E extracta mathematicae Vol. 33, Núm. 1, 33 – 50 (2018) Representing Matrices, M-ideals and Tensor Products of L1-predual Spaces S. Dutta, D. Khurana, A. Sensarma Department of Mathematics and Statistics Indian Institute of Technology Kanpur, India sudipta@iitk.ac.in , divyakh@iitk.ac.in , aryasen@iitk.ac.in Presented by Jesús M.F. Castillo Received November 28, 2016 Abstract: Motivated by Bratteli diagrams of Approximately Finite Dimensional (AF) C∗- algebras, we consider diagrammatic representations of separable L1-predual spaces and show that, in analogy to a result in AF C∗-algebra theory, in such spaces, every M-ideal corre- sponds to directed sub diagram. This allows one, given a representing matrix of a L1-predual space, to recover a representing matrix of an M-ideal in X. We give examples where the converse is true in the sense that given an M-ideal in a L1-predual space X, there exists a diagrammatic representation of X such that the M-ideal is given by a directed sub diagram and an algorithmic way to recover a representing matrix of M-ideals in these spaces. Given representing matrices of two L1-predual spaces we construct a representing matrix of their injective tensor product. Key words: representing matrix, generalized diagram, directed sub diagram, M-ideals, ten- sor products. AMS Subject Class. (2010): 46B25, 46B20. 1. Introduction In 1971 Lazar and Lindenstrauss (see [3]) introduced notion of representing matrices for separable L1-predual spaces. The idea to construct representing matrix of a L1-predual space depends on following result in [3, Theorem 3.2], which essentially says that any separable L1-predual space is built up by putting together increasing union of `n∞, n = 1, 2, . . . ∞’s. Theorem 1.1. Let X be a separable infinite dimensional Banach space such that X∗ is isometric to L1(µ) for some positive measure µ. Let F be a finite dimensional space whose unit ball is a polytope. Then there exists a sequence {En}∞n=1 of finite dimensional subspaces of X such that E1 ⊃ F, En+1 ⊃ En and En = `mn∞ for every n and X = ∪∞n=1En. We now describe the notion of representing matrices. By Theorem 1.1 any separable L1-predual space is ∪∞n=1`n∞ and different such spaces are con- 33 34 s. dutta, d. khurana, a. sensarma structed depending on how one embeds `n∞ → `n+1∞ . Let {ei}ni=1 denote the standard unit vector basis of ` n ∞. By admissible basis of `n∞ we mean a basis of the form {θieπ(i)}ni=1 where θi = ±1 and π is a permutation of {1, . . . ,n}. It is easy to see that if {ui} is an admissible basis of `n∞ then for any m > n a linear operator T : `n∞ → `m∞ is an isometry if and only if there exists an admissible basis {vi}mi=1 of ` m ∞ such that Tui = vi + Σ m j=n+1a i jvj with Σni=1|a i j| ≤ 1 for every n + 1 ≤ j ≤ m. Now for any separable L1-predual space with the representation X = ∪n∈NEn where En ⊆ En+1 and each En is isometric to `n∞ we may choose admissible basis {ein}ni=1 of En such that, after relabelling, Tne i n = e i n+1 + a i ne n+1 n+1 with Σni=1|a i n| ≤ 1. A triangular matrix A = ( ain )1≤i≤n n≥1 associated with X in this manner is called a representing matrix of X. The construction of the representing matrix is best understood in the context of C(K), K is totally disconnected. For use in the later part of this paper, we illustrate this with an example by constructing of representing matrix for such a space. Let K be a totally disconnected compact metric space. Then there exists a sequence { ∏ n} ∞ n=1 of partitions of K into disjoint closed sets so that for every n, { ∏ n} has n elements, { ∏ n+1} is a refinement of { ∏ n} and %n = max A∈ ∏ n d(A) → 0 where d(A) denotes diameter of A. Let En be the linear span of the characteristic functions of the sets in ∏ n. Then it follows trivially that each En is isometric to ` n ∞, En ⊆ En+1 and C(K) = ⋃∞ n=1 En. Let us denote ∏ n = {K 1 n,K 2 n, . . . ,K n n} for all n ∈ N. We may write 1K11 = 1K12 + 1K22 . Now ∏ 3 = {K 1 3,K 2 3,K 3 3}, 1K12 = 1K13 + 1K33 and 1K22 = 1K23 . We continue this procedure to get a representing matrix of C(K) which is 0, 1-valued [3, Theorem 5.1]. A L1-predual space X has a rich collection of structural subspaces of X, namely M-ideals. M-ideals in a L1-predual space are themselves L1- preduals and in some sense deterministic for the isometric