1. Introduction and Priliminaries

In the next chapter we are going to use locally trivial subfactors to construct a set of middle subfactors. The important of locally trivial subfactos was indicated by S.Popa at ^[9]. The simplest locally trivial subfactors are those having only two orthogonal projections in their relative commutant. It is well known that these subfactors are isomorphic to Jones subfactors.. Also it is easy easy to show that these kind of subfactors do not posses middle subfactors. For the locally trivial subfactors that their relative commutant have dimension larger than two we believe that the above result is still valid ie’there are no middle subfactors. Suppose we are Given a pair L ⊂ M of subfactors that are limiting algebras of a tower of commuting squares.

Then using the results in ^[1] we can show that if the inclusion graphs of the corresponding finite C* algebras are A graphs, then the subfactors L ⊂ M do not have middle subfactors. One of the problems in the index theory, is to find the set of all the values for the indices of hyperfinite irreducible subfactors. Using the above constructions, we are going to show that the above set contains the interval (37.0037,∞).

2. Main Results

For a given pair of subfactors N ⊂ M, With [M:N] = λ^-1 <∞. Let e be a projection in M that induces the expectation N onto Q = (e)′∩N. Let P₁,P₂ = 1-P₁ be a partition of unity in Q. Using standard argu-ments as in ^[5], there exists an isomorphism Ф, taking NP1 onto NP2. Let L(P₁) be the set of all the elements of the form L(P₁) = (x +Ф(x); x∈NP1 ). Then it is well known that L(P₁), is a locally trivial subfactor of N. Also by (Lemma2.2.1) ^[5], [N : LP ] = 1=tr(P1) + 1=tr(P2). Where tr is a unique normalized trace on M. Suppose P1, does not communicate with e₀ = e. Set y = P1eP2 ≠ 0. Note that the relative commutant of L(P₁), inside M is spanned by the projection P₁. Since y, does not communicate with P₁, we have that the algebra, H(P₁) =< L(P₁); Y > is a middle subfac-tor which is strictly larger than L(P₁) If under certain conditions H(P₁) becomes strictly smaller than M, then H(P₁) becomes a proper middle sub-factor. In this case by the above arguments it is easy to see that the inclusions H(P₁)⊃ L(P₁) and H(P₁)⊂M are irreducible inclusion of subfactors. Let us denote r₁ = [ H(P₁) : L(P₁)] and r₂ = [M : H(P₁)]. Let IR, be the set of all irreducible subfactors of finite index. Let us denote by IIR, the set of indices of all subfactors in IR. Then by (Proposition2.1.15) ^[5] r₁r₂ is in IIR. It is easy to check that the set of of the elements f of the form f =∑j∈_J u_jgz_jegw_j, with g = E_Q(P₂), z_j∈Q, u_j = P₁x_jP₁, w_j = P₁y_jP₁, x_j∈N y_j∈N where J is a set of indices is dense in (H(P₁))P₁. Let e₁ be a projection in N, such that E_Q(e₁) = λ. Then e₁ induces the expectation of Q onto the subfactor Q₁ = (e)′∩Q. Next for a number r, r∈IIR, construct an irreducible subfac-tor Q₂, Q₂⊆Q₁, with [Q₁ : Q₂] = r. We can define the projection e₃, Using (corollary 1.8) ^[13], there exits a projection e₃ in Q₁, such that e₃ induces the expectation of Q₂ onto The subfactor Q₃, with Q₃ = (e₃)′∩Q₂. This process will induce the following tonnel, M⊃N ⊃ Q ⊃ Q₁ ⊃ Q₂ ⊃ Q₃. Let us set P₁ = qe₁e₃ with q a projection in Q₃. Now we can check that the following set of elements, f of the form, with z = , with as in the above and J a set of indices will be a dense subset of (H(P₁))P₁. In particular assuming now that H(P₁) = M implies that the above set of elements are dense in MP₁. Furthermore as we mentioned in the above for any number r∈IIR; e₃∈Q₁ can be chosen such that tr(e₃) = r. For example suppose tr(e₁) = tr(e₀) = .5. Then it is easy to see that there exists a unitary V ∈(e₃)′∩N = L, with L a type Von Neumann algebra, such that V e₁V* = 1 − e₁. Then we can express the f from the above as f = λ(1 −λ)²e₁e₃ze₁e₃ + (e₁e₃e₀V e₁e₃)(e₁e₃V*ze₁e₃). For a given real number S, let [S], be the largest integer which is smaller or equal to [S]. Let us set S_r = S − [S]. Let us assume now that H(P₁) = M, We will get the following results.

