and [10, Proposition 3.2.3] that tr M | N ′∩ M is indeed the Markov trace for the inclusion C ⊂ N ′ ∩ M . ✷ Proof of Theorem 3.21:
Consider the intermediate von Neumann subalgebra R := N ∨ ( N ′ ∩ M ).Then, as in the proof of [2, Lemma 3.4], we see that ( C , N ′ ∩ M, N, R ) is a non-degenerate commut-ing square. By Lemma 3.24, tr N ′ ∩ M is the Markov trace for C ⊂ N ′ ∩ M ; so, by Proposition 3.23,there exists a unitary orthonormal basis, say, { u i : i ∈ I } for N ′ ∩ M over C . Then, by Lemma 3.8, { u i : i ∈ I } is a unitary orthonormal basis for R /N as well.On the other hand, since N ⊂ M is regular, from [2, Proposition 3.7], we know that M/ R alsohas a unitary orthonormal basis, say, { v j : j ∈ J } . We assert that { v j u i : i ∈ I, j ∈ J } is a unitaryorthonormal basis for M/N . It is easy to see that { v j u i : i ∈ I and j ∈ J } is a Pimsner-Popabasis for M/N . Also, E MN ( u ∗ i v ∗ j v k u l ) = E R N ◦ E M R ( u ∗ i v ∗ j v k u l ) = E R N (cid:0) u ∗ i E M R ( v ∗ j v k ) u l (cid:1) = δ j,k δ i,l . Thus, { v j u i : i ∈ I and j ∈ J } is a unitary orthonormal basis for M/N . ✷ Two-sided basis versus unitary orthonormal basis.
Some preliminary observations suggest that the above questions of Jones (Question 3.1) andPopa (Question 3.16) may be intimately interrelated in the case of integer index (extremal) sub-factors. Below, we illustrate some such connections.The following fact is implicit in [6].
Lemma 3.25. [6]
Let N ⊂ M be a subfactor of finite index. If M/N has a unitary orthonormalbasis, then M /M admits a two-sided unitary orthonormal basis.In particular, N ⊂ M is extremal.Proof. This proof is extracted verbatim from [6]. Suppose { λ i : 1 ≤ i ≤ n } is a unitary orthonor-mal basis for M/N . Thus, P i λ i e λ ∗ i = 1 . Now, put v k = n − X i =0 ω ki λ i e λ ∗ i , ≤ k ≤ n − , where ω is an n -th root of unity. In [6, Proposition 3.24], it has been shown that { v k } is a unitaryorthonormal basis for M /M . Clearly this is two-sided.Next, recall that N ⊂ M is extremal if and only if M ⊂ M is extremal - see, for instance, [19].Since M /M has a two-sided basis, it is easily seen (see [2]) that it is extremal. ✷ Proposition 3.26.
Let N ⊂ M be a finite index hyperfinite subfactor of type II with finitedepth. If M/N has a unitary orthonormal basis, then it also has a two-sided unitary orthonormalbasis.Proof.
By Lemma 3.25, it follows that M /M , and hence, M /M has a two-sided unitary or-thonormal basis. It is known that the standard invariants of the extremal subfactors N ⊂ M and M ⊂ M are isomorphic. Thus, by Popa’s classification result (see [19]), N ⊂ M and M ⊂ M ,both being hyperfinite, are isomorphic. Hence, N ⊂ M has a two-sided basis. This completes theproof. ✷ It will be good to know an answer of the following natural question.
Question 3.27. If N ⊂ M is a finite depth integer index subfactor of type II , then is it true that M/N has a unitary orthonormal basis if and only if
M/N has a two-sided Pimsner-Popa basis?Remark . Note that even if it can be shown that a finite index subfactor with a two-sided basisalso admits a unitary orthonormal basis, then in view of [2], it will follow that Conjecture 3.20holds true.