properties of the representing matrices, M-ideals, tensor products 35 space, meaning, any isometric property of a L1-predual space can be read off from some isometric properties of its M-ideals. On the other hand, rep- resenting matrices ‘encode’ every possible information of the structure of a L1-predual space. A separable predual X of L1 may be thought of as an isometric version (commutative, where *-isomorphism is replaced by linear isometry) of Approx- imately Finite Dimensional (AF) real C∗-algebras. Two sided norm closed ideals in an AF C∗-algebra are completely determined by hereditary directed sub diagrams of its Bratteli diagram (see [1]). The analogous notion of closed two sided ideals in a C∗-algebra in Banach space category is M-ideals. Here we present a representing diagram of a separable L1-predual space, the dia- gram itself arise out of representing matrix of such a space. We show that every directed sub diagram of a representing diagram represents an M-ideal in the corresponding space. Since by definition of representing diagram, it is always hereditary, this is an exact analogy to the corresponding result for AF C∗-algebras. We believe the converse is also true and we establish it in some cases. We now briefly describe the plan of this paper. In section 2 we present our main idea of diagrammatic representation of a separable L1-predual space X and directed sub diagram. We show any directed sub diagram corresponds to an M-ideal in X and the residual diagram corresponds to X/M. If M is an M-summand then we show the diagram for X splits into two directed sub diagram. This recovers the result in [7]. We believe that the converse, that any M-ideal in a L1-predual space X is represented by a directed sub diagram of some diagram is true. However there is a problem here. There are M-ideals which have empty sub diagram. Nevertheless we present converse for C(K) spaces (with extra assumption for general K). We also observe that for A(K) -the space of affine continuous function on K, where K is a separable Poulsen simplex (note that A(K) is isometric to the Gurariy space in this case) given any M-ideal, there exists a diagrammatic representation of corresponding space such that the given M-ideal is represented by a directed sub diagram. In Section 3 we describe a ‘Fill in the Gaps’ algorithm for construction of representing matrix from information that X = ∪∞n=1` mn ∞ . This in one hand provides way to construct representing matrix for an M-ideal given by di- rected sub diagram and on the other, allows one to write down representing matrix of X⊗̌Y , X, Y L1-preduals, knowing the representing matrix of X and Y . We also show that for C[0, 1], given an M-ideal, there exists a diagram- 36 s. dutta, d. khurana, a. sensarma matic representation of C[0, 1] such that the given M-ideal is represented by a directed sub diagram. Through out this work we only consider separable L1-predual spaces. Re- call that a subspace M of Banach space is called an M-ideal if there exists a projection (called L-projection) P : X∗ → X∗ such that ker P = M⊥ and X∗ = Range P ⊕1 ker P , where ⊕1 denote the `1-sum. In this case Range P is isometric to M∗. M is said to be an M-summand in X if X = M ⊕∞ N. Trivially any M-summand is an M-ideal. Acknowledgements. A Sensarma wishes to acknowledge the support received from CSIR, India, Senior Research Fellowship (Award letter no. 09/092(0872)/2013-EMR-I) 2. Directed diagrams and M-ideals For a L1-predual space X with a representing matrix A = ( ain )1≤i≤n n≥1 we will consider the following diagrammatic representation of X. A diagram D of a L1-predual space X = ∪∞n=1`n∞, and representing matrix A = ( ain )1≤i≤n n≥1 consists of nodes and weighted arrows. The nodes at the n-th level of the diagram are {ein : 1 ≤ i ≤ n} where span{ein : 1 ≤ i ≤ n} is isometric to `n∞, n ∈ N. For a node ein, there can be at most two arrows from ein one reaching to e i n+1 and another to e n+1 n+1. Any arrow from e i n to e i n+1 has weight 1 and there is an arrow from ein to e n+1 n+1, then it has a weight a i n. For example if all ain 6= 0 then we have the following diagram: e11 e12 e22 e13 e23 e33 e14 e24 e34 e44 a1 1 a 1 2 a2 2 a 13 a 2 3 a3 3 representing matrices, M-ideals, tensor products 37 In case some ain’s are zero we do not put