Lemma 1 Keeping the same notations as in the above let Then there exist unitary operators U, U₂, U₃ in L and projection p≤e₁, such that f can be expresses as in the following, With σ1, will be equal to 1 if odd integer and equal to zero otherwise. Simi-larly σ2 is equal to 1 if is not an integer and equal to zero otherwise.

Proof First suppose tr(e₀)−¹ = λ−¹ = 2n, for some positive integer n, then let be a partition of unity by orthogonal projections in L, such that for any odd integer k < 2n, there exists a unitary , with . Using our definition of f₁ = e₁, this implies, Let Then U is a unitary in L. Set The we have, Next for each 1 ≤ i ≤ n, there exists a unitary such that Hence Next since for Thus we get For let us define Then it is easy to check This implies that y = 0. Another useful relation that we will need later is the following equality, that can be checked easily. Let us define the operator h, with with, 1≤ i, k ≤ n and . Then using the above relations we can see that h is a projection, tr(h) = tr(g), and is orthogonal to h. Hence we get, tr(h) = tr() = .5. This will implies that f can be expressed as, Suppose λ−¹ = 2n+1. Then we have the following partition of unity, . Where the above projections have equal traces. Furthermore ther exists a unitary such that f can be expressed as, then we have the following partition of unity by the following projections, Where σ₁ and σ₂, can only take values 0 or 1, depending if the corresponding projections f_2n+1 and f₀ are or are not equal to zero. Furthermore for k ≠ 0, all non zero projections fk's, have equal traces and tr(f₀) < λ. Hence generally f can be expressed as, Where U and U₂ are as in the above, U₃ a unitary in and p, is a sub projection of e₁ and is in . At this point note that we can extend U₂ and U₃ to be unitaries in L. Finally we can express f, as

Now let us set the following notations. and .

Lemma 2 Keeping the same notations as in the above, and assuming that σ₁ and σ₂ both different from zero and without loss of generality, we have the following equalities.

Proof Note that since commues with all the above operators, drop-ping from all the operations does not makes any different from the final outcome. So in the following operations we ignore the existence of . In particular we can identify with . Since the proof of the above equalities are very similar, we only show some of the equalities. We have Hence we get, Next Thus . But we have, This implies, . But is orthogonal which implies that . Fur-thermore Hence we get, Next This implies, But using the above relations, which implies Now let us calculate . Now using the above relations we can show that hence

Assuming that Me₁e₃ is acting standardly on H = [Me₁e₃ ] and L(P₁) = M, the above lemma implies that the operator identity can be spanned by at most four orthogonal projections in G each of trace less or equal to λ. Hence by Remarks(1.4) ^[13], we get the following Corollary,

Corollary 3 Keeping the same notations as before, for [M:N] > 4, H(P₁) is a proper middle subfactor.

As before let IIR represents the set of all indices of irreducible hyperfinite subfactors.

Suppose H(P₁) is a proper middle subfactor, ie' L(P₁) ⊂ H(P₁) ⊂ M, where the inclusions are restrict. Let us denote r₁ = [H(P₁):L(P₁)] and r₂ = [M:L(P₁)]. Also using the fundamental property of the index of subfactors, r₁r₂ = r = [M:L(P₁)] = [M : N][N : L(P₁)]. But [M : N] =λ−¹ and [N:L(P₁)] = (tr(P1)(tr(P2))−¹ Hence r = r₁r₂ = λ−¹(tr(P1)tr(P2))−¹ And by the results of ^[5], r∈IIR. Now notice that by the result of S.Popa in "Subfactors and classification in von Neumann algebras" Corollary(4.4) of the above article indicate the gap in IIR between the values 4 and . In fact corresponds to the square of the norm of Coxeter graph E10 and there exists a subfactor of such an index. By its definition P1 = qe₁e₃ and we had λ = tr(e₁) λ1 = tr(e₃). Hence tr(P1) = tr(q)λλ1. Let us set c = tr(q), and then c can take any value in the interval [0,1]. Note also that (λ1)−¹ can take any value in IIR larger than .5. Let us denote, , then = tr(P1). Note that can take any value in IIR larger than or equal to . Hence if we set then we have Since q can be taken to be any projection in Q₃, tr(P1), can get any value in the interval . This implies the following theorem.