ASES AND REGULAR SUBFACTORS 13 Regular subfactors and weak Kac algebras
As recalled in the introduction, a finite index irreducible regular inclusion of II -factors isalways of the form N ⊂ N ⋊ G with respect to an outer action of a finite group G . It is thennatural to ask what happens if we drop the irreducibility condition. Remark . Employing Szyma´nski’s characterization of depth 2 (irreducible) subfactors, Ceccherini-Silberstein (in [6, Theorem 4.6]) asserted that every finite index regular subfactor N ⊂ M of type II is of the form N ⊂ N ⋊ H with respect to an outer action of a finite dimensional Hopf ∗ -algebra H on N . However, it had the following obvious oversight:If his assertion is true, then it will automatically force N ⊂ M to be irreducible, whereas hehas claimed to have characterized regular subfactors sans irreducibility.In fact, Ceccherini-Silberstein’s oversight stems from an incomplete proof of an assertion madein [6, Theorem 4.5], as explained below:In the proof of [6, Theorem 4.6], in view of [6, Theorem 4.5], a unitary orthonormal basis { λ i } is chosen for M/N and then it is deduced that N ′ ∩ M = Alg { λ i e λ ∗ i : i ∈ I } . Note that, the family { λ i e λ ∗ i } consists of mutually orthogonal projections with P i λ i e λ ∗ i = 1.Thus, if N ⊂ M is regular with finite index, then according to [6, Theorem 4.6], N ′ ∩ M is alwayscommutative. However, this is known to be untrue. For instance, taking an irreducible regularsubfactor K ⊂ L and putting N = C ⊗ K and M = M n ( C ) ⊗ L , it can be seen that N ⊂ M is regular (see Lemma 4.2) with integer index and N ′ ∩ M ∼ = M n ( C ); so that N ′ ∩ M is notcommutative.So, the question of characterizing (finite index) regular subfactors of type II is still unresolved. Lemma 4.2.
Let N ⊂ M be a regular inclusion of von Neumann algebras. Then, C ⊗ N ⊂ M n ⊗ M is also regular.Proof. Note that { u ⊗ v : u ∈ U ( n ) , v ∈ N M ( N ) } ⊆ N M n ⊗ M ( C ⊗ N ). Thus,[ ∗ -alg U ( n )] ⊗ [ ∗ -alg N M ( N )] ⊆ ∗ -alg N M n ⊗ M ( C ⊗ N ) . It is enough to show that { u ⊗ z : u ∈ U ( n ) , z ∈ M } ⊂ N M n ⊗ M ( C ⊗ N ) ′′ . Let z ∈ M and u ∈ U ( n ). Then, there exists a net { x i } in ∗ -alg N M ( N ) such that x i WOT −→ z . Thus, { u ⊗ x i } is a net in [ ∗ -AC U ( n )] ⊗ [ ∗ -alg N M ( N )], which is a ∗ -subalgebra of ∗ -alg N M n ⊗ M ( C ⊗ N ).Also, u ⊗ x i WOT −→ u ⊗ z . Hence, u ⊗ z ∈ ( ∗ -alg N M n ⊗ M ( C ⊗ N )) ′′ . ✷ Theorem 4.3.
Let N ⊂ M be a finite index regular inclusion of type II -factors such that N ′ ∩ M is either simple or commutative. Then, N ⊂ M is of depth at most .Proof. Note that, by Theorem 3.21, M admits a unitary orthonormal basis over N . More precisely,taking R := N ∨ ( N ′ ∩ M ), we saw that R /N admits a unitary orthonormal basis, say, { u i : i ∈ I } ⊂ U ( N ′ ∩ M ) ⊂ N M ( N ); M/ R admits a unitary orthonormal basis { v j : j ∈ J } ⊂ N M ( N );and then, taking w i,j = v j u i we saw that { w i,j : i ∈ I, j ∈ J } is a unitary orthonormal basis for M/N and { w i,j : ( i, j ) ∈ I × J } ⊂ N M ( N ) . In particular, we have P i,j w i,j e w ∗ i,j = 1 . Now, note that for any unitary u ∈ N we have w ∗ i,j uw i,j = v i,j for some unitary v i,j ∈ N .Thus, u ( w i,j e w ∗ i,j ) u ∗ = w i,j v i,j e v ∗ i,j w ∗ i,j = w i,j e w ∗ i,j . This implies that w i,j e w ∗ i,j ∈ N ′ ∩ M for all ( i, j ) ∈ I × J . Further, we readily see that( w i,j e w ∗ i,j ) e ( w i,j e w ∗ i,j ) = τ w i,j e w ∗ i,j ∀ ( i, j ) ∈ I × J. Hence, w i,j e w ∗ i,j ∈ ( N ′ ∩ M ) e ( N ′ ∩ M ) for every ( i, j ) ∈ I × J . Now, since 1 = P i,j w i,j e w ∗ i,j and that ( N ′ ∩ M ) e ( N ′ ∩ M ) is an ideal in N ′ ∩ M , it follows that that ( N ′ ∩ M ) e ( N ′ ∩ M ) = N ′ ∩ M . Thus, in view of [8, Theorem 4.6.3], N ⊂ M has depth at most 2. ✷ Corollary 4.4. If N ⊂ M is a finite index regular inclusion of type II factors such that N ′ ∩ M is commutative, then it has depth . Few remarks are in order which tell that the converse of the above result need not be true.