arrows from e i n to e n+1 n+1. For example diagram for a space with a11,a 2 2,a 3 3 = 0, will look like the following: e11 e12 e22 e13 e23 e33 e14 e24 e34 e44 a 1 2 a 2 3 a 13 In the following we describe the diagram for the space c with representing matrix A such that a1n = 1, n ≥ 1 and a j n = 0,j 6= 1 (see [3]): e11 e12 e22 e13 e23 e33 e14 e24 e34 e44 1 1 1 Note that every representing matrix of a L1-predual space corresponds to a unique diagram D and vice-versa. For a given diagram D we will denote the corresponding space by XD. Now we will introduce the notion of generalized diagram for a L1-predual space X, where X = ∪Xn and Xn is isometric to `mn∞ for an increasing 38 s. dutta, d. khurana, a. sensarma sequence (mn). Let {e1mn, . . . ,e mn mn } be the admissible basis of Xn. Any isom- etry Tmn : ` mn ∞ → ` mn+1 ∞ is uniquely specified by scalars (a i mn+j ), 1 ≤ j ≤ mn+1 −mn, 1 ≤ i ≤ mn such that Tmne i mn = eimn+1 + a i mn+1 emn+1mn+1 + · · · + a i mn+1 emn+1mn+1 , i = 1, 2, . . . ,mn. For a node eimn , there will be one arrow from e i mn to eimn+1 . If a i mn+j 6= 0, then there will be a weighted arrow from eimn to e mn+j mn+1 , 1 ≤ j ≤ mn+1 −mn with weight aimn+j. Definition 2.1. A sub diagram S of D will be called a directed sub dia- gram if whenever ein ∈S for some n,i ∈ N, i ≤ n then (a) ein+1 ∈S, (b) if ain 6= 0, e n+1 n+1 ∈S. A sub diagram S ⊆ D is directed if whenever ein ∈ S for some n,i ∈ N, i ≤ n and there is an arrow from ein to e j n+1 then e j n+1 ∈S. We define directed sub diagram of a generalized diagram similarly. If we take S ⊆D, and, S is directed then the original isometric embedding of Xn into Xn+1 is preserved (see introduction). Hence XS will be an isometric subspace of XD. Moreover there exists a norm one projection P : X ∗ → X∗ with ker P = X⊥S . To see this observe that XS = ∪ ∞ n=1` mn ∞ , hence XS is itself a L1-predual space which is an isometric subspace of XD. We prove that for any directed sub diagram S the space XS is an M-ideal in XD and the diagram D\S represents the space XD/XS. Theorem 2.2. Let X be a L1-predual space with a given diagram D. Then for any directed sub diagram S of D the subspace XS is an M-ideal in X. Proof. Let X = ∪Xn, where Xn ⊂ Xn+1, Xn is isometric to `n∞ for each n. Let P : X∗ → X∗ be a norm one projection with ker P = XS⊥, that is, X∗ = XS ⊥⊕F where F = Range P . We need to prove that X∗ = XS⊥⊕1 F. Let Mn = span{ein : ein ∈S, 1 ≤ i ≤ n} and Fn = span{ein : ein 6∈ S, 1 ≤ i ≤ n} for each n ∈ N. Then Xn = Mn⊕∞Fn and X∗n = M⊥n ⊕1F⊥n . For any x∗ ∈ X∗ we can write x∗ = x∗1 + x ∗ 2 where x ∗ 1 ∈ XS ⊥ and x∗2 ∈ F. Then x ∗|Xn = x∗1|Xn + x ∗ 2|Xn and ||x∗|Xn|| = ||x∗1|Xn|| + ||x ∗ 2|Xn||. For given � > 0, we can choose some m ∈ N representing matrices, M-ideals, tensor products 39 such that ||x∗|Xn|| ≥ ||x∗||− �, ||x∗1|Xn|| ≥ ||x ∗ 1||− � and ||x ∗ 2|Xn|| ≥ ||x ∗ 2||− � for all n ≥ m. Now ||x∗1|| + ||x ∗ 2|| ≥ ||x ∗|| ≥ ||x∗|Xn|| = ||x ∗ 1|Xn|| + ||x ∗ 2|Xn|| ≥ ||x ∗ 1|| + ||x ∗ 2||− 2�. Thus it follows that ||x∗|| = ||x∗1|| + ||x ∗ 2|| for all x ∗ ∈ X∗. From this we can conclude that X∗ = XS ⊥ ⊕1 F. Remark 2.3. Let X be a L1-predual space with a given generalized dia- gram D. Same proof as in Theorem 2.2 shows that directed sub diagram S of D represents the subspace XS which is an M-ideal in X. Next Theorem is analogous to [1, Theorem III.4.4]) in the case of L1- predual spaces. Theorem 2.4. Let X be a L1-predual space with a given diagram D and S a directed sub diagram of D. Then the diagram D\S represents the space X/XS. Proof. Let X = ∪Xn where Xn is isometric to `n∞. As before, let Mn = span{ein : ein ∈ S, 1 ≤ i ≤ n} and Fn = span{ein : ein 6∈ S, 1 ≤ i ≤ n} for each n ∈ N. Then XS = ∪Mn, Mn = `m∞ for some m ≤ n, is the M-ideal corresponding to the directed diagram S and Xn = Mn ⊕∞ Fn. Consider the norm one projection Pn : Xn → Fn where Pn ( n∑ i=1 aie i n ) = ∑ ein /∈S aie i n. Let in : Fn → Fn+1 be the isometry determined by arrows of the diagram D\S, that is, for ein ∈D\S, in(e i n) = e i n+1 + a i ne n+1 n+1 if e i n+1, e n+1 n+1 ∈D\S, in(e i n) = e i n+1 if e i n+1 ∈D\S, e n+1 n+1 6∈ D\S, in(e i n) = a i ne n+1 n+1 if e n+1 n+1 ∈D\S, e i n+1 6∈ D\S, in(e i n) = 0 if e i n+1, e n+1 n+1 6∈ D\S. It is straightforward to verify that Pn+1|Xn = in ◦Pn. We now define P : ∪Xn →∪Fn by Px = Pnx if x ∈ Xn. It follows that P is well defined and extends as a quotient map from X to the space determined by ∪Fn which is the space determined by the diagram D\S. This completes the proof. 