Theorem 4 Keeping the same notations as in the above, suppose and Let N ⊃ M, [M:N]=λ−¹ be a pair of irreducible subfactors and lets define a projection p = qe₁e₃, for some projection q in Q₃. Then H(P₁) =<L(P₁), p₁ep₂> is a proper irreducible subfactor of M. In particular letting q to vary in Q₃, we will get that IIR includes the interval .

At this end note that by the above arguments there exists a function Ф, acting on the above interval

References

[1]	D.Bisch. A note on intermediate subfactors. Pac.J.Math.163(1994), 201-216.
	In article		View Article
[2]	E.Christensen, Subalgebras of finite algebra, Math.Ann.243(1979), 17-29.
	In article		View Article
[3]	F. Goodman, P. S. Delaharpe and V. Jones, Coxter graph and towers of algebras. Math. Sci. Res. Hist.Publ., Springer-Verlag, V. 14, 1989.
	In article		View Article
[4]	U.Haagerup, Principal graphs of a subfactors, Proceeding of the Tan-guchi Simposium on operator algebras, World scientific, 1999.
	In article
[5]	V.Jones, Index for subfactors, Invent Math 72 (1983), 1-25.
	In article		View Article
[6]	V.Jones, V.S.Sunder, Introduction to Subfactors. Combridge Univer-sity Press. 1997.
	In article		View Article
[7]	M.Khoshkam and B.Mashood, Rigidity in commuting squres, New Ze-land Journal Of Math. V. 32(2003), 37-56.
	In article
[8]	M.Khoshkam and B.Mashood, On the relative commutants of subfac-tors, Proceeding of A.M.S 126(1998)2725-2732.
	In article
[9]	B.Mashood, On a ladder of commuting squares and a pair of factors generated by them, preprint(1990)Dept.Math.University of Victoria. Canada.BC.V8W 2Y2.
	In article
[10]	B.Mashood, On the tower of relative commutants part-I, Math Japonica 37, No.5(1992)901-906.
	In article
[11]	B.Mashood, and K.Taylor, On the continuity of the index of subfactor of a finite factor, J.Func.anal.76 no.11(1988)56-66.
	In article
[12]	A.Ocneanu, Quantized groups, string algreba, operator algebra and application, London Math Society. Lecture notes ser. 136 119-172, Cambridge University Press, 1988.
	In article
[13]	M.Pimsner, S.Popa, Entropy and Index for subfactors., Ann.Sci.Ecole Norm.Sup. 19 (1986).
	In article
[14]	M.Pimsner, S.Popa, Finite dimensional approximation for pair of algebras and obstruction for index, J.Func.Anal.98(1991), 270-291.
	In article		View Article
[15]	M.Pimsner, S.Popa, Sur les sous facteurs d'indice fini d'facteur de type II ayant propriet T,C.R.Acad.Sci, 303(1986), 359-361.
	In article
[16]	S.Popa, Classifcation of amenable subfactors of type II., Acta Math., 172(1994), 163-255.
	In article		View Article
[17]	S.Popa, Sur la classification d' facteur hyperfini, Comtes rendus (1990), 95-100.
	In article
[18]	N.Sato, Two subfactors arising from a non-degenerate commuting squares, Pac.J.Math.Vol.180, No 2, 1997, 369-376.
	In article		View Article
[19]	H.Wenzl, Hecke algebras of type An and subfactors, Invent, Math.92(1998) 349-383.
	In article