Remark . (1) A depth 2 subfactor having commutative first relative commutant need notbe regular. For example, consider a finite dimensional Hopf C ∗ -algebra (that is, a Kacalgebra) K , which is not a group algebra, acting minimally on a type II factor N . Then, N ⊂ N ⋊ K is a depth 2 subfactor. Being irreducible, this subfactor is not regular.(2) Notice that a depth 2 regular subfactor N ⊂ M may have a non-commutative first relativecommutant. As an example, one may look at the subfactor illustrated in Remark 4.1. Theorem 4.6.
Let N ⊂ M be a finite index regular inclusion of II -factors with commutativerelative commutant N ′ ∩ M . Then, there exists a biconnected weak Kac algebra K and a minimalaction of K on M such that N ⊂ M is isomorphic to N ⊂ N ⋊ K .Proof. By Theorem 4.3 and Corollary 4.4, we observe that N ⊂ M has depth 2. Choose a 2-stepdownward basic construction N − ⊂ N − ⊂ N ⊂ M . Then, N − ⊂ N − is also of depth two -see Corollary 3.12. Let K := N ′− ∩ M . From [12], it will follow that K admits a biconnectedweak Kac algebra structure (with an appropriate action on N ) provided the Watatani index oftr N | N ′− ∩ N is a scalar.By Proposition 3.6, we know that Ind(tr M | N ′∩ M ) is a scalar. Let J : L ( N ) → L ( N ) denotethe modular conjugation operator. Since N − ⊂ N ⊂ M is an instance of basic construction,the map B ( L ( N )) ∋ x JxJ ∈ B ( L ( N )) is an anti-isomorphism that maps N ′− onto M andtr M = tr N ′− ◦ [ J ( · ) J ]; so that tr M | N ′∩ M = (cid:0) tr N ′− ◦ [ J ( · ) J ] (cid:1) | N ′∩ M . Also, J ( N ′ ∩ M ) = N ′− ∩ N ;so that, N ′− ∩ N is commutative and ( C ⊂ N ′ ∩ M ) ∼ = ( C ⊂ N ′− ∩ N ). Further, since N − ⊂ N is extremal (being of depth 2), we havetr N | N ′− ∩ N = tr N ′− | N ′− ∩ N . Thus, tr M | N ′∩ M and tr N | N ′− ∩ N have same trace vectors and henceInd(tr N | N ′− ∩ N ) = Ind(tr M | N ′∩ M ) , which is a scalar. Thus, by [12, Corollary 4.7 and Theorem 4.17], K admits a weak Kac algebrastructure, which is also biconnected, by [12, Remark 5.8 (ii)]. Further, by [12, Propositions 6.1 and6.3, and Remark 6.4 (i)], K acts minimally on N such that N ⊂ M is isomorphic to N ⊂ N ⋊ K .This completes the proof. ✷ We end our discussion with a few well-known classes of reducible regular subfactors.
Example 4.7. (1) If a finite group G acts innerly on a II factor N in such a way that M = N ⋊ G is a II factor, then the inclusion N ⊂ M is regular and N ′ ∩ M is non-trivial. (2) Suppose N is a type II factor. Then, the depth 1 subfactor C ⊗ N ⊂ M n ( C ) ⊗ N is anexample of a regular subfactor with simple first relative commutant ( ∼ = M n ( C )). https://mathoverflow.net/questions/364547/action-of-a-finite-group-on-a-finite-factor ASES AND REGULAR SUBFACTORS 15
Acknowledgements.
The authors would like to thank Yongle Jiang, Vijay Kodiyalam, SebastienPalcoux and Leonid Vainerman for many fruitful exchanges. The authors would also like to marktheir note of appreciation to T. Ceccherini-Silberstein for acknowledging our concerns related tothe oversights in some of his proofs (from [6]).
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