40 s. dutta, d. khurana, a. sensarma We now investigate the converse of Theorem 2.2. Explicitly stated the problem is the following. Problem 2.5. Let X be a L1-predual space and M an M-ideal in X. Then there exists a diagram D representing X and a directed sub diagram S of D such that M = XS. We believe the answer to Problem 2.5 is affirmative. We will present evidences towards this for M-summands in general and M-ideals in some class of L1-predual spaces. The following proposition shows that any M-summand in a L1-predual space is represented by a directed sub diagram. Proposition 2.6. Let X be a L1-predual space and M be an M-summand in X. Then there exists a diagram D representing X such that M corresponds to some directed sub diagram S of D. Proof. Let N be the complement of M in X, that is, X = M⊕∞N. Then by [7, Proposition 2.4] it follows that X has a representing matrix of the form A =   0 a12 0 a 1 4 0 a 1 6 . . . 0 a23 0 a 2 5 0 . . . 0 a34 0 a 3 6 . . . 0 a45 0 . . . 0 a56 . . . 0 . . . . . . . . . . . . . . . . . . . . . . . .   where BA = (b i n) with b i n = a 2i−1 2n , CA = (c i n) with c i n = a 2i 2n, n ∈ N, 1 ≤ i ≤ n, are the matrices for M and N respectively. Let S1 and S2 be the diagrams corresponding to matrices BA and CA respectively. Now it follows that S1 and S2 are directed sub diagrams of the diagram of X corresponding to the representing matrix A. Remark 2.7. Directed sub diagrams S1 and S2 considered in Proposi- tion 2.6 are disjoint in the sense that no arrows of S1 enters into S2 and vice-versa. We now consider M-ideals in C(K)-spaces. We need to recall few notation and a result from [7]. representing matrices, M-ideals, tensor products 41 Let X be a L1-predual space with X = ∪∞n=1`n∞ and {e i n : 1 ≤ i ≤ n} are admissible bases of `n∞, n ∈ N. Define φj ∈ X∗, j ∈ N, by φj(e i n) = { 0 if i 6= j, 1 if i = j; i = 1, . . . ,n; j ≤ n; n ∈ N. By ext BX∗ we will denote the extreme point of BX∗ . Lemma 2.8. [7, Lemma 1.2] Let X and {φj} be as above. Then (a) φj ∈ ext BX∗ for all j ∈ N, and (b) {±φi : i ∈ N} = ext BX∗ , where closure is taken in weak∗-topology of BX∗ . Remark 2.9. For each i, ker φi represents the space XSi for some directed sub diagram Si of a given diagram D of X where the line passing through eii is a part of the diagram D\Si. The idea of the proof for the following result is to use the flexibility pro- vided by Lemma 2.8 for the choice of φi in a totally disconnected compact metric space K. Recall that for any C(K) space where K is a compact metric space, an M-ideal is given by JD = {f ∈ C(K) : f|D = 0}, where D is some closed subset of K. Proposition 2.10. Let K be a totally disconnected compact metric space and D a closed subset of K. Then there exists a diagram D representing C(K) and a directed sub diagram D ⊆S such that JD = XS. Proof. Since K be a totally disconnected we can get a sequence { ∏ n} ∞ n=1,∏ n = {K 1 n,K 2 n, . . . ,K n n} of partitions of K into disjoint closed sets, ∏ n+1 is a refinement of ∏ n and %n = maxA∈ ∏ n diam (A) → 0 (see introduction). Let D0 = {dn : n ∈ N} be a countable dense set in D. Choose φ1 = δd1 . For n ≥ 2 by renaming the elements in ∏ n we assume that d1 ∈ K 1 n. For n = 2 if (a) D0 ∩K22 6= ∅, we find the least n0 such that dn0 ∈ D0 ∩K 2 2 and choose φ2 = δdn0 . We will assume for all n ≥ 3, dn0 ∈ K 2 n, again by possibly renaming the members of ∏ n, (b) otherwise choose and fix any k ∈ K22 and take φ2 = δk. We will assume for all n ≥ 3, k ∈ K2n. 42 s. dutta, d. khurana, a. sensarma We will follow the same procedure for n ≥ 3. We need to ensure that each dn will be chosen. Let N be the least number among all k’s such that dk ∈ Kin for some i,n. Let m 6= N and dm ∈ Kin as well. Since diam (Kin) → 0 we can choose some suitable large M ∈ N such that dm ∈ KMM and m is the least among all k’s such that dk ∈ K M M . So following the algorithm above we define φM = δdm . Let D be the diagram representing C(K) given by the partition { ∏ n} after renaming the elements of { ∏ n} as considered above. Since D0 is dense in D, we have JD = ⋂ d∈D ker δd = ⋂ d∈D0 ker δd. Thus JD = XS, where S is the intersection of directed diagrams corresponding to kernel of φi = δdi , di ∈ D0. Next result shows affirmative answer to Problem 2.5 for general C(K) space with additional assumption on an M-ideal. By int D we mean interior of a set D. Proposition 2.11. Let K be any compact metric space and D a closed subset of K such that D = int D. Then the M-ideal JD corresponds to the space XS for some directed sub diagram S of given diagram D of C(K), provided, S is not an empty diagram. Proof. Let φj = δkj , kj ∈ K. Since {φj} are weak*-dense in extreme points of the dual unit ball of C(K) and D = int D, we have a sub collection φji ⊆ int D such that φji = δkji and kji is dense in D. It follows JD = ⋂ ker φji and hence JD is represented by the directed sub diagram S of D which is generated by intersection of directed sub diagram representing ker φji , where φji = δkji . Remarks 2.12. (1) If we assume K to be a ‘nice’ compact metric space, then given D a closed subset in K, we can construct a diagrammatic representation of C(K) such that JD corresponds to a directed sub di- agram. We will do it in next section as we need algorithm to construct representing matrix of a L1-predual space X when it is given in the form X = ∪n≥1`mn∞ . (2) Let K be (the) separable Poulsen simplex. Then the space A(K) - the space of real valued affine continuous functions on K is the separable representing matrices, M-ideals, tensor products 43 Gurariy space. It was proved in ([8]) that any infinite dimensional M- ideal in separable Gurariy space is isometric to itself. Thus any rep- resenting diagram of the Gurariy space represents M-ideals in it and Problem 2.5 has affirmative solution for the Gurariy space. We note that an empty diagram is always a directed sub diagram of any given diagram D. It may be the case that an M-ideal in a L1-predual space corresponds to an empty diagram. We give an easy example towards this. Example 2.13. Consider the matrix A such that a1n = 1 for all n and ain = 0 for all i > 1. It is proved in [3] that A represents c. Consider the M-ideal J = {(xn) ∈ c : xn = 0, n ≥ 2}. Then J = ∩ker φn, φn = δn, n ≥ 2. In second figure on page 5, except the line segment starting from the node enn and the line segment that starts from the node e 1 1 and ends at e 1 n−1, all the diagram represents the space ker φn. It is straightforward to verify that ∩n≥2 ker φn is empty. Another difficulty in solving Problem 2.5 affirmatively in general is empty diagram may represent a space which is not an M-ideal. We give an example of this in a typical non G-space. Note that an empty diagram is always directed. Example 2.14. Let X = { f ∈ C[1,ω0] : f(ω0) = f(1)+f(2) 2 } . Then X is a L1 -predual space which is not a G-space (see [5]). We will consider the following admissible basis for X (see [2]): e11 = ( 1, 1, 1, 1, . . . ) , e12 = ( 1, 0, 1 2 , 1 2 , 1 2 , . . . ) , e22 = ( 0, 1, 1 2 , 1 2 , 1 2 , . . . ) , e13 = ( 1, 0, 0, 1 2 , 1 2 , 1 2 , . . . ) , e23 = ( 0, 1, 0, 1 2 , 1 2 , 1 2 , . . . ) , e33 = ( 0, 0, 0, 1, 0, . . . ) , . . . . For n ∈ [1,ω0], we denote by Jn the M-ideal {f ∈ C[1,ω0] : f(n) = 0}. Each Jn is of codimension 1 in C[1,ω0]. We consider the M-ideal in C[1,ω0], J[3,ω0] = {f ∈ C[1,ω0] : f(n) = 0, n ≥ 3}. Now consider the subspace J[3,ω0]∩X = ∩n≥3Jn∩X of X. As in Example 2.13 it is easy to check that the intersection of corresponding directed sub diagrams of Jn ∩X for n ≥ 3 is empty diagram. However, J[3,ω0] ∩ X is not an M-ideal in X. To see this we observe that J[3,ω0] ∩ X is the range of norm one projection P : X → X given by P(f) = (f(1),−f(1), 0, 0, . . . ). Thus if J[3,ω0] ∩ X is an M-ideal then it is an M-summand as well. So for any f ∈ X, ‖f‖ = max{‖Pf‖,‖(I − P)f‖}. 44 s. dutta, d. khurana, a. sensarma However if we consider the element f ∈ X where f(1) = 1, f(2) = 0 and f(n) = 1/2 for all n ≥ 3, i.e, f = (1, 0, 1/2, 1/2, 1/2, . . . ) then ( 1, 0, 1 2 , 1 2 , 1 2 , . . . ) = ( 1 2 ,−1 2 , 0, 0, 0, . . . ) + ( 1 2 , 1 2 , 1 2 , 1 2 , 1 2 , . . . ) and the norm of both side will not match. Thus J[3,ω0] ∩ X is not an M-summand in X. 3. Fill in the Gaps In this section we provide an algorithm to construct representing matrix of a L1-predual space X where X is given by X = ∪Xn and Xn is isometric to `mn∞ for an increasing sequence (mn). This construction is implicit in the description of representing matrix given in [3]. However we fix an algorithm (there may be several as seen below) and use it for finding representing matrix of X⊗̌Y - the injective tensor product of two separable L1-predual spaces, knowing the representing matrices of X and Y . First we need to provide following justification to our construction. FACT: Let X be a L1-predual space such that X = ∪Xn, where Xn ⊆ Xn+1 and Xn is isometric to ` mn ∞ for some increasing sequence (mn). If Z is a L1-predual space with Z = ∪Zn, where Zn ⊆ Zn+1, Zn is isometric to `n∞, Zmn = Xn and the isometry Tn : Xn → Xn+1 is same as composition of isometries of Zmn to Zmn+1, Zmn+1 to Zmn+2, . . . ,Zmn+1−1 to Zmn+1 given by the representing matrix of Z, then Z is isometric to X. We now describe the proposed algorithm. Let X = ∪Xn where for each n, Xn = `mn∞ with admissible basis {eimn} mn i=1. Any isometry Tmn from ` mn ∞ to ` mn+1 ∞ in terms of admissible basis is given by Tmne i mn = eimn+1 + a i mn+1 emn+1mn+1 + . . . + a i mn+1 emn+1mn+1 , i = 1, 2, . . . ,mn . (1) Hence given X as above and isometric embeddings `mn∞ → ` mn+1 ∞ we know exactly (mn+1 −mn)mn numbers of( aij ) , i = 1, . . . ,mn , j = mn + 1, . . . ,mn+1 −mn . Let us assume C = ( cin )1≤i≤n n≥1 is a representing matrix for X. We will write representing matrices, M-ideals, tensor products 45 {eimn} in terms of {e i mn+1 } according to isometries given by C: eimn = e i mn+1 + cimne mn+1 mn+1 = eimn+2 + c i mn+1 emn+2mn+2 + c i mn (emn+1mn+2 + c mn+1 mn+1 emn+2mn+2) = eimn+2 + c i mn emn+1mn+2 + (c i mn+1 + cimnc mn+1 mn+1 )emn+2mn+2 = . . . . This way we will have (mn+1−mn) 2 (mn+1 + mn−1) numbers of (c j n) unknowns. We will put cmn+imn+j = 0 , i = 1, . . . ,mn+1 −mn , 1 ≤ j ≤ i . It is a straight forward verification that this way we will have (mn+1−mn) 2 (mn+1 −mn) of cin’s zero. Thus remaining (mn+1 −mn)mn of cin’s equal the number of known variables ain’s and can be expressed in terms of linear equations. We emphasize that the above way of choosing (cin) is not unique and different ways will give us different representing matrices. Note that here we can not recover first m1 − 1 columns of the representing matrix by the above algorithm so it can be chosen arbitrarily (see [6, Theorem 4.7]). Remark 3.1. Let the admissible basis of Xn is {eimn : 1 ≤ i ≤ mn}. If we follow the above algorithm of ‘Fill in the Gaps’ from Xn to Xn+1 where Xn is isometric to `mn∞ and Xn+1 is isometric to ` mn+1 ∞ then the basis elements emn+imn+1, e mn+i mn+2 , . . . , emn+imn+1−1 are same as e mn+i mn+1 for all i ≥ 1. We illustrate this procedure by considering two special cases. First one is simple trial case with mn = 2n and our second example provides us with representing matrix of C[0, 1] with entries 0 and 1 2 . Example 3.2. Let C = ( cin )1≤i≤n n≥1 be a representing matrix of X and Xn = span { e12n, . . . ,e 2n 2n } and Tn : Xn → Xn+1 is an isometric embedding with ei2n = e i 2(n+1) + a i 2n+1e 2n+1 2(n+1) + ai2(n+1)e 2(n+1) 2(n+1) , 1 ≤ i ≤ 2n, n ≥ 1 . If we write the expression for ei2n according to the matrix C then we get ei2n = e i 2(n+1) + c i 2ne 2n+1 2(n+1) + ( ci2n+1 + c i 2nc 2n+1 2n+1 ) e 2(n+1) 2(n+1) . From above two expressions for ei2n we have a i 2n+1 = c i 2n and a i 2(n+1) = ci2n+1 + ci2nc 2n+1 2n+1. 46 s. dutta, d. khurana, a. sensarma Now if we proceed by above algorithm and put c2n+12n+1 = 0, n ∈ N we get ai2n+1 = c i 2n, a i 2(n+1) = ci2n+1, 1 ≤ i ≤ 2n, n ≥ 1, and, we have the following representing matrix for X, C =   − a13 a 1 4 a 1 5 a 1 6 . . . ... a23 a 2 4 a 2 5 a 2 6 . . . ... ... 0 a35 a 3 6 . . . ... ... ... a45 a 4 6 . . . ... ... ... ... 0 . . . ... ... ... ... ... ...   . The Fact stated above indeed justifies that the resulting matrix is a repre- senting matrix of X. Example 3.3. Consider the function φ : R → R, φ(t) = 1 + t for t ∈ [−1, 0], φ(t) = 1 − t for t ∈ [0, 1], and φ(t) = 0 for t /∈ [−1, 1]. Define gk,2n = φ(2 nt−k), t ∈ [0, 1]. We can write C[0, 1] = ∪Xn, Xn = span{gk,2n : k = 0, 1, . . . , 2n} where {gk,2n : k = 0, 1, . . . , 2n} is an admissible basis of Xn. Then for all n = 0, 1, . . . and k = 1, 2, . . . , 2n − 1 we have (see [4]) gk,2n = 1 2 g2k−1,2n+1 + g2k,2n+1 + 1 2 g2k+1,2n+1 , g0,2n = g0,2n+1 + 1 2 g1,2n+1 , g2n,2n = 1 2 g2n+1−1,2n+1 + g2n+1,2n+1 . Let C = ( cin )1≤i≤n n≥1 be a representing matrix of C[0, 1]. First we have to write the expression for gk,2n according to C. Now comparing the equations with the above and put c 2n+j 2n+i = 0, 1 ≤ i ≤ 2 n+1 − 2n − 1, 1 ≤ j ≤ i, we will get a representing matrix of C[0, 1] with entries 0 and 1 2 only. We now answer Problem 2.5 in affirmative for C[0, 1]. Theorem 3.4. Let D be a closed subset of [0, 1]. Then there exists a diagram D representing C[0, 1] such that the M-ideal JD corresponds to the space XS for some directed sub diagram S of D, provided S is not an empty diagram. representing matrices, M-ideals, tensor products 47 Proof. Let D0 = {dn : n ∈ N} be a countable dense set in D. We can extend D0 to a set M = {ki : i ∈ N} such that M = [0, 1]. Consider e12 = 1−t, t ∈ [0, 1] and e22 = t, t ∈ [0, 1]. Without loss of generality choose an element k1 ∈ [0, 1] and consider e13 = 1 − 1 k1 t if t ∈ [0,k1] , e13 = 0 if t ∈ [k1, 1] ; e23 = 0 if t ∈ [0,k1] , e 2 3 = t−k1 1 −k1 if t ∈ [k1, 1] and e33 = 1 k1 t if t ∈ [0,k1] , e33 = 1 − t 1 −k1 if t ∈ [k1, 1] . Here e12, e 2 2, e 1 3, e 2 3, e 3 3 satisfy the following equations; e 1 2 = e 1 3 + (1 − k1)e 3 3, e22 = e 2 3 + k1e 3 3. Now with out loss of generality choose k2 ∈ [0,k1] and k3 ∈ [k1, 1]. Consider e15 = 1 − 1 k2 t if t ∈ [0,k2] , e15 = 0 if t ∈ [k2, 1] ; e25 = 0 if t ∈ [0,k3] , e 2 5 = t−k3 1 −k3 if t ∈ [k3, 1] ; e35 = 0 if t ∈ [0,k2] , e 3 5 = t−k2 k1 −k2 if t ∈ [k2,k1] , e35 = k3 − t k3 −k1 if t ∈ [k1,k3] , e35 = 0 if t ∈ [k3, 1] ; e45 = 1 k2 t if t ∈ [0,k2] , e45 = k1 − t k1 −k2 if t ∈ [k2,k1] , e45 = 0 if t ∈ [k1, 1] and e 5 5 = 0 if t ∈ [0,k1] , e55 = t−k1 k3 −k1 if t ∈ [k1,k3] , e55 = 1 − t 1 −k3 if t ∈ [k3, 1] . By the construction e13, e 2 3, e 3 3, e 1 5, e 2 5, e 3 5, e 4 5, e 5 5 satisfy the following equations: e13 = e 1 5 + k1 −k2 k2 e45 , e 2 3 = e 2 5 + k3 −k1 1 −k1 e55 and e 3 3 = e 3 5 + k2 k1 e45 + 1 −k3 1 −k1 e55 . Similarly we can construct ei2n+1, 1 ≤ i ≤ 2 n + 1. Take an element f ∈ C[0, 1]. Define a sequence (pn) ∞ n=0 in the following way. Let p0 = f(0)e 1 2, p1 = p0 + (f(1) −p0(1))e22 , p2 = p1 + (f(k1) −p1(k1))e 3 3 , p3 = p2 + (f(k2) −p2(k2))e45 , p4 = p3 + (f(k3) −p3(k3))e 5 5 , and so on. Here p0 and f takes the same value at 0 while p1 and f takes the same value at 0 and 1 and interpolates linearly in between, p2 and f takes 48 s. dutta, d. khurana, a. sensarma same value at 0, 1 and k1 and interpolates linearly in between, and so on. It is straightforward to check that limn ||pn −f||∞ = 0. Therefore we can write C[0, 1] = ∪En, where En = span { ei2n+1 : 1 ≤ i ≤ 2 n + 1 } and En is isometric to ` 2n+1 ∞ . We know that the support of e i 2n+1 is going to zero as n approaches to infinity and it consists a single element of {ki : i ∈ N}. Each i ∈ N, ki will be in some Tji = ∩ ∞ n=1 supp{e j 2n+1} and any two Tji ’s are disjoint. Here we consider the generalized diagram of C[0, 1] with respect to above basis and from n-th to (n + 1)-th step we choose 2n−1 of ki’s and these ki’s lie in the support of exactly one of the basis elements of e 2n+1 2n+1 , . . . , e2 n+1 2n+1 . Now if we follow the algorithm for ‘Fill in the Gaps’ from n-th to (n + 1)-th step and consider ki ∈ supp(e 2n+j 2n+1 ) chosen above, then ki ∈ supp(e 2n+j m ) and ki /∈ supp(elm), l 6= 2n + j for 2n + 1 ≤ m ≤ 2n+1 − 1, j ≥ 1 (see Remark 3.1). So by following the same procedure of choosing φi as in Proposition 2.10 we will get for any ki there exists a φm such that φm = δki and the set {ki}∞i=1 is dense in [0, 1] (see [2, Lemma 2]). Given that D0 is dense in D so JD = ∩d∈D ker δd = ∩d∈D0 ker δd. Thus JD = XS, where S is the intersection of directed diagrams corresponding to kernel of φi = δdi , di ∈ D0. This completes the proof. Representing matrix for X⊗̌Y : If X and Y are separable L1-predual spaces, then it is known that X⊗̌Y is also a separable L1-predual space. We adopt the above algorithm to find a representing matrix for X⊗̌Y . Let X and Y has representing matrices ( ain )1≤i≤n n≥1 and ( bin )1≤i≤n n≥1 respectively corre- sponding to the admissible basis {ein : 1 ≤ i ≤ n} and {fin : 1 ≤ i ≤ n}. Then X⊗̌Y = ∪∞n=1En2 , where En2 is isometric to ` n2 ∞ with admissible bases {ein ⊗ f j n, i = 1, . . . ,n; j = 1, . . . ,n}. We will denote this collection as{ Ei n2 : 1 ≤ i ≤ n2 } with the following convention: (a) First n2 terms of the admissible basis of E(n+1)2 is same as the admissible basis of En2 . For example if E i n2 = ek (n−1)2 ⊗ e l (n−1)2 then E i (n+1)2 = ek n2 ⊗el n2 . (b) We will choose { En 2+i (n+1)2 : 1 ≤ i ≤ (n + 1)2 −n2 } by the following way. Take En 2+1 (n+1)2 = e1n+1 ⊗ f n+1 n+1 , E n2+2 (n+1)2 = en+1n+1 ⊗ f 1 n+1. For i = 2k + 1, k ∈ N, En 2+2k+1 (n+1)2 = ekn+1 ⊗f n+1 n+1 and for i = 2k + 2, k ∈ N, E n2+2k+2 (n+1)2 = en+1n+1 ⊗f k n+1. representing matrices, M-ideals, tensor products 49 We will now follow algorithm for ‘Fill in the Gaps’ described above. Let us illustrate this with first few steps. Let C = ( cin )1≤i≤n n≥1 be the representing matrix of X⊗̌Y . According to the above convention E11 = e 1 1⊗f 1 1 and E 1 4 = e 1 2⊗f 1 2 , E 2 4 = e 1 2⊗f 2 2 , E 3 4 = e 2 2⊗f 1 2 , E44 = e 2 2 ⊗ f 2 2 . By expanding E 1 1 in terms of {E i 4} 4 i=1 according to the given representing matrix of X and Y we get E11 = E 1 4 + b 1 1E 2 4 + a 1 1E 3 4 + a 1 1b 1 1E 4 4 . Similarly expansion of E11 in terms of {E i 4} 4 i=1 according to the representing matrix C of X⊗̌Y , E11 = E 1 4 + c 1 1E 2 4 + (c 1 2 + c 1 1c 2 2)E 3 4 + (c 1 2c 3 3 + c 1 1c 2 3 + c 1 3 + c 1 1c 2 2c 3 3)E 4 4. By following the algorithm we will get c11 = b 1 1, c 1 2 = a 1 1, c 2 2 = 0, c 1 3 = a 1 1b 1 1, c23 = 0, c 3 3 = 0. By expanding {E i 4} 4 i=1 in terms of {E i 9} 9 i=1 according to given representing matrices for X, Y and matrix C we will get c14 = b 1 2 , c 2 4 = b 2 2 , c 3 4 = 0 , c 4 4 = 0 , c15 = a 1 2 , c 2 5 = 0 , c 3 5 = a 2 2 , c 4 5 = 0 , c 5 5 = 0 , c16 = 0 , c 2 6 = 0 , c 3 6 = b 1 2 , c 4 6 = b 2 2 , c 5 6 = 0 , c 6 6 = 0 , c17 = 0 , c 2 7 = a 1 2 , c 3 7 = 0 , c 4 7 = a 2 2 , c 5 7 = 0 , c 6 7 = 0 , c 7 7 = 0 , c18 = a 1 2b 1 2 , c 2 8 = a 1 2b 2 2 , c 3 8 = a 2 2b 1 2 , c 4 8 = a 2 2b 2 2 , c 5 8 = 0 , c68 = 0 , c 7 8 = 0 , c 8 8 = 0 . Proceeding as above we will get representing matrix of X⊗̌Y as C =   b11 a 1 1 a 1 1b 1 1 b 1 2 a 1 2 0 0 a 1 2b 1 2 . . . ... 0 0 b22 0 0 a 1 2 a 1 2b 2 2 . . . ... ... 0 0 a22 b 1 2 0 a 2 2b 1 2 . . . ... ... . . . 0 0 b22 a 2 2 a 2 2b 2 2 . . . ... ... . . . . . . 0 0 0 0 . . . ... ... . . . . . . . . . 0 0 0 . . . ... ... ... ... . . . 0 0 . . . ... ... ... ... ... ... ... 0 . . . ... ... ... ... ... ... ... ... . . .   . 50 s. dutta, d. khurana, a. sensarma Remark 3.5. From above description of representing matrix for X⊗̌Y we can actually read off representing matrices ( ain )1≤i≤n n≥1 and ( bin )1≤i≤n n≥1 for X and Y respectively. For example representing matrix of Y is given by B =   c11 c 1 4 c 1 9 c 1 42 . . . c1 n2 . . . ... c24 c 2 9 c 2 42 . . . c2 n2 . . . ... ... c59 c 5 42 . . . . . . c2 2+1 n2 . . . ... ... . . . c10 42 . . . . . . c3 2+1 n2 . . . ... ... . . . . . . . . . . . . . . . . . . . . . ... ... . . . . . . . . . . . . . . . . . . . . . ... ... ... ... . . . . . . . . . . . . . . . ... ... ... ... ... . . . . . . . . . . . . ... ... ... ... ... ... c (n−1)2+1 n2 . . . ... ... ... ... ... ... ... . . .   . Thus if a L1-predual space has a representing matrix like C, it is actually tensor product of two L1-predual spaces with representing matrices A and B. References [1] K.R. Davidson, “ C∗-Algebras by Example ”, Fields Institute Monographs, 6, American Mathematical Society, Providence, RI, 1996. [2] A.B. Hansen, Y. Sternfeld, On the characterization of the dimension of a compact metric space K by the representing matrices of C(K), Israel J. Math. 22 (2) (1975), 148 – 167. [3] A.J. Lazar, J. Lindenstraus, Banach Spaces whose duals are L1 spaces and their representing matrices, Acta Math. 126 (1971), 165 – 193. [4] A. Lima, V. Lima, E. Oja, Absolutely summing operators on C[0,1] as a tree space and the bounded approximation property, J. Funct. Anal. 259 (11) (2010), 2886 – 2901. [5] J. Lindenstraus, “ Extension of Compact Operators ”, Mem. Amer. Math. Soc. no. 48, American Mathematical Society, Providence, RI, 1964. [6] J. Lindenstrauss, G. Olsen, Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier (Grenoble) 28 (1) (1978), vi, 91 – 114. [7] W. Lusky, On Separable Lindenstrauss Spaces, J. Functional Analysis 26 (2) (1977), 103 – 120. [8] T.S.S.R.K. Rao, On almost isometric ideals in Banach spaces, Monatsh. Math. 181 (1) (2016), 169 